unsolved problems in mathematical systems and control theory

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UnsolvedProblemsinMathematicalSystemsandControlTheoryEditedbyVincentD.BlondelAlexandreMegretskiPRINCETONUNIVERSITYPRESSPRINCETONANDOXFORD ivCopyrightc2004byPrincetonUniversityPressPublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540,USAIntheUnitedKingdom:PrincetonUniversityPress,3MarketPlace,Wood-stock,OxfordshireOX201SY,UKAllrightsreservedLibraryofCongressCataloging-in-PublicationDataUnsolvedproblemsinmathematicalsystemsandcontroltheoryEditedbyVincentD.Blondel,AlexandreMegretski.p.cm.Includesbibliographicalreferences.ISBN0-691-11748-9(cl:alk.paper)1.Systemanalysis.2.Controltheory.I.Blondel,Vincent.II.Megretski,Alexandre.QA402.U53520042003064802003—dc22Thepublisherwouldliketoacknowledgetheeditorsofthisvolumeforpro-vidingthecamera-readycopyfromwhichthisbookwasprinted.PrintedintheUnitedStatesofAmerica10987654321 Ihaveyettoseeanyproblem,howevercomplicated,which,whenyoulookedatitintherightway,didnotbecomestillmorecompli-cated.PoulAnderson ContentsPrefacexiiiAssociateEditorsxvWebsitexviiPART1.LINEARSYSTEMS1Problem1.1.StabilityandcompositionoftransferfunctionsGuillermoFern´andez-Anaya,JuanCarlosMart´ınez-Garc´ıa3Problem1.2.TherealizationproblemforHerglotz-NevanlinnafunctionsSeppoHassi,HenkdeSnoo,EduardTsekanovski˘ı8Problem1.3.DoesanyanalyticcontractiveoperatorfunctiononthepolydiskhaveadissipativescatteringnDrealization?DmitryS.Kalyuzhniy-Verbovetzky14Problem1.4.PartialdisturbancedecouplingwithstabilityJuanCarlosMart´ınez-Garc´ıa,MichelMalabre,VladimirKuˇcera18Problem1.5.IsMonopoli’smodelreferenceadaptivecontrollercorrect?A.S.Morse22Problem1.6.ModelreductionofdelaysystemsJonathanR.Partington29Problem1.7.SchurextremalproblemsLevSakhnovich33Problem1.8.Theelusiveifftestfortime-controllabilityofbehaviorsAmolJ.Sasane36 viiiCONTENTSProblem1.9.AFarkaslemmaforbehavioralinequalitiesA.A.(Tonny)tenDam,J.W.(Hans)Nieuwenhuis40Problem1.10.RegularfeedbackimplementabilityoflineardifferentialbehaviorsH.L.Trentelman44Problem1.11.RiccatistabilityErikI.Verriest49Problem1.12.StateandfirstorderrepresentationsJanC.Willems54Problem1.13.ProjectionofstatespacerealizationsAntoineVandendorpe,PaulVanDooren58PART2.STOCHASTICSYSTEMS65Problem2.1.OnerrorofestimationandminimumofcostforwidebandnoisedrivensystemsAgamirzaE.Bashirov67Problem2.2.OnthestabilityofrandommatricesGiuseppeC.Calafiore,FabrizioDabbene71Problem2.3.AspectsofFishergeometryforstochasticlinearsystemsBernardHanzon,RalfPeeters76Problem2.4.OntheconvergenceofnormalformsforanalyticcontrolsystemsWeiKang,ArthurJ.Krener82PART3.NONLINEARSYSTEMS87Problem3.1.MinimumtimecontroloftheKeplerequationJean-BaptisteCaillau,JosephGergaud,JosephNoailles89Problem3.2.LinearizationoflinearlycontrollablesystemsR.Devanathan93Problem3.3.BasesforLiealgebrasandacontinuousCBHformulaMatthiasKawski97 CONTENTSixProblem3.4.AnextendedgradientconjectureLuisCarlosMartinsJr.,GeraldoNunesSilva103Problem3.5.OptimaltransactioncostsfromaStackelbergperspectiveGeertJanOlsder107Problem3.6.Doescheapcontrolsolveasingularnonlinearquadraticproblem?YuriV.Orlov111Problem3.7.Delta-SigmamodulatorsynthesisAndersRantzer114Problem3.8.Determiningofvariousasymptoticsofsolutionsofnonlineartime-optimalproblemsviarightidealsinthemomentalgebraG.M.Sklyar,S.Yu.Ignatovich117Problem3.9.DynamicsofprincipalandminorcomponentflowsU.Helmke,S.Yoshizawa,R.Evans,J.H.Manton,andI.M.Y.Mareels122PART4.DISCRETEEVENT,HYBRIDSYSTEMS129Problem4.1.L2-inducedgainsofswitchedlinearsystemsJo˜aoP.Hespanha131Problem4.2.ThestatepartitioningproblemofquantizedsystemsJanLunze134Problem4.3.FeedbackcontrolinflowshopsS.P.SethiandQ.Zhang140Problem4.4.DecentralizedcontrolwithcommunicationbetweencontrollersJanH.vanSchuppen144PART5.DISTRIBUTEDPARAMETERSYSTEMS151Problem5.1.InfinitedimensionalbacksteppingfornonlinearparabolicPDEsAndrasBalogh,MiroslavKrstic153Problem5.2.ThedynamicalLamesystemwithboundarycontrol:onthestruc-tureofreachablesetsM.I.Belishev160 xCONTENTSProblem5.3.Null-controllabilityoftheheatequationinunboundeddomainsSorinMicu,EnriqueZuazua163Problem5.4.Istheconservativewaveequationregular?GeorgeWeiss169Problem5.5.ExactcontrollabilityofthesemilinearwaveequationXuZhang,EnriqueZuazua173Problem5.6.SomecontrolproblemsinelectromagneticsandfluiddynamicsLorellaFatone,MariaCristinaRecchioni,FrancescoZirilli179PART6.STABILITY,STABILIZATION187Problem6.1.CopositiveLyapunovfunctionsM.K.C¸amlıbel,J.M.Schumacher189Problem6.2.Thestrongstabilizationproblemforlineartime-varyingsystemsAvrahamFeintuch194Problem6.3.RobustnessoftransientbehaviorDiederichHinrichsen,ElmarPlischke,FabianWirth197Problem6.4.LiealgebrasandstabilityofswitchednonlinearsystemsDanielLiberzon203Problem6.5.RobuststabilitytestforintervalfractionalorderlinearsystemsIvoPetr´aˇs,YangQuanChen,BlasM.Vinagre208Problem6.6.Delay-independentanddelay-dependentAizermanproblemVladimirR˘asvan212Problem6.7.Openproblemsincontroloflineardiscretemultidimensionalsys-temsLiXu,ZhipingLin,Jiang-QianYing,OsamiSaito,YoshihisaAnazawa221Problem6.8.AnopenprobleminadaptativenonlinearcontroltheoryLeonidS.Zhiteckij229Problem6.9.GeneralizedLyapunovtheoryanditsomega-transformableregionsSheng-GuoWang233 CONTENTSxiProblem6.10.SmoothLyapunovcharacterizationofmeasurementtoerrorsta-bilityBrianP.Ingalls,EduardoD.Sontag239PART7.CONTROLLABILITY,OBSERVABILITY245Problem7.1.Timeforlocalcontrollabilityofa1-DtankcontainingafluidmodeledbytheshallowwaterequationsJean-MichelCoron247Problem7.2.AHautustestforinfinite-dimensionalsystemsBirgitJacob,HansZwart251Problem7.3.ThreeproblemsinthefieldofobservabilityPhilippeJouan256Problem7.4.ControloftheKdVequationLionelRosier260PART8.ROBUSTNESS,ROBUSTCONTROL265Problem8.1.H∞-normapproximationA.C.Antoulas,A.Astolfi267Problem8.2.NoniterativecomputationofoptimalvalueinH∞controlBenM.Chen271Problem8.3.DeterminingtheleastupperboundontheachievabledelaymarginDanielE.Davison,DanielE.Miller276Problem8.4.Stablecontrollercoefficientperturbationinfloatingpointimple-mentationJunWu,ShengChen280PART9.IDENTIFICATION,SIGNALPROCESSING285Problem9.1.AconjectureonLyapunovequationsandprincipalanglesinsub-spaceidentificationKatrienDeCock,BartDeMoor287 xiiCONTENTSProblem9.2.StabilityofanonlinearadaptivesystemforfilteringandparameterestimationMasoudKarimi-Ghartemani,AlirezaK.Ziarani293PART10.ALGORITHMS,COMPUTATION297Problem10.1.Root-clusteringformultivariatepolynomialsandrobuststabilityanalysisPierre-AlexandreBliman299Problem10.2.Whenisapairofmatricesstable?VincentD.Blondel,JacquesTheys,JohnN.Tsitsiklis304Problem10.3.FreenessofmultiplicativematrixsemigroupsVincentD.Blondel,JulienCassaigne,JuhaniKarhum¨aki309Problem10.4.Vector-valuedquadraticformsincontroltheoryFrancescoBullo,JorgeCort´es,AndrewD.Lewis,SoniaMart´ınez315Problem10.5.NilpotentbasesofdistributionsHenryG.Hermes,MatthiasKawski321Problem10.6.Whatisthecharacteristicpolynomialofasignalflowgraph?AndrewD.Lewis326Problem10.7.OpenproblemsinrandomizedµanalysisOnurToker330 PrefaceFiveyearsago,afirstvolumeofopenproblemsinMathematicalSystemsandControlTheoryappeared.1Someofthe53problemsthatwerepublishedinthisvolumeattractedconsiderableattentionintheresearchcommunity.Thebookinfrontofyoucontainsanewcollectionof63openproblems.Thecontentsofbothvolumesshowtheevolutionofthefieldinthehalfdecadesincethepublicationofthefirstvolume.Onenoticeablefeatureistheshifttowardawiderclassofquestionsandmoreemphasisonissuesdrivenbyphysicalmodeling.EarlyversionsofsomeoftheproblemsinthisbookhavebeenpresentedattheOpenProblemsessionsoftheOberwolfachTagungonRegelungstheorie,onFebruary27,2002,andoftheConferenceonMathematicalTheoryofNetworksandSystems(MTNS)inNotreDame,Indiana,onAugust12,2002.Theeditorsthanktheorganizersofthesemeetingsfortheirwillingnesstoprovidetheproblemsthiswelcomeexposure.Sincetheappearanceofthefirstvolume,openproblemshavecontinuedtomeetwithlargeinterestinthemathematicalcommunity.Undoubtedly,themostspectaculareventinthisarenawastheannouncementbytheClayMathematicsInstitute2oftheMillenniumPrizeProblemswhosesolutionwillberewardedbyonemillionU.S.dollarseach.Modestyandmodestyofmeanshavepreventedtheeditorsofthepresentvolumefromofferingsimilarrewardstowardthesolutionoftheproblemsinthisbook.However,wetrustthat,notwithstandingthisabsenceofafinancialincentive,theintellectualchallengewillstimulatemanyreaderstoattacktheproblems.Theeditorsthankinthefirstplacetheresearcherswhohavesubmittedtheproblems.WearealsoverythankfultothePrincetonUniversityPress,andinparticularVickieKearn,fortheirwillingnesstopublishthisvol-ume.Thefulltextoftheproblems,togetherwithcomments,additions,andsolutions,willbepostedonthebookwebsiteatPrincetonUniver-sityPress(linkavailablefromhttp://pup.princeton.edu/math/)andonhttp://www.inma.ucl.ac.be/∼blondel/op/.Readersareencouragedtosubmitcontributionsbyfollowingtheinstructionsgivenonthesewebsites.Theeditors,Louvain-la-Neuve,March15,2003.1VincentD.Blondel,EduardoD.Sontag,M.Vidyasagar,andJanC.Willems,OpenProblemsinMathematicalSystemsandControlTheory,SpringerVerlag,1998.2Seehttp://www.claymath.org. AssociateEditorsRogerBrockett,HarvardUniversity,USAJean-MichelCoron,UniversityofParis(Orsay),FranceRolandHildebrand,UniversityofLouvain(Louvain-la-Neuve),BelgiumMiroslavKrstic,UniversityofCalifornia(SanDiego),USAAndersRantzer,LundInstituteofTechnology,SwedenJoachimRosenthal,UniversityofNotreDame,USAEduardoSontag,RutgersUniversity,USAM.Vidyasagar,TataConsultancyServices,IndiaJanWillems,UniversityofLeuven,Belgium WebsiteThefulltextoftheproblemspresentedinthisbook,togetherwithcom-ments,additionsandsolutions,arefreelyavailableinelectronicformatfromthebookwebsiteatPrincetonUniversityPress:http://pup.princeton.edu/math/andfromaneditorwebsite:http://www.inma.ucl.ac.be/∼blondel/op/Readersareencouragedtosubmitcontributionsbyfollowingtheinstruc-tionsgivenonthesewebsites. PART1LinearSystems Problem1.1StabilityandcompositionoftransferfunctionsG.Fern´andez-AnayaDepartamentodeCienciasB´asicasUniversidadIberoam´ericanaLomasdeSantaFe01210M´exicoD.F.M´exicoguillermo.fernandez@uia.mxJ.C.Mart´ınez-Garc´ıaDepartamentodeControlAutom´aticoCINVESTAV-IPNA.P.14-74007300M´exicoD.F.M´exicomartinez@ctrl.cinvestav.mx1INTRODUCTIONAsfarasthefrequency-describedcontinuouslineartime-invariantsystemsareconcerned,thestudyofcontrol-orientedproperties(likestability)re-sultingfromthesubstitutionofthecomplexLaplacevariablesbyrationaltransferfunctionshavebeenlittlestudiedbytheAutomaticControlcom-munity.However,someinterestingresultshaverecentlybeenpublished:Concerningthestudyoftheso-calleduniformsystems,i.e.,LTIsystemsconsistingofidenticalcomponentsandamplifiers,itwasestablishedin[8]ageneralcriterionforrobuststabilityforrationalfunctionsoftheformD(f(s)),whereD(s)isapolynomialandf(s)isarationaltransferfunction.Byapplyingsuchacriterium,itgaveageneralizationofthecelebratedKharitonov’stheorem[7],aswellassomerobuststabilitycriteriaunderH∞-uncertainty.Theresultsgivenin[8]arebasedontheso-calledH-domains.1Asfarasrobuststabilityofpolynomialfamiliesisconcerned,someKharito-1TheH-domainofafunctionf(s)isdefinedtobethesetofpointshonthecomplexplaneforwhichthefunctionf(s)−hhasnozerosontheopenright-halfcomplexplane. 4PROBLEM1.1nov’slikeresults[7]aregivenin[9](foraparticularclassofpolynomials),wheninterpretingsubstitutionsasnonlinearlycorrelatedperturbationsonthecoefficients.Morerecently,in[1],someresultsforproperandstablerealrationalSISOfunctionsandcoprimefactorizationswereproved,bymakingsubstitutionswithα(s)=(as+b)/(cs+d),wherea,b,c,anddarestrictlypositiverealnumbers,andwithad−bc6=0.Buttheseresultsarelimitedtothebilineartransforms,whichareveryrestricted.In[4]isstudiedthepreservationofpropertieslinkedtocontrolproblems(likeweightednominalperformanceandrobuststability)forSingle-InputSingle-Outputsystems,whenperformingthesubstitutionoftheLaplacevariable(intransferfunctionsassociatedtothecontrolproblems)bystrictlypositiverealfunctionsofzerorelativedegree.Someresultsconcerningthepreservationofcontrol-orientedpropertiesinMulti-InputMulti-Outputsystemsaregivenin[5],while[6]dealswiththepreservationofsolvabilityconditionsinalgebraicRiccatiequationslinkedtorobustcontrolproblems.Followingourinterestinsubstitutionsweproposeinsection22.2threein-terestingproblems.Themotivationsconcerningtheproposedproblemsarepresentedinsection22.3.2DESCRIPTIONOFTHEPROBLEMSInthissectionweproposethreecloselyrelatedproblems.Thefirstonecon-cernsthecharacterizationofatransferfunctionasacompositionoftransferfunctions.Thesecondproblemisamodifiedversionofthefirstproblem:thecharacterizationofatransferfunctionastheresultofsubstitutingtheLaplacevariableinatransferfunctionbyastrictlypositiverealtransferfunctionofzerorelativedegree.Thethirdproblemisinfactaconjectureconcerningthepreservationofstabilitypropertyinagivenpolynomialre-sultingfromthesubstitutionofthecoefficientsinthegivenpolynomialbyapolynomialwithnon-negativecoefficientsevaluatedinthesubstitutedco-efficients.Problem1:LetaSingleInputSingleOutput(SISO)transferfunctionG(s)begiven.FindtransferfunctionsG0(s)andH(s)suchthat:1.G(s)=G0(H(s));2.H(s)preservesproperstabletransferfunctionsundersubstitutionofthevariablesbyH(s),and:3.ThedegreeofthedenominatorofH(s)isthemaximumwiththeprop-erties1and2. STABILITYANDCOMPOSITIONOFTRANSFERFUNCTIONS5Problem2:LetaSISOtransferfunctionG(s)begiven.FindatransferfunctionG0(s)andaStrictlyPositiveRealtransferfunctionofzerorelativedegree(SPR0),sayH(s),suchthat:1.G(s)=G0(H(s))and:2.ThedegreeofthedenominatorofH(s)isthemaximumwiththeprop-erty1.Problem3:(Conjecture)Givenanystablepolynomial:asn+asn−1+···+as+ann−110andgivenanypolynomialq(s)withnon-negativecoefficients,thenthepoly-nomial:q(a)sn+q(a)sn−1+···+q(a)s+q(a)nn−110isstable(see[3]).3MOTIVATIONSConsidertheclosed-loopcontrolscheme:y(s)=G(s)u(s)+d(s),u(s)=K(s)(r(s)−y(s)),where:P(s)denotestheSISOplant;K(s)denotesastabilizingcontroller;u(s)denotesthecontrolinput;y(s)denotesthecontrolinput;d(s)denotesthedisturbanceandr(s)denotesthereferenceinput.Weshalldenotetheclosed-looptransferfunctionfromr(s)toy(s)asFr(G(s),K(s))andtheclosed-looptransferfunctionfromd(s)toy(s)asFd(G(s),K(s)).•Considertheclosed-loopsystemFr(G(s),K(s)),andsupposethattheplantG(s)resultsfromaparticularsubstitutionofthesLaplacevariableinatransferfunctionG0(s)byatransferfunctionH(s),i.e.,G(s)=G0(H(s)).IthasbeenprovedthatacontrollerK0(s)whichstabilizestheclosed-loopsystemFr(G0(s),K0(s))issuchthatK0(H(s))stabilizesFr(G(s),K0(H(s)))(see[2]and[8]).Thus,thesimplificationofproceduresforthesynthesisofstabilizingcontrollers(profitingfromtransferfunctioncompositions)justifiesproblem1.•Asfarasproblem2isconcerned,considerthesynthesisofacontrollerK(s)stabilizingtheclosed-looptransferfunctionFd(G(s),K(s)),andsuchthatkFd(G(s),K(s))k∞<γ,forafixedgivenγ>0.IfweknownthatG(s)=G0(H(s)),beingH(s)aSPR0transferfunction,thesolutionofproblem2wouldarisetothefollowingprocedure:1.FindacontrollerK0(s)whichstabilizestheclosed-looptransferfunctionFd(G0(s),K0(s))andsuchthat:kFd(G0(s),K0(s))k∞<γ. 6PROBLEM1.12.ThecomposedcontrollerK(s)=K0(H(s))stabilizestheclosed-loopsystemFd(G(s),K(s))and:kFd(G(s),K(s))k∞<γ(see[2],[4],and[5]).Itisclearthatcondition3inthefirstproblem,orcondition2inthesecondproblem,canberelaxedtothefollowingcondition:thedegreeofthedenominatorofH(s)isashighasbepossiblewiththeappropriateconditions.Withthisnewcondition,theopenproblemsareabitlessdifficult.•Finally,problem3canbeinterpretedintermsofrobustnessunderpositivepolynomialperturbationsinthecoefficientsofastabletransferfunction.BIBLIOGRAPHY[1]G.Fern´andez,S.Mu˜noz,R.A.S´anchez,andW.W.Mayol,“Simulta-neousstabilizationusingevolutionarystrategies,”Int.J.Contr.,vol.68,no.6,pp.1417-1435,1997.[2]G.Fern´andez,“PreservationofSPRfunctionsandstabilizationbysub-stitutionsinSISOplants,”IEEETransactiononAutomaticControl,vol.44,no.11,pp.2171-2174,1999.[3]G.Fern´andezandJ.Alvarez,“Onthepreservationofstabilityinfam-iliesofpolynomialsviasubstitutions,”Int.J.ofRobustandNonlinearControl,vol.10,no.8,pp.671-685,2000.[4]G.Fern´andez,J.C.Mart´ınez-Garc´ıa,andV.Kuˇcera,“H∞-RobustnessPropertiesPreservationinSISOSystemswhenapplyingSPRSubstitu-tions,”SubmittedtotheInternationalJournalofAutomaticControl.[5]G.Fern´andezandJ.C.Mart´ınez-Garc´ıa,“MIMOSystemsPropertiesPreservationunderSPRSubstitutions,”InternationalSymposiumontheMathematicalTheoryofNetworksandSystems(MTNS’2002),UniversityofNotreDame,USA,August12-16,2002.[6]G.Fern´andez,J.C.Mart´ınez-Garc´ıa,andD.Aguilar-George,“Preserva-tionofsolvabilityconditionsinRiccatiequationswhenapplyingSPR0substitutions,”submittedtoIEEETransactionsonAutomaticControl,2002.[7]V.L.Kharitonov,“Asymptoticstabilityoffamiliesofsystemsoflineardifferentialequations,”Differential’nyeUravneniya,vol.14,pp.2086-2088,1978. STABILITYANDCOMPOSITIONOFTRANSFERFUNCTIONS7[8]B.T.PolyakandYa.Z.Tsypkin,“Stabilityandrobuststabilityofuni-formsystems,”AutomationandRemoteContr.,vol.57,pp.1606-1617,1996.[9]L.Wang,“Robuststabilityofaclassofpolynomialfamiliesundernon-linearlycorrelatedperturbations,”SystemandControlLetters,vol.30,pp.25-30,1997. Problem1.2TherealizationproblemforHerglotz-NevanlinnafunctionsSeppoHassiDepartmentofMathematicsandStatisticsUniversityofVaasaP.O.Box700,65101VaasaFinlandsha@uwasa.fiHenkdeSnooDepartmentofMathematicsUniversityofGroningenP.O.Box800,9700AVGroningenNederlanddesnoo@math.rug.nlEduardTsekanovski˘ıDepartmentofMathematicsNiagaraUniversity,NY14109USAtsekanov@niagara.edu1MOTIVATIONANDHISTORYOFTHEPROBLEMRoughlyspeaking,realizationtheoryconcernsitselfwithidentifyingagivenholomorphicfunctionasthetransferfunctionofasystemorasitslinearfrac-tionaltransformation.Linear,conservative,time-invariantsystemswhosemainoperatorisboundedhavebeeninvestigatedthoroughly.However,manyrealizationsindifferentareasofmathematicsincludingsystemtheory,elec-tricalengineering,andscatteringtheoryinvolveunboundedmainoperators,andacompletetheoryisstilllacking.TheaimofthepresentproposalistooutlinethenecessarystepsneededtoobtainageneralrealizationtheoryalongthelinesofM.S.Brodski˘ıandM.S.Livˇsic[8],[9],[16],whohave THEREALIZATIONPROBLEMFORHERGLOTZ-NEVANLINNAFUNCTIONS9consideredsystemswithaboundedmainoperator.Anoperator-valuedfunctionV(z)actingonaHilbertspaceEbelongstotheHerglotz-NevanlinnaclassN,ifoutsideRitisholomorphic,symmetric,i.e.,V(z)∗=V(¯z),andsatisfies(Imz)(ImV(z))≥0.HereandinthefollowingitisassumedthattheHilbertspaceEisfinite-dimensional.EachHerglotz-NevanlinnafunctionV(z)hasanintegralrepresentationoftheformZ1tV(z)=Q+Lz+−dΣ(t),(1)Rt−z1+t2whereQ=Q∗,L≥0,andΣ(t)isanondecreasingmatrix-functiononRwithRdΣ(t)/(t2+1)<∞.Conversely,eachfunctionoftheform(1)belongsRtotheclassN.Ofspecialimportance(cf.[15])aretheclassSofStieltjesfunctionsZ∞dΣ(t)V(z)=γ+,(2)0t−zR∞whereγ≥0anddΣ(t)/(t+1)<∞,andtheclassS−1ofinverseStieltjes0functionsZ∞11V(z)=α+βz+−dΣ(t),(3)0t−ztR∞whereα≤0,β≥0,anddΣ(t)/(t2+1)<∞.02SPECIALREALIZATIONPROBLEMSOnewaytocharacterizeHerglotz-Nevanlinnafunctionsistoidentifythemas(linearfractionaltransformationsof)transferfunctions:V(z)=i[W(z)+I]−1[W(z)−I]J,(4)whereJ=J∗=J−1andW(z)isthetransferfunctionofsomegeneral-izedlinear,stationary,conservativedynamicalsystem(cf.[1],[3]).TheapproachbasedontheuseofBrodski˘ı-LivˇsicoperatorcolligationsΘyieldstoasimultaneousrepresentationofthefunctionsW(z)andV(z)intheformW(z)=I−2iK∗(T−zI)−1KJ,(5)ΘV(z)=K∗(T−zI)−1K,(6)ΘRwhereTRstandsfortherealpartofT.ThedefinitionsandmainresultsassociatedwithBrodski˘ı-LivˇsictypeoperatorcolligationsinrealizationofHerglotz-Nevanlinnafunctionsareasfollows,cf.[8],[9],[16].LetT∈[H],i.e.,TisaboundedlinearmappinginaHilbertspaceH,andassumethatImT=(T−T∗)/2iofTisrepresentedasImT=KJK∗,whereK∈[E,H],andJ∈[E]isself-adjointandunitary.ThenthearrayTKJΘ=(7)HE 10PROBLEM1.2definesaBrodski˘ı-Livˇsicoperatorcolligation,andthefunctionWΘ(z)givenby(5)isthetransferfunctionofΘ.InthecaseofthedirectingoperatorJ=Ithesystem(7)iscalledascatteringsystem,inwhichcasethemainoperatorTofthesystemΘisdissipative:ImT≥0.InsystemtheoryWΘ(z)isinterpretedasthetransferfunctionoftheconservativesystem(i.e.,ImT=KJK∗)oftheform(T−zI)x=KJϕandϕ=ϕ−2iK∗x,−+−whereϕ−∈Eisaninputvector,ϕ+∈Eisanoutputvector,andxisastatespacevectorinH,sothatϕ+=WΘ(z)ϕ−.ThesystemissaidtobeminimalifthemainoperatorTofΘiscompletelynonself-adjoint(i.e.,therearenonontrivialinvariantsubspacesonwhichTinducesself-adjointoperators),cf.[8],[16].AclassicalresultduetoBrodski˘ıandLivˇsic[9]statesthatthecompactlysupportedHerglotz-NevanlinnafunctionsoftheRbformdΣ(t)/(t−z)correspondtominimalsystemsΘoftheform(7)viaa(4)withW(z)=WΘ(z)givenby(5)andV(z)=VΘ(z)givenby(6).Nextconsideralinear,stationary,conservativedynamicalsystemΘoftheformAKJΘ=.(8)H+⊂H⊂H−EHereA∈[H+,H−],whereH+⊂H⊂H−isariggedHilbertspace,A⊃T⊃A,A∗⊃T∗⊃A,AisaHermitianoperatorinH,Tisanon-HermitianoperatorinH,K∈[E,H],J=J∗=J−1,andImA=KJK∗.Inthiscase−ΘissaidtobeaBrodski˘ı-Livˇscriggedoperatorcolligation.ThetransferfunctionofΘin(8)anditslinearfractionaltransformaregivenbyW(z)=I−2iK∗(A−zI)−1KJ,V(z)=K∗(A−zI)−1K.(9)ΘΘRThefunctionsV(z)in(1)whichcanberealizedintheform(4),(9)withatransferfunctionofasystemΘasin(8)havebeencharacterizedin[2],[5],[6],[7],[18].ForthesignificanceofriggedHilbertspacesinsystemtheory,see[14],[16].Systems(7)and(8)naturallyappearinelectricalengineeringandscatteringtheory[16].3GENERALREALIZATIONPROBLEMSIntheparticularcaseofStieltjesfunctionsorofinverseStieltjesfunctionsgeneralrealizationresultsalongthelinesof[5],[6],[7]remaintobeworkedoutindetail,cf.[4],[10].Thesystems(7)and(8)arenotgeneralenoughfortherealizationofgeneralHerglotz-Nevanlinnafunctionsin(1)withoutanyconditionsonQ=Q∗andL≥0.However,ageneralizationoftheBrodski˘ı-Livˇsicoperatorcolligation(7)leadstoanalogousrealizationresultsforHerglotz-NevanlinnafunctionsV(z)oftheform(1)whosespectralfunctioniscompactlysupported:suchfunctionsV(z)admitarealizationvia(4)withW(z)=W(z)=I−2iK∗(M−zF)−1KJ,Θ(10)V(z)=W(z)=K∗(M−zF)−1K,ΘR THEREALIZATIONPROBLEMFORHERGLOTZ-NEVANLINNAFUNCTIONS11whereM=M+iKJK∗,M∈[H]istherealpartofM,Fisafinite-RRdimensionalorthogonalprojector,andΘisageneralizedBrodski˘ı-LivˇsicoperatorcolligationoftheformMFKJΘ=,(11)HEsee[11],[12],[13].Thebasicopenproblemsare:Determinetheclassoflinear,conservative,time-invariantdynamicalsys-tems(newtypeofoperatorcolligations)suchthatanarbitrarymatrix-valuedHerglotz-NevanlinnafunctionV(z)actingonEcanberealizedasalinearfractionaltransformation(4)ofthematrix-valuedtransferfunctionWΘ(z)ofsomeminimalsystemΘfromthisclass.Findcriteriaforagivenmatrix-valuedStieltjesorinverseStieltjesfunctionactingonEtoberealizedasalinearfractionaltransformationofthematrix-valuedtransferfunctionofaminimalBrodski˘ı-LivˇsictypesystemΘin(8)with:(i)anaccretiveoperatorA,(ii)anα-sectorialoperatorA,or(iii)anextremaloperatorA(accretivebutnotα-sectorial).Thesameproblemforthe(compactlysupported)matrix-valuedStieltjesorinverseStieltjesfunctionsandthegeneralizedBrodski˘ı-Livˇsicsystemsoftheform(11)withthemainoperatorMandthefinite-dimensionalorthogonalprojectorF.Thereisacloseconnectiontotheso-calledregularimpedanceconserva-tivesystems(wherethecoefficientofthederivativeisinvertible)thatwererecentlyconsideredin[17](seealso[19]).ItisshownthatanyfunctionD(s)withnon-negativerealpartintheopenrighthalf-planeandforwhichD(s)/s→0ass→∞hasarealizationwithsuchanimpedanceconservativesystem.BIBLIOGRAPHY[1]D.Alpay,A.Dijksma,J.Rovnyak,andH.S.V.deSnoo,“Schurfunc-tions,operatorcolligations,andreproducingkernelPontryaginspaces,”Oper.TheoryAdv.Appl.,96,Birkh¨auserVerlag,Basel,1997.[2]Yu.M.Arlinski˘ı,“Ontheinverseproblemofthetheoryofcharacteristicfunctionsofunboundedoperatorcolligations”,DopovidiAkad.NaukUkrain.RSR,2(1976),105–109(Russian).[3]D.Z.Arov,“Passivelinearsteady-statedynamicalsystems,”Sibirsk.Mat.Zh.,20,no.2,(1979),211–228,457(Russian)[Englishtransl.:SiberianMath.J.,20no.2,(1979)149–162]. 12PROBLEM1.2[4]S.V.Belyi,S.Hassi,H.S.V.deSnoo,andE.R.Tsekanovski˘ı,“OntherealizationofinverseStieltjesfunctions,”Proceedingsofthe15thInternationalSymposiumonMathematicalTheoryofNetworksandSystems,EditorsD.GillianandJ.Rosenthal,UniversityofNotreDame,SouthBend,Idiana,USA,2002,http://www.nd.edu/∼mtns/papers/201606.pdf[5]S.V.BelyiandE.R.Tsekanovski˘ı,“Realizationandfactorizationprob-lemsforJ-contractiveoperator-valuedfunctionsinhalf-planeandsys-temswithunboundedoperators,”SystemsandNetworks:Mathemati-calTheoryandApplications,AkademieVerlag,2(1994),621–624.[6]S.V.BelyiandE.R.Tsekanovski˘ı,“Realizationtheoremsforoperator-valuedR-functions,”Oper.TheoryAdv.Appl.,98(1997),55–91.[7]S.V.BelyiandE.R.Tsekanovski˘ı,“Onclassesofrealizableoperator-valuedR-functions,”Oper.TheoryAdv.Appl.,115(2000),85–112.[8]M.S.Brodski˘ı,“TriangularandJordanrepresentationsoflinearop-erators,”Moscow,Nauka,1969(Russian)[Englishtrans.:Vol.32ofTransl.Math.Monographs,Amer.Math.Soc.,1971].[9]M.S.Brodski˘ıandM.S.Livˇsic,“Spectralanalysisofnon-selfadjointoperatorsandintermediatesystems,”UspekhiMat.Nauk,13no.1,79,(1958),3–85(Russian)[Englishtrans.:Amer.Math.Soc.Transl.,(2)13(1960),265–346].[10]I.DovshenkoandE.R.Tsekanovski˘ı,“ClassesofStieltjesoperator-functionsandtheirconservativerealizations,”Dokl.Akad.NaukSSSR,311no.1(1990),18–22.[11]S.Hassi,H.S.V.deSnoo,andE.R.Tsekanovski˘ı,“AnaddendumtothemultiplicationandfactorizationtheoremsofBrodski˘ı-Livˇsic-Potapov,”Appl.Anal.,77(2001),125–133.[12]S.Hassi,H.S.V.deSnoo,andE.R.Tsekanovski˘ı,“Oncommuta-tiveandnoncommutativerepresentationsofmatrix-valuedHerglotz-Nevanlinnafunctions,”Appl.Anal.,77(2001),135–147.[13]S.Hassi,H.S.V.deSnoo,andE.R.Tsekanovski˘ı,“RealizationsofHerglotz-NevanlinnafunctionsviaF-systems,”Oper.Theory:Adv.Appl.,132(2002),183–198.[14]J.W.Helton,“Systemswithinfinite-dimensionalstatespace:theHilbertspaceapproach,”Proc.IEEE,64(1976),no.1,145–160.[15]I.S.Ka˘candM.G.Kre˘ın,“TheR-functions:Analyticfunctionsmap-pingtheupperhalf-planeintoitself,”SupplementItotheRussianedi-tionofF.V.Atkinson,DiscreteandContinuousBoundaryProblems,Moscow,1974[Englishtrans.:Amer.Math.Soc.Trans.,(2)103(1974),1–18]. THEREALIZATIONPROBLEMFORHERGLOTZ-NEVANLINNAFUNCTIONS13[16]M.S.Livˇsic,“Operators,Oscillations,Waves,”Moscow,Nauka,1966(Russian)[Englishtrans.:Vol.34ofTrans.Math.Monographs,Amer.Math.Soc.,1973].[17]O.J.Staffans,“Passiveandconservativeinfinite-dimensionalimpedanceandscatteringsystems(fromapersonalpointofview),”Pro-ceedingsofthe15thInternationalSymposiumonMathematicalTheoryofNetworksandSystems,Ed.,D.GillianandJ.Rosenthal,Univer-sityofNotreDame,SouthBend,Indiana,USA,2002,Plenarytalk,http://www.nd.edu/∼mtns[18]E.R.Tsekanovski˘ıandYu.L.Shmul’yan,“ThetheoryofbiextensionsofoperatorsinriggedHilbertspaces:Unboundedoperatorcolligationsandcharacteristicfunctions,”UspekhiMat.Nauk,32(1977),69–124(Russian)[Englishtransl.:RussianMath.Surv.,32(1977),73–131].[19]G.Weiss,“Transferfunctionsofregularlinearsystems.PartI:charac-terizationsofregularity”,Trans.Amer.Math.Soc.,342(1994),827–854. Problem1.3DoesanyanalyticcontractiveoperatorfunctiononthepolydiskhaveadissipativescatteringnDrealization?DmitryS.Kalyuzhniy-VerbovetzkyDepartmentofMathematicsTheWeizmannInstituteofScienceRehovot76100Israeldmitryk@wisdom.weizmann.ac.il1DESCRIPTIONOFTHEPROBLEMLetX,U,Ybefinite-dimensionalorinfinite-dimensionalseparableHilbertspaces.ConsidernDlinearsystemsoftheformPnx(t)=(Akx(t−ek)+Bku(t−ek)),Xnα:k=1(t∈Zn:t>0)Pnky(t)=(Ckx(t−ek)+Dku(t−ek)),k=1k=1(1)wheree:=(0,...,0,1,0,...,0)∈Zn(hereunitisonthek-thplace),forallkPt∈Znsuchthatnt≥0onehasx(t)∈X(thestatespace),u(t)∈Uk=1k(theinputspace),y(t)∈Y(theoutputspace),Ak,Bk,Ck,Dkareboundedlinearoperators,i.e.,Ak∈L(X),Bk∈L(U,X),Ck∈L(X,Y),Dk∈L(U,Y)forallk∈{1,...,n}.Weusethenotationα=(n;A,B,C,D;X,U,Y)forsuchasystem(hereA:=(A,...,A),etc.).ForT∈L(H,H)nandP1n12z∈CndenotezT:=nzT.Thenthetransferfunctionofαisk=1kkθ(z)=zD+zC(I−zA)−1zB.αXClearly,θisanalyticinsomeneighbourhoodofz=0inCn.LetαAkBkGk:=∈L(X⊕U,X⊕Y),k=1,...,n.CkDkWecallα=(n;A,B,C,D;X,U,Y)adissipativescatteringnDsystem(see[5,6])ifforanyζ∈Tn(theunittorus)ζGisacontractiveoperator,i.e., DISSIPATIVESCATTERINGNDREALIZATION15kζGk≤1.Itisknown[5]thatthetransferfunctionofadissipativescatter-ingnDsystemα=(n;A,B,C,D;X,U,Y)belongstothesubclassB0(U,Y)noftheclassBn(U,Y)ofallanalyticcontractiveL(U,Y)-valuedfunctionsontheopenunitpolydiskDn,whichissegregatedbytheconditionofvanishingofitsfunctionsatz=0.Thequestionwhethertheconverseistruewasimplicitlyaskedin[5]andstillhasnotbeenanswered.Thus,weposethefollowingproblem.Problem:Eitherprovethatanarbitraryθ∈B0(U,Y)canberealizednasthetransferfunctionofadissipativescatteringnDsystemoftheform(1)withtheinputspaceUandtheoutputspaceY,orgiveanexampleofafunctionθ∈B0(U,Y)(forsomen∈N,andsomefinite-dimensionalnorinfinite-dimensionalseparableHilbertspacesU,Y)thathasnosucharealization.2MOTIVATIONANDHISTORYOFTHEPROBLEMForn=1thetheoryofdissipative(orpassive,inotherterminology)scatter-inglinearsystemsiswelldeveloped(see,e.g.,[2,3])andrelatedtovariousproblemsofphysics(inparticular,scatteringtheory),stochasticprocesses,controltheory,operatortheory,and1Dcomplexanalysis.Itiswellknown(essentially,dueto[8])thattheclassoftransferfunctionsofdissipativescat-tering1Dsystemsoftheform(1)withtheinputspaceUandtheoutputspaceYcoincideswithB0(U,Y).Moreover,thisclassoftransferfunctions1remainsthesamewhenoneisrestrictedwithintheimportantspecialcaseofconservativescattering1Dsystems,forwhichthesystemblockmatrixGisunitary,i.e.,G∗G=I,GG∗=I.LetusnotethatintheX⊕UX⊕Ycasen=1asystem(1)canberewritteninanequivalentform(withoutaunitdelayinoutputsignaly)thatisthestandardformofalinearsystem,thenatransferfunctiondoesnotnecessarilyvanishatz=0,andtheclassoftransferfunctionsturnsintotheSchurclassS(U,Y)=B1(U,Y).TheclassesB0(U,Y)andB(U,Y)arecanonicallyisomorphicduetotherelation11B0(U,Y)=zB(U,Y).11In[1]animportantsubclassSn(U,Y)inBn(U,Y)wasintroduced.ThissubclassconsistsofanalyticL(U,Y)-valuedfunctionsonDn,say,θ(z)=Pθzt(hereZn={t∈Zn:t≥0,k=1,...,n},zt:=Qnztkfort∈Znt+kk=1k+z∈Dn,t∈Zn)suchthatforanyn-tupleT=(T,...,T)ofcommuting+1ncontractionsonsomecommonseparableHilbertspaceHandanypositivePr<1onehaskθ(rT)k≤1,whereθ(rT)=θ⊗(rT)t∈L(U⊗t∈ZntQ+H,Y⊗H),and(rT)t:=n(rT)tk.Forn=1andn=2onehask=1kSn(U,Y)=Bn(U,Y).However,foranyn>2andanynon-zerospacesUandYtheclassSn(U,Y)isapropersubclassofBn(U,Y).J.Aglerin[1]constructedarepresentationofanarbitraryfunctionfromSn(U,Y),whichinasystem-theoreticallanguagewasinterpretedin[4]asfollows:Sn(U,Y) 16PROBLEM1.3coincideswiththeclassoftransferfunctionsofnDsystemsofRoessertypewiththeinputspaceUandtheoutputspaceY,andcertainconservativityconditionimposed.Theanalogousresultisvalidforconservativesystemsoftheform(1).Asystemα=(n;A,B,C,D;X,U,Y)iscalledaconservativescatteringnDsystemifforanyζ∈TntheoperatorζGisunitary.Clearly,aconservativescatteringsystemisaspecialcaseofadissipativeone.By[5],theclassoftransferfunctionsofconservativescatteringnDsystemscoincideswiththesubclassS0(U,Y)inS(U,Y),whichissegregatedfromthelatterbynntheconditionofvanishingofitsfunctionsatz=0.Sinceforn=1andn=2onehasS0(U,Y)=B0(U,Y),thisgivesthewholeclassoftransferfunctionsnnofdissipativescatteringnDsystemsoftheform(1),andthesolutiontotheproblemformulatedaboveforthesetwocases.In[6]thedilationtheoryfornDsystemsoftheform(1)wasdeveloped.Itwasproventhatα=(n;A,B,C,D;X,U,Y)hasaconservativedilationifandonlyifthecorrespondinglinearfunctionLG(z):=zGbelongstoS0(X⊕U,X⊕Y).Systemsthatsatisfythiscriterionarecalledn-dissipativenscatteringones.Inthecasesn=1andn=2thesubclassofn-dissipativescatteringsystemscoincideswiththewholeclassofdissipativeones,andinthecasen>2thissubclassisproper.Sincetransferfunctionsofasystemandofitsdilationcoincide,theclassoftransferfunctionsofn-dissipativescatteringsystemswiththeinputspaceUandtheoutputspaceYisS0(U,Y).nAccordingto[7],foranyn>2thereexistp∈N,m∈N,operatorsDk∈L(Cp)andcommutingcontractionsT∈L(Cm),k=1,...,n,suchthatkXnXnmaxkzkDkk=10andRrational,stable,andstrictlyproper,thusboundedandanalyticontherighthalfplaneC+.Itisafundamentalprobleminrobustcontroldesigntoapproximatesuchsystemsbyfinite-dimensionalsystems.Thus,forafixednaturalnumbern,wewishtofindarationalapproximantGn(s)ofdegreeatmostninordertomakesmalltheapproximationerrorkG−Gnk,wherek.kdenotesanappropriatenorm.See[9]forsomerecentworkonthissubject.Commonlyusednormsonalineartime-invariantsystemwithimpulsere-sponseg∈L1(0,∞)andtransferfunctionG∈H∞(C)aretheH∞+pR∞p
1/pnormkGk∞=supRes>0|G(s)|,theLnormskgkp=0|g(t)|dt(1≤p<∞),andtheHankelnormkΓk,whereΓ:L2(0,∞)→L2(0,∞)istheHankeloperatordefinedbyZ∞(Γu)(t)=g(t+τ)u(τ)dτ.0ThesenormsarerelatedbykΓk≤kGk∞≤kgk1≤2nkΓk,wherethelastinequalityholdsforsystemsofdegreeatmostn.Twoparticularapproximationtechniquesforfinite-dimensionalsystemsarewell-establishedintheliterature[14],andtheycanalsobeusedforsomeinfinite-dimensionalsystems[5]: 30PROBLEM1.6•Truncatedbalancedrealizations,or,equivalently,outputnormalreal-izations[11,13,5];•OptimalHankel-normapproximants[1,4,5].Asweexplaininthenextsection,thesetechniquesareknowntoproduceH∞-convergentsequencesofapproximantsformanyclassesofdelaysystems(systemsofnucleartype).Wearethusledtoposethefollowingquestion:DothesequencesofreducedordermodelsproducedbytruncatedbalancedrealizationsandoptimalHankel-normapproximationsconvergeforallstabledelaysystems?2MOTIVATIONANDHISTORYOFTHEPROBLEMBalancedrealizationswereintroducedin[11],andmanypropertiesoftrun-cationsofsuchrealizationsweregivenin[13].AnH∞errorboundforthereduced-ordersystemproducedbytruncatingabalancedrealizationwasgivenforfinite-dimensionalsystemsin[3,4],andextendedtoinfinite-di-mensionalsystemsin[5].Thiscommonlyusedboundisexpressedintermsofthesequence(σ)∞ofsingularvaluesoftheHankeloperatorΓcorre-kk=1spondingtotheoriginalsystemG;inourcaseΓiscompact,andsoσk→0.12P∞Providedthatg∈L∩LandΓisnuclear(i.e.,k=1σk<∞)withdistinctsingularvalues,thentheinequalitykG−Gbk≤2(σ+σ+...)n∞n+1n+2holdsforthedegree-nbalancedtruncationGbofG.TheelementarylowernboundkG−Gnk≥σn+1holdsforanydegree-napproximationtoG.AnothernumericallyconvenientapproximationmethodistheoptimalHan-kel-normtechnique[1,4,5],whichinvolvesfindingabestrank-nHankelapproximationΓHtoΓ,intheHankelnorm,sothatkΓ−ΓHk=σ.Innnn+1thiscasetheboundkG−GH−Dk≤σ+σ+...n0∞n+1n+2isavailableforthecorrespondingtransferfunctionGHwithasuitablecon-nstantD0.Again,werequirethenuclearityofΓforthistobemeaningful.3AVAILABLERESULTSInthecaseofadelaysystemG(s)=e−sTR(s)asspecifiedabove,itisknownrTthattheHankelsingularvaluesσkareasymptotictoA,whererisπk MODELREDUCTIONOFDELAYSYSTEMS31therelativedegreeofRand|srR(s)|tendstothefinitenonzerolimitAas|s|→∞.HenceΓisnuclearifandonlyiftherelativedegreeofRisatleast2.(Equivalently,ifandonlyifgiscontinuous.)Wereferto[6,7]fortheseandmorepreciseresults.e−sTEvenforaverysimplenon-nuclearsystemsuchasG(s)=,forwhichs+1kσ→T/π,notheoreticalupperboundisknownfortheH∞errorsinktherationalapproximantsproducedbytruncatedbalancedrealizationsandoptimalHankel-normapproximation,althoughnumericalevidencesuggeststhattheyshouldstilltendtozero.ArelatedquestionistofindthebesterrorboundsinL1approximationofadelaysystem.Forexample,asmoothingtechniquegivesanL1approx-lnnimationerrorOnforsystemsofrelativedegreer=1(see[8]),anditispossiblethattheoptimalHankelnormmightyieldasimilarrateofconvergence.(AlowerboundofC/nforsomeconstantC>0followseasilyfromtheabovediscussion.)OneapproachthatmaybeusefulintheseanalysesistoexploitBonsall’stheoremthataHankelintegraloperatorΓisboundedifandonlyifitisuniformlyboundedonthesetofallnormalizedL2functionswhoseLaplacetransformsarerationalofdegreeone[2,12].AnexplicitconstantinBon-sall’stheoremisnotknown,andwouldbeofgreatinterestinitsownright.AnotherapproachwhichmayberelevantisthatofMegretski[10],whointroducesmaximalrealpartnorms.TheirintereststemsfromtheinequalitykGk∞≥kReGk∞≥kΓk/2.BIBLIOGRAPHY[1]V.M.Adamjan,D.Z.Arov,andM.G.Kre˘ın,“AnalyticpropertiesofSchmidtpairsforaHankeloperatorandthegeneralizedSchur–Takagiproblem,”Math.USSRSbornik,15:31–73,1971.[2]F.F.Bonsall,“BoundednessofHankelmatrices”,J.LondonMath.Soc.(2),29(2):289–300,1984.[3]D.Enns,ModelReductionforControlSystemDesign,Ph.D.disserta-tion,StanfordUniversity,1984.[4]K.Glover,“AlloptimalHankel-normapproximationsoflinearmul-tivariablesystemsandtheirL∞-errorbounds,Internat.J.Control,39(6):1115–1193,1984. 32PROBLEM1.6[5]K.Glover,R.F.Curtain,andJ.R.Partington,“Realisationandap-proximationoflinearinfinite-dimensionalsystemswitherrorbounds,”SIAMJ.ControlOptim.,26(4):863–898,1988.[6]K.Glover,J.Lam,andJ.R.Partington,“Rationalapproximationofaclassofinfinite-dimensionalsystems.I.SingularvaluesofHankeloper-ators,”Math.ControlSignalsSystems,3(4):325–344,1990.[7]K.Glover,J.Lam,andJ.R.Partington,“Rationalapproximationofaclassofinfinite-dimensionalsystems.II.OptimalconvergenceratesofL∞approximants,”Math.ControlSignalsSystems,4(3):233–246,1991.[8]K.GloverandJ.R.Partington,“Boundsontheachievableaccuracyinmodelreduction,”In:Modelling,RobustnessandSensitivityReductioninControlSystems(Groningen,1986),pp.95–118.Springer,Berlin,1987.[9]P.M.M¨akil¨aandJ.R.Partington,“Shiftoperatorinducedapproxi-mationsofdelaysystems,”SIAMJ.ControlOptim.,37(6):1897–1912,1999.[10]A.Megretski,“Modelorderreductionusingmaximalrealpartnorms,”PresentedatCDC2000,Sydney,2000.http://web.mit.edu/ameg/www/images/lund.ps.[11]B.C.Moore,“Principalcomponentanalysisinlinearsystems:control-lability,observability,andmodelreduction,”IEEETrans.Automat.Control,26(1):17–32,1981.[12]J.R.PartingtonandG.Weiss,“Admissibleobservationoperatorsfortheright-shiftsemigroup,”Math.ControlSignalsSystems,13(3):179–192,2000.[13]L.PerneboandL.M.Silverman,“Modelreductionviabalancedstatespacerepresentations,”IEEETrans.Automat.Control,27(2):382–387,1982.[14]K.Zhou,J.C.Doyle,andK.Glover,RobustandOptimalControl,UpperSaddleRiver,NJ:PrenticeHall1996. Problem1.7SchurextremalproblemsLevSakhnovichCourantInstituteofMathematicalScienceNewYork,NY11223USALev.Sakhnovich@verizon.net1DESCRIPTIONOFTHEPROBLEMInthispaperweconsiderthewell-knownSchurproblemthesolutionofwhichsatisfyinadditiontheextremalconditionw?(z)w(z)≤ρ2,|z|<1,(1)minwherew(z)andρminarem×mmatricesandρmin>0.Herethematrixρminisdefinedbyacertainminimal-rankcondition(seeDefinition1).WeremarkthattheextremalSchurproblemisaparticularcase.Thegeneralcaseisconsideredinbook[1]andpaper[2].Ourapproachtotheextremalproblemsdoesnotcoincidewiththesuperoptimalapproach[3],[4].Inpaper[2]wecompareourapproachtotheextremalproblemswiththesuperoptimalapproach.Interpolationhasfoundgreatapplicationsincontroltheory[5],[6].SchurExtremalProblem:Them×mmatricesa0,a1,...,anaregiven.Describethesetofm×mmatrixfunctionsw(z)holomorphicinthecircle|z|<1andsatisfyingtherelationw(z)=a+az+...+azn+...(2)01nandinequality(1.1).AnecessaryconditionofthesolvabilityoftheSchurextremalproblemistheinequalityR2−S≥0,(3)minwherethe(n+1)m×(n+1)mmatricesSandRminaredefinedbytherelationsS=CC?,R=diag[ρ,ρ,...,ρ],(4)nnminminminmin 34PROBLEM1.7a00...0a1a0...0Cn=.............(5)anan−1...a0Definition1:Weshallcallthematrixρ=ρmin>0minimalifthefollowingtworequirementsarefulfilled:1.TheinequalityR2−S≥0(6)minholds.2.Ifthem×mmatrixρ>0issuchthatR2−S≥0,(7)thenrank(R2−S)≤rank(R2−S),(8)minwhereR=diag[ρ,ρ,...,ρ].Remark1:Theexistenceofρminfollowsdirectlyfromdefinition1.Question1:Isρminunique?Remark2:Ifm=1thenρisuniqueandρ2=λ,whereλisminminmaxmaxthelargesteigenvalueofthematrixS.Remark3:Undersomeassumptionstheuniquenessofρminisprovedinthecasem>1,n=1(see[2],[7]).Ifρminisknownthenthecorrespondingwmin(ξ)isarationalmatrixfunc-tion.Thisgeneralizesthewell-knownfactforthescalarcase(see[7]).Question2:Howtofindρmin?InordertodescribesomeresultsinthisdirectionwewritethematrixS=CC?inthefollowingblockformnnS11S12,(9)S21S22whereS22isanm×mmatrix.Proposition1:[1]Ifρ=q>0satisfiesinequality(1.7)andtherelationq2=S+S?(Q2−S)−1S,(10)22121112whereQ=diag[q,q,...,q],thenρmin=q.Weshallapplythemethodofsuccessiveapproximationwhenstudyingequa-tion(1.10).Weputq2=S,q2=S+S?(Q2−S)−1S,wherek≥0,022k+12212k1112Qk=diag[qk,qk,...,qk].WesupposethatQ2−S>0.(11)011Theorem1:[1]Thesequenceq2,q2,q2,...monotonicallyincreasesandhas024thelimitm.Thesequenceq2,q2,q2,...monotonicallydecreasesandhasthe1135limitm.Theinequalitym≤mholds.Ifm=mthenρ2=q2.21212minQuestion3:Supposerelation(1.11)holds.Isthereacasewhenm16=m2?Theansweris“no”ifn=1(see[2],[8]).Remark4:Inbook[1]wegiveanexampleinwhichρminisconstructedinexplicitform. SCHUREXTREMALPROBLEMS35BIBLIOGRAPHY[1]LevSakhnovich.InterpolationTheoryandApplications,KluwerAcad.Publ.(1997).[2]J.HeltonandL.Sakhnovich,ExtremalProblemsofInterpolationTheory(forthcoming).[3]N.J.Young,“TheNevanlinna-PickProblemformatrix-valuedfunc-tions,”J.OperatorTheory,15,pp.239–265,(1986).[4]V.V.PellerandN.J.Young,“Superoptimalanalyticapproximationsofmatrixfunctions,”JournalofFunctionalAnalysis,120,pp.300–343,(1994).[5]M.GreenandD.Limebeer,LinearRobustControl,Prentice-Hall,(1995).[6]“LectureNotesinControlandInformation,”Sciences,SpringerVerlag,135(1989).[7]N.I.Akhiezer,“OnaMinimuminFunctionTheory,”OperatorTheory,Adv.andAppl.95,pp.19–35,(1997).[8]A.FerranteandB.C.Levy,“HermitianSolutionsoftheEquationsX=Q+NX−1N?,”LinearAlgebraandApplications247,pp.359-373,(1996). Problem1.8Theelusiveifftestfortime-controllabilityofbehavioursAmolSasaneFacultyofMathematicalSciencesUniversityofTwente7500AE,EnschedeTheNetherlandsA.J.Sasane@math.utwente.nl1DESCRIPTIONOFTHEPROBLEMProblem:LetR∈C[η,...,η,ξ]g×wandletBbethebehaviorgivenby1mthekernelrepresentationcorrespondingtoR.FindanalgebraictestonRcharacterizingthetime-controllabilityofB.Intheabove,weassumeBtocompriseofonlysmoothtrajectories,thatis,
B=w∈C∞Rm+1,Cw|Dw=0,R

whereD:C∞Rm+1,Cw→C∞Rm+1,CgisthedifferentialmapthatactsRasfollows:ifR=rij,theng×wPwr∂,...,∂,∂ww1k=11k∂x1∂xm∂tk..DR..=...wwPwr∂,...,∂,∂wk=1gk∂x1∂xm∂tkTime-controllabilityisapropertyofthebehavior,definedasfollows.ThebehaviorBissaidtobetime-controllableifforanyw1andw2inB,thereexistsaw∈Bandaτ≥0suchthatw1(•,t)forallt≤0w(•,t)=.w2(•,t−τ)forallt≥τ THEELUSIVEIFFTESTFORTIME-CONTROLLABILITY372MOTIVATIONANDHISTORYOFTHEPROBLEMThebehavioraltheoryforsystemsdescribedbyasetoflinearconstantcoef-ficientpartialdifferentialequationshasbeenachallengingandfruitfulareaofresearchforquitesometime(see,forinstance,PillaiandShankar[5],Oberst[3]andWoodetal.[4]).Anexcellentelementaryintroductiontothebehavioraltheoryinthe1−Dcase(correspondingtosystemsdescribedbyasetoflinearconstantcoefficientordinarydifferentialequations)canbefoundinPoldermanandWillems[6].In[5],[3]and[4],thebehavioursarisingfromsystemsofpartialdifferentialequationsarestudiedinageneralsettinginwhichthetime-axisdoesnotplayadistinguishedroleintheformulationofthedefinitionspertinenttocontroltheory.Sinceinthestudyofsystemswith“dynamics,”itisusefultogivespecialimportancetotimeindefiningsystemtheoreticconcepts,recentattemptshavebeenmadeinthisdirection(see,forexample,CotroneoandSasane[2],Sasaneetal.[7],andC¸amlıbelandSasane[1]).Theformulationofdefinitionswithspecialemphasisonthetime-axisisstraightforward,sincetheycanbeseenquiteeasilyasextensionsofthepertinentdefinitionsinthe1−Dcase.However,thealgebraiccharacterizationofthepropertiesofthebehavior,suchastime-controllability,turnouttobequiteinvolved.Althoughthetraditionaltreatmentofdistributedparametersystems(inwhichoneviewsthemasanordinarydifferentialequationwithaninfinite-dimensionalHilbertspaceasthestate-space)isquitesuccessful,thestudyofthepresentproblemwillhaveitsadvantages,sinceitwouldgiveatestthatisalgebraicinnature(andhencecomputationallyeasy)forapropertyofthesetsoftrajectories,namelytime-controllability.Anothermotivationforconsideringthisproblemisthattheproblemofpatchingupofsolutionsofpartialdifferentialequationsisalsoaninterestingquestionfromapurelymathematicalpointofview.3AVAILABLERESULTSInthe1−Dcase,itiswell-known(see,forexample,theorem5.2.5onpage154of[6])thattime-controllabilityisequivalentwiththefollowingcondition:Thereexistsar0∈N∪{0}suchthatforallλ∈C,rank(R(λ))=r0.ThisconditionisinturnequivalentwiththetorsionfreenessoftheC[ξ]-moduleC[ξ]w/C[ξ]gR.LetusconsiderthefollowingstatementsA1.TheC(η,...,η)[ξ]-moduleC(η,...,η)[ξ]w/C(η,...,η)[ξ]gRistor-1m1m1msionfree.A2.Thereexistsaχ∈C[η,...,η,ξ]wC(η,...,η)[ξ]gRandthereexists1m1manonzerop∈C[η,...,η,ξ]suchthatp·χ∈C(η,...,η)[ξ]gR,and1m1m 38PROBLEM1.8deg(p)=deg((p)),wheredenotesthehomomorphismp(ξ,η1,...,ηm)7→p(ξ,0,...,0):C[ξ,η1,...,ηm]→C[ξ].In[2],[7]and[1],thefollowingimplicationswereproved:Bistime-controllable⇓6⇑⇑⇐¬A2A16⇒AlthoughitistemptingtoconjecturethattheconditionA1mightbetheifftestfortime-controllability,thediffusionequationrevealstheprecariousnessofhazardingsuchaguess.In[1]itwasshownthatthediffusionequationistime-controllablewithrespectto1thespaceWdefinedbelow.BeforedefiningthesetW,werecallthedefinitionofthe(small)Gevreyclassoforder2,denotedbyγ(2)(R):γ(2)(R)isthesetofallϕ∈C∞(R,C)suchthatforeverycompactsetKandevery>0thereexistsaconstantCsuchthatforeveryk∈N,|ϕ(k)(t)|≤Ck(k!)2forallt∈K.Wisthendefinedtobethesetofallw∈Bsuchthatw(0,•)∈γ(2)(R).Furthermore,itwasalsoshownin[1],thatthecontrolcouldthenbeimplementedbythetwopointcontrolinputfunctionsactingatthepointx=0:u1(t)=w(0,t)andu2(t)=∂w(0,t)forallt∈R.ThesubsetWofC∞(R2,C)functionscomprises∂xalargeclassofsolutionsofthediffusionequation.Infact,aninterestingopenproblemistheproblemofconstructingatrajectoryinthebehaviorthatisnotintheclassW.Alsowhetherthewholebehavior(andnotjusttrajectoriesinW)ofthediffusionequationistime-controllableornotisanopenquestion.Theanswerstothesequestionswouldeitherstrengthenordiscardtheconjecturethatthebehaviorcorrespondingtop∈C[η1,...,ηm,ξ]istime-controllableiffp∈C[η1,...,ηm],whichwouldeventuallyhelpinsettlingthequestionoftheequivalenceofA1andtime-controllability.BIBLIOGRAPHY[1]M.K.C¸amlıbelandA.J.Sasane,“Approximatetime-controllabilityversustime-controllability,”submittedtothe15thMTNS,U.S.A.,June2002.[2]T.CotroneoandA.J.Sasane,“Conditionsfortime-controllabilityofbehaviours,”InternationalJournalofControl,75,pp.61-67(2002).[3]U.Oberst,“Multidimensionalconstantlinearsystems,”ActaAppl.Math.,20,pp.1-175(1990).1Thatis,foranytwotrajectoriesinW∩B,thereexistsaconcatenatingtrajectoryinW∩B. THEELUSIVEIFFTESTFORTIME-CONTROLLABILITY39[4]D.H.Owens,E.RogersandJ.Wood,“Controllableandautonomousn−Dlinearsystems,”MultidimensionalSystemsandSignalProcessing,10,pp.33-69(1999).[5]H.K.PillaiandS.Shankar,“ABehaviouralApproachtotheControlofDistributedSystems,”SIAMJournalonControlandOptimization,37,pp.388-408(1998).[6]J.W.PoldermanandJ.C.Willems,IntroductiontoMathematicalSys-temsTheory,Springer-Verlag,1998.[7]A.J.Sasane,E.G.F.ThomasandJ.C.Willems,“Time-autonomyversustime-controllability,”acceptedforpublicationinTheSystemsandControlLetters,2001. Problem1.9AFarkaslemmaforbehavioralinequalitiesA.A.(Tonny)tenDamInformation,CommunicationandTechnologyDivisionNationalAerospaceLaboratoryNLRP.O.Box90502,1006BMAmsterdamTheNetherlandstendam@nlr.nlJ.W.(Hans)NieuwenhuisFacultyofEconomicsUniversityofGroningenPostbus800,9700AVGroningenTheNetherlandsj.w.nieuwenhuis@eco.rug.nl1DESCRIPTIONOFTHEPROBLEMWithinthesystemsandcontrolcommunitytherehasalwaysbeenaninterestinminimalityissues.InthischapterweconjectureaFarkasLemmaforbehavioralinequalitiesthat,whentrue,willallowtostudyminimalityandelimationissuesforbehavioralsystemsdescribedbyinequalities.LetRn×m[s,s−1]denotethe(n×m)polynomialmatriceswithrealco-efficientsandpositiveandnegativepowersintheindeterminates.LetRn×m[s,s−1]denotethesetofmatricesinRn×m[s,s−1]withnon-negative+coefficientsonly.Inthischapterweconsiderdiscrete-timesystemswithtime-axisZ.Letσdenotethe(backward)shiftoperator,andletR(σ,σ−1)denotepolynomialoperatorsintheshift.OfinterestistherelationbetweentwopolynomialmatricesR(s,s−1)andR0(s,s−1)whentheysatisfyR(σ,σ−1)w≥0⇒R0(σ,σ−1)w≥0.(1)Basedonthestaticcase,onemayexpectthatsucharelationshouldbetheextensionofFarkas’slemmatothebehavioralcase.Thisleadstotheraisond’treofthischapter. AFARKASLEMMAFORBEHAVIORALINEQUALITIES410Conjecture:LetR∈Rg×q[s,s−1]andR0∈Rg×q[s,s−1].Thenwehave{R(σ,σ−1)w≥0⇒R0(σ,σ−1)w≥0}ifandonlyifthereexistsapolynomial0matrixH∈Rg×g[s,s−1]suchthatR0(s,s−1)=H(s,s−1)R(s,s−1).+Inordertoprovethisconjecture,onecouldtrytoextendtheoriginalproofgivenbyFarkasin[4].However,thisproofexplicitlyusesthefactthateveryscalarthatisunequaltozeroisinvertible.SuchageneralstatementdoesnotholdforelementsofRg×q[s,s−1].ThemostpromisingapproachforthedynamiccaseseemstobetheuseofmathematicaltoolssuchastheseparationtheoremofHahn-Banach(see,forinstance,[5]).Thebasicmathematicalpreliminariesreadasfollows.DenoteE:=(Rq)Zwiththetopologyofpoint-wiseconvergence.ThedualofE,denotedbyE∗,consistsofallRq-valuedsequencesthathavecom-pactsupport.LetR∈Rg×q[s,s−1].LetB={w∈Eq|R(σ,σ−1)w≥0}.ThepolarconeofB,denotedbyB#,isgivenby{w∈E∗|∀w∈B:Pw∗(t)w(t)≥0}.WewouldliketoestablishthatB#={w∈E|∃α∈t∈ZE∗,α≥0suchthatw∗=RT(σ−1,σ)α},butwehavesofarnotbeenabletoproveordisprovethesestatements.Thesestatements,togetherwiththe##factthat{B1⊆B2}implies{B2⊆B1},arebelievedtobeusefulinaproofoftheconjecture.2MOTIVATIONANDHISTORYOFTHEPROBLEMIntheearlyninetiesthefirstauthorstartedtoinvestigateminimalityissuesforso-calledbehaviorinequalitysystems,e.g.,systemswhosebehaviorBallowsadescriptionB={w∈Rq|R(σ,σ−1)w≥0}.Examplescanbefoundin[2].Thefirstpublicationthatweareawareofthatdealswiththisclassofsys-temsis[1].Andtheconjecturementionedabovecanalreadybefoundinthatpaper.Astheproblemprovedhardtosolve,anumberofinvestigationswherecarriedoutinthecontextoflinearstaticinequalities,wheretheprob-lemofminimalrepresentationsofsystemscontainingbothequalitiesandinequalitieswassolved[2].Theconjecture,however,withstoodourefforts,anditbecameapartofthePh.D.thesisofthefirstauthor[2].Asthestudyisplacedinthecontextofbehaviors,theFarkaslemmaforbehavioralinequalitiesisalsodiscussedintheWillem’sFestschrift[3](chapter16).UntiltheFarkaslemmaforbehavioralinequalitieshasbeenproven,issueslikeminimalrepresentations,eliminationoflatentvariablesetceteracannotbesolvedintheirfullgenerality.ItisourbeliefthattheFarkaslemmaforbehaviorinequalities,asconjecturedhere,willbeacornerstoneforfurtherinvestigationsinatheoryforbehavioralinequalities. 42PROBLEM1.93AVAILABLERESULTSForthestaticcase,theconjectureisnothingelsethanthefamousFarkaslemmaforlinearinqualities.Forthedynamiccase,theconjectureholdstrueforaspecialcase.Proposition:LetR∈Rg×q[s,s−1]beafull-rowrankpolynomialmatrix.0LetR0∈Rg×q[s,s−1].Then:{R(σ,σ−1)w≥0⇒R0(σ,σ−1)w≥0}ifand0onlyifthereexistsauniquepolynomialmatrixH∈Rg×g[s,s−1]suchthat+R0(s,s−1)=H(s,s−1)R(s,s−1).Theproofofthispropositioncanbefoundin[2](proposition4.5.12).4ARELATEDCONJECTUREItisofinteresttopresentarelatedconjecture,whoseresolutioniscloselylinkedtotheFarkaslemmaforbehavioralinequalities.Recallfrom[6]thatamatrixU∈Rg×g[s,s−1]issaidtobeunimodularifithasaninverseU−1∈Rg×g[s,s−1].WewillcallamatrixH∈Rg×g[s,s−1]+posimodularifitisunimodularandH−1∈Rg×g[s,s−1].Omittingthe+formaldefinitions,wewillcallarepresentationminimalifthenumberofequationsusedtodescribethebehaviorisminimal.Conjecture:Let{w∈(Rq)Z|R(σ,σ−1)w=0andR(σ,σ−1)w≥0}and12{w∈(Rq)Z|R0(σ,σ−1)w=0andR0(σ,σ−1)w≥0}bothbetwominimal12representations.TheyrepresentthesamebehaviorifandonlyiftherearepolynomialmatricesU(s,s−1),H(s,s−1)andS(s,s−1)suchthatR0(s,s−1)U(s,s−1)0R(s,s−1)1=1(2)R0(s,s−1)S(s,s−1)H(s,s−1)R(s,s−1)22withUunimodular,HposimodularandnoconditionsonS.Weremarkthatthisconjectureholdstrueforstaticinequalitiesandforthatcaseisgivenasproposition3.4.5in[2].BIBLIOGRAPHY[1]A.A.tenDam,“Representationsofdynamicalsystemsdescribedbybehaviouralinequalities,”In:ProceedingsEuropeanControlConferenceECC’93,vol.3,pp.1780-1783,June28-July1,Groningen,TheNether-lands,1993.[2]A.A.tenDam,UnilaterallyConstrainedDynamicalSystems,Ph.D.Dissertation,UniversityGroningen,TheNetherlands,1997.URL:http://www.ub.rug.nl/eldoc/dis/science/a.a.ten.dam/ AFARKASLEMMAFORBEHAVIORALINEQUALITIES43[3]A.A.tenDamandJ.W.Nieuwenhuis,“Onbehaviouralinequalities,”In:TheMathematicsfromSystemsandControl:FromIntelligentControltoBehavioralSystems,Groningen,1999,pp.165-176.[4]J.Farkas,“DiealgebraischeGrundlagederAnwendungendesMechanis-chenPrincipsvonFourier,”MathematischeundnaturwissenschaftlicheBerichteausUngarn,16,pp.154-157,1899.(Translationof:Gy.Farkas,“AFourier-f´elemechanikaielvalkalmaz´as´anakalgebraialapja,”Mathe-matikai´esTerm´eszettudom´anyiErtes´ıt¨o´,16,pp.361-364,1898.)[5]W.Rudin,FunctionalAnalysis,TataMcGraw-HillPublishingCompanyLtd.,1973.[6]J.C.Willems,“Paradigmsandpuzzlesinthetheoryofdynamicalsys-tems,”IEEETransactionsonAutomaticControl,vol.36,no.3,pp.259-294,1991. Problem1.10RegularfeedbackimplementabilityoflineardifferentialbehaviorsH.L.TrentelmanMathematicsInstituteUniversityofGroningenP.O.Box800,9700AVGroningenTheNetherlandsH.L.Trentelman@math.rug.nl1INTRODUCTIONInthisshortpaper,wewanttodiscussanopenproblemthatappearsinthecontextofinterconnectionofsystemsinabehavioralframework.Givenasystembehavior,playingtheroleofplanttobecontrolled,theproblemistocharacterizeallsystembehaviorsthatcanbeachievedbyinterconnectingtheplantbehaviorwithacontrollerbehavior,wheretheinterconnectionshouldbearegularfeedbackinterconnection.Morespecifically,wewilldealwithlineartime-invariantdifferentialsystems,i.e.,dynamicalsystemsΣgivenasatriple{R,Rw,B},whereRisthetime-axis,andwhereB,calledthebehaviorofthesystemΣ,isequaltothesetofallsolutionsw:R→Rwofasetofhigherorder,linear,constantcoefficient,differentialequations.Moreprecisely,∞wdB={w∈C(R,R|R()w=0},dtforsomepolynomialmatrixR∈R•×w[ξ].ThesetofallsuchsystemsΣisdenotedbyLw.Often,wesimplyrefertoasystembytalkingaboutitsbe-havior,andwewriteB∈LwinsteadofΣ∈Lw.BehaviorsB∈LwcanhencebedescribedbydifferentialequationsoftheformR(d)w=0,typicallywithdtthenumberofrowsofRstrictlylessthanitsnumberofcolumns.Mathemat-ically,R(d)w=0isthenanunder-determinedsystemofequations.Thisdtresultsinthefactthatsomeofthecomponentsofw=(w1,w2,...,ww)areunconstrained.Thisnumberofunconstrainedcomponentsisaninteger“in-variant”associatedwithB,andiscalledtheinputcardinalityofB,denotedbym(B),itsnumberoffree,“input,”variables.Theremainingnumberof REGULARFEEDBACKIMPLEMENTABILITY45variables,w−m(B),iscalledtheoutputcardinalityofBandisdenotedbyp(B).Finally,athirdintegerinvariantassociatedwithasystembehaviorB∈LwisitsMcMillandegree.Itcanbeshownthat(modulopermutationofthecomponentsoftheexternalvariablew)anyB∈Lwcanberepresentedbyastatespacerepresentationoftheformdx=Ax+Bu,y=Cx+Du,dtw=(u,y).Here,A,B,C,andDareconstantmatriceswithrealcompo-nents.Theminimalnumberofcomponentsofthestatevariablexneededinsuchaninput/state/outputrepresentationofBiscalledtheMcMillandegreeofB,andisdenotedbyn(B).SupposenowΣ={R,Rw1×Rw2,B}∈Lw1+w2andΣ={R,Rw2×Rw3,B}∈1122Lw2+w3arelineardifferentialsystemswithcommonfactorRw2inthesignalspace.ThemanifestvariableofΣ1is(w1,w2)andthatofΣ2is(w2,w3).Thevariablew2issharedbythesystems,anditisthroughthisvariable,calledtheinterconnectionvariable,thatwecaninterconnectthesystems.WedefinetheinterconnectionofΣ1andΣ2throughw2asthesystemΣ∧Σ:={R,Rw1×Rw2×Rw3,B∧B},1w221w22withinterconnectionbehaviorB1∧w2B2:={(w1,w2,w3)|(w1,w2)∈B1and(w2,w3)∈B2}.TheinterconnectionΣ1∧w2Σ2iscalledaregularinterconnectioniftheoutputcardinalitiesofΣ1andΣ2adduptothatofΣ1∧w2Σ2:p(B1∧w2B2)=p(B1)+p(B2).Itiscalledaregularfeedbackinterconnectionif,inaddition,thesumoftheMcMillandegreesofB1andB2isequaltotheMcMilandegreeoftheinterconnection:n(B1∧w2B2)=n(B1)+n(B2).ItcanbeproventhattheinterconnectionofΣ1andΣ2isaregularfeedbackinterconnectionif,possiblyafterpermutationofcomponentswithinw1,w2andw3,thereexistsacomponent-wisepartitionofw2intow2=(u,y1,y2),ofw1intow1=(v1,z1),andofw3intow3=(v2,z2)suchthatthefollowingfourconditionshold:1.inthesystemΣ1,(v1,y2,u)isinputand(z1,y1)isoutput,andthetransfermatrixfrom(v1,y2,u)to(z1,y1)isproper.2.inthesystemΣ2,(v2,y1,u)isinputand(z2,y2)isoutput,andthetransfermatrixfrom(v2,y1,u)to(z2,y2)isproper.3.inthesystemΣ1∧w2Σ2,(v1,v2,u)isinputand(z1,z2,y1,y2)isoutput,andthetransfermatrixfrom(v1,v2,u)to(z1,z2,y1,y2)isproper.4.ifweintroducenew(“perturbationsignals”)e1ande2and,insteadofy1andy2weapplyinputsy1+e2andy2+e1toΣ2andΣ1respec-tively,thenthetransfermatrixfrom(v1,v2,u,e1,e2)to(z1,z2,y1,y2)isproper. 46PROBLEM1.10Thefirstthreeoftheseconditionsstatethat,intheinterconnectionofΣ1andΣ2,alongtheterminalsoftheinterconnectedsystemonecanidentifyasignalflowthatiscompatiblewiththesignalflowdiagramofafeedbackcon-figurationwithpropertransfermatrices.Thefourthconditionstatesthatthisfeedbackinterconnectionis“well-posed.”Theequivalenceoftheprop-ertyofbeingaregularfeedbackinterconnectionwiththesefourconditionswasstudiedforthe“fullinterconnectioncase”in[8]and[2].2STATEMENTOFTHEPROBLEMSupposeP∈Lw+cisasystem(theplant)withtwotypesofexternalfullvariables,namelycandw.Thefirstofthese,c,istheinterconnectionvariablethroughwhichitcanbeinterconnectedtoasecondsystemC∈Lc(thecontroller)withexternalvariablec.TheexternalvariablecissharedbyPfullandC.TheremainingvariablewisthevariablethroughwhichPfullinteractswiththerestofitsenvironment.Afterinterconnectingplantandcontrollerthroughthesharedvariablec,weobtainthefullcontrolledbehaviorP∧C∈Lw+c.ThemanifestcontrolledbehaviorK∈Lwisobtainedbyfullcprojectingalltrajectories(w,c)∈Pfull∧cContheirfirstcoordinate:K:={w|thereexistscsuchthat(w,c)∈Pfull∧cC}.(1)Ifthisholds,thenwesaythatCimplementsK.If,foragivenK∈LwthereexistsC∈LcsuchthatCimplementsK,thenwecallKimplementable.If,inaddition,theinterconnectionofPfullandCisregular,wecallKregularlyimplementable.Finally,iftheinterconnectionofPfullandCisaregularfeedbackinterconnection,wecallKimplementablebyregularfeedback.Thisnowbringsustothestatementofourproblem:theproblemistocharacterize,foragivenP∈Lw+c,thesetofallbehaviorsK∈Lwthatfullareimplementablebyregularfeedback.Inotherwords:Problemstatement:LetP∈Lw+cbegiven.LetK∈Lw.FindfullnecessaryandsufficientconditionsonKunderwhichthereexistsC∈Lcsuchthat1.CimplementsK[meaningthat(1)holds],2.p(Pfull∧cC)=p(Pfull)+p(C),3.n(Pfull∧cC)=n(Pfull)+n(C).Effectively,acharacterizationofallsuchbehaviorsK∈Lwgivesacharac-terizationofthe“limitsofperformance”ofthegivenplantunderregularfeedbackcontrol. REGULARFEEDBACKIMPLEMENTABILITY473BACKGROUNDOuropenproblemistofindconditionsforagivenK∈Lwtobeimple-mentablebyregularfeedback.AnobviousnecessaryconditionforthisisthatKisimplementable,i.e.,itcanbeachievedbyinterconnectingtheplantwithacontrollerby(justany)interconnectionthroughtheintercon-nectionvariablec.Necessaryandsufficientconditionsforimplementabilityhavebeenobtainedin[7].TheseconditionsareformulatedintermsoftwobehaviorsderivedfromthefullplantbehaviorPfull:P:={w|thereexistscsuchthat(w,c)∈Pfull}andN:={w|(w,0)∈Pfull}.PandNarebothinLw,andarecalledthemanifestplantbehaviorandhiddenbehaviorassociatedwiththefullplantbehaviorPfull,respectively.In[7]ithasbeenshownthatK∈LwisimplementableifandonlyifN⊆K⊆P,(2)i.e.,KcontainsN,andiscontainedinP.Thiselegantcharacterizationofthesetofimplementablebehaviorsstillholdstrueif,insteadof(ordinary)lineardifferentialsystembehaviors,wedealwithnDlinearsystembehaviors,whicharesystembehaviorsthatcanberepresentedbypartialdifferentialequationsoftheform∂∂∂R(,,...,)w(x1,x2,...,xn)=0,∂x1∂x2∂xnwithR(ξ1,ξ2,...,ξn)apolynomialmatrixinnindeterminates.Recently,in[6]avariationofcondition(2)wasshowntobesufficientforimplementabilityofsystembehaviorsinamoregeneral(includingnonlinear)context.ForasystembehaviorK∈Lwtobeimplementablebyregularfeedback,anothernecessaryconditionisofcoursethatKisregularlyimplementable,i.e.,itcanbeachievedbyinterconnectingtheplantwithacontrollerbyreg-ularinterconnectionthroughtheinterconnectionvariablec.Alsoforregularimplementabilitynecessaryandsufficientconditionscanalreadybefoundintheliterature.In[1]ithasbeenshownthatagivenK∈Lwisregu-larlyimplementableifandonlyif,inadditiontocondition(2),thefollowingconditionholds:K+Pcont=P.(3)Condition(3)statesthatthesumofKandthecontrollablepartofPisequaltoP.ThecontrollablepartPcontofthebehaviorPisdefinedasthelargestcontrollablesubbehaviorofP,whichistheuniquebehaviorPcontwiththepropertiesthat1.)P⊆P,and2.)P0controllableandP0⊆PimpliescontP0⊆P.Clearly,ifthemanifestplantbehaviorPiscontrollable,thencontP=Pcont,socondition(3)automaticallyholds.Inthiscase,implementabil-ityandregularimplementabilityareequivalentproperties.Forthespecial 48PROBLEM1.10caseN=0(whichisequivalenttothe“fullinterconnectioncase”),condi-tions(2)and(3)forregularimplementabilityinthecontextofnDsystembehaviorscanalsobefoundin[4].Inthesamecontext,resultsonregularimplementabilitycanalsobefoundin[9].Wefinallynotethat,againforthefullinterconnectioncase,theopenproblemstatedinthispaperhasrecentlybeenstudiedin[3],usingasomewhatdifferentnotionoflinearsystembehavior,indiscretetime.Uptonow,however,thegeneralproblemhasremainedunsolved.BIBLIOGRAPHY[1]M.N.BelurandH.L.Trentelman,“Stabilization,poleplacementandregularimplementability,”IEEETransactionsonAutomaticControl,May2002.[2]M.Kuijper,“Whydostabilizingcontrollersstabilize?”Automatica,vol.31,pp.621-625,1995.[3]V.Lomadze,Oninterconnectionsandcontrol,manuscript,2001.[4]P.RochaandJ.Wood,“TrajectorycontrolandinterconnectionofnDsystems,”SIAMJournalonContr.andOpt.,vol.40,no1,pp.107-134,2001.[5]J.W.PoldermanandJ.C.Willems,IntroductiontoMathematicalSys-temsTheory:ABehavioralApproach,SpringerVerlag,1997.[6]A.J.vanderSchaft,Achievablebehaviorofgeneralsystems,manuscript,submittedforpublication,2002.[7]J.C.WillemsandH.L.Trentelman,“Synthesisofdissipativesystemsusingquadraticdifferentialforms,Part1,”IEEETransactionsonAu-tomaticControl,vol.47,no.1,pp.53-69,2002.[8]J.C.Willems,“OnInterconnections,ControlandFeedback,”IEEETransactionsonAutomaticControl,vol.42,pp.326-337,1997.[9]E.ZerzandV.Lomadze,“Aconstructivesolutiontointerconnectionanddecompositionproblemswithmultidimensionalbehaviors,”SIAMJournalonContr.andOpt.,vol.40,no4,pp.1072-1086,2001. Problem1.11RiccatistabilityErikI.Verriest1SchoolofElectricalandComputerEngineeringGeorgiaInstituteofTechnologyAtlanta,GA,30332-0250USAerik.verriest@ee.gatech.edu1DESCRIPTIONOFTHEPROBLEMGiventwon×nrealmatrices,AandB,considerthematrixRiccatiequationA0P+PA+Q+PBQ−1B0P+R=0.(1)Canonecharacterizethepairs(A,B)forwhichtheaboveequationhasasolutionforpositivedefinitesymmetricmatricesP,Q,andR?In[8]apair(A,B)wasdefinedtobeRiccatistableifatripleofpositivedefinitematricesP,Q,Rexistssuchthat(1)holds.TheproblemmaybestatedequivalentlyasanLMI:Canonecharacterizeallpairs(A,B)withoutinvokingadditionalmatrices,forwhichthereexistpositivedefinitematricesPandQsuchthatA0P+PA+QPB0<0.(2)BP−Q2MOTIVATIONANDHISTORYOFTHEPROBLEMEquation(1)playsanimportantroleinthestabilityanalysisoflineartime-invariantdelay-differentialsystems.Itisknown[9]thattheautonomoussystemx˙(t)=Ax(t)+Bx(t−τ)(3)1SupportbytheNSF-CNRScollaborativegrantINT-9818312isgratefullyacknowl-edged. 50PROBLEM1.11isasymptoticallystable,forallvaluesofτ≥0,ifthepair(A,B)isRiccatistable.Notethatsince(3)hastobestableforτ=0andτ→∞,thematricesA+BandAhavetobeHurwitzstable,i.e.,hasitsspectrumintheopenlefthalfplane.RecallalsothatamatrixCisSchur-Cohnstable,ifitsspectrumliesintheopenunitdisk.IfB=0,thusreducingtheproblemtoafinitedimensionaltime-invariantsystem,theRiccatiequationreducestotheubiquitousLyapunovequation,A0P+PA+S=0,(4)wherewehavesetQ+R=S.Itiswellknownthatapositivedefinitepair(P,S)existsifandonlyifAisHurwitz.Thisconditionisnecessaryandsufficient.TheaboveresultanditsequivalentLMIformulation,initiatedawholesetofextensions:formultipledelays,distributeddelays,time-variantsystems(withtime-variantdelays)[3,5].Inadditionalltheabovevariantscanfur-therbeextendedtoincludeparametervariations(robuststability)andnoise(stochasticstability).Alsoothertypesoffunctionaldifferentialequations(scaledelay)leadtosuchconditions[7].ThemainideainderivingtheseresultsistheuseoftheLyapunov-KrasovskiitheorywithappropriateLya-punovfunctionals.Theequation(1)appearsalsoinH∞controltheoryandingametheory.3AVAILABLERESULTSIn[8],whereRiccati-stabilitywascalled“d-stability,”referringto“delay,”thefollowingconnectionswithspectralpropertiesofAandBwereobtainedTheorem1:IfthereexistsatripleofsymmetricpositivedefinitematricesP,Q,andR,satisfying(1),thenAisHurwitzandA−1BisSchur-Cohn.Thereisnocompleteconverseofthistheorem,however,twopartialcon-versesareeasilyproven:Theorem2:IfthematrixproductA−1BisSchur-Cohn,thenthereexistsanorthogonalmatrixΘsuchthatΘAisHurwitz,andthepair(ΘA,ΘB)isRiccati-stable.Theorem3:IfthematrixAisHurwitz,thenthereexistsamatrixBsuchthatA−1BisSchur-Cohnand(A,B)isRiccati-stable.Inadditionthefollowingscalingpropertiesareshownin[8].Lemma1:If(A,B)isRiccati-stable,then(αA,αB)isRiccati-stableforallα>0.Lemma2:If(A,B)isRiccati-stable,then(SAS−1,SBS−1)isRiccati-stable,forallnonsingularS.Lemma3:If(A,B)isRiccati-stable,andBhasfullrank,then(A0,B0)is RICCATISTABILITY51Riccati-stable.ThefullrankconditiononBcanberelaxed.Lemma3isadualityresult.In[8]adetailedconstructionwasgivenforasubsetofRiccati-stablepairsforthecasen=2.Itleadstoan(over-)parameterization,buttheconstructionreadilyextendstoarbitrarydimensions,byusingTheorem4:Assumethatthepairs{(Ai,Bi)|i=1...N}areRiccati-stableforthesameP-matrix.i.e.,thereexistQi,Ri>0,i=1...NsuchthatA0P+PA+Q+PBQ−1B0P+R=0.iiiiiiiThenallpairsinthepositiveconegeneratedbytheabovepairsareRiccati-PPstable.i.e.,∀αi≥0,butnotallzero,thepair(iαiAi,iαiBi)isRiccati-stable.TheinvarianceofRiccati-stabilityundersimilarity(lemma2)ensuresthatif(A,B)isRiccati-stable,onecantransformthesystemtooneforwhichthenewPmatrix,i.e.,S−TPS−1istheidentity.Thusmotivated,weprovideasimplifiedform:GivenB,denotebyABthesetofmatricesAforwhich(A,B)isRiccatistable,withP=I,i.e.,A={A|∃Q=Q0>0,s.t.A+A0+Q+BQ−1B0<0}.BHence,anecessaryconditionforA∈ABisthatitssymmetricpartAssatisfies1−10As<−(Q+BQB),2forsomeQ>0.IfforeachBthesetABcanbedetermined,theproposedproblemwillbesolved.Thefollowingspecialcaseisproven:Theorem5:IfBisinthereal-diagonalform,B=Blockdiag{Λ+,0,−Λ−,B1,...,Bc}whereΛ+=diag{λ1,...,λp}arethepositiverealeigenvalues,−Λ−=diag{−λp+1,...,−λp+m}arethenegativerealones,andtheBk’sare2×2σkωkblocksBk=,associatedwiththecomplexeigenvaluesσk±iωk,−ωkσkthenthesetABischaracterizedbythesetofallmatrices,A,whosesym-metricpartsatisfiesAs<−2Blockdiag{Λ+,0,Λ−,|σ1|I2...,|σc|I2}.Proof:Inthisblockdiagonalform,itisclearthatitsufficestochoosethesameblockdiagonalstructureforQandtheproblemdecouples.Forreal2eigenvaluesinthesetsΛand−Λ,observethatq+λk≥2|λ|andequal-+−qkityisobtainedforq=|λk|.Likewiseforazeroeigenvalue,thecorrespondingqmaybetakeninfinitesimallysmall.Forcomplexconjugateeigenvaluepairs,observethat−1q1qσkωkq1qσkωk10+≥2|σk|.qq2−ωkσkqq2−ωkσk01 52PROBLEM1.11q1q|σ|ωEqualityisachievedwith=,if|σ|≥|ω|and,qq2ω|σ|ρρswitchingtopolarform,withqq1=|cosφ|(1+cosχ),q2=|cosφ|(1−cosχ),2cos2χandq=ρtanφ−cos2φ,whereρandφarerespectivelythemodulusandtheargumentofthecomplexeigenvalueσ+iω,andχarbitrarywith|cosχ|<|sinφ|if|σ|<|ω|.Inthelattercase,thesolutionwasobtainedbydirectoptimizationoftheminimaleigenvalueofthematrixQ+BQ−1B0overallkkpositivedefinitematricesQ.HenceQ+BQ−1B0≥0ifBissingular,andQ+SBQ−1B0≥2zI,wherez=min({|λ|;k=1...p+m}{|σ|,`=1,...,c})k`ifBhasfullrank,fromwhichthetheoremfollows.utEquationsrelatedto(1)arealsodiscussedin[1,2,4,6].BIBLIOGRAPHY[1]W.N.Anderson,Jr.,T.D.MorleyandG.E.Trapp,“PositivesolutionstoX=A−B∗X−1B∗,”LAA,134,pp.53-62,1990.[2]R.Datko,“SolutionsoftheoperatorequationA∗K+KA+KRK=−W,”In:SemigroupsofOperators:TheoryandApplications,A.V.Bal-akrishnan,Ed.,Birkh¨auser,2000.[3]L.DugardandE.I.Verriest,StabilityandControlofTime-DelaySys-tems,Springer-Verlag,LNCIS,vol.228,1998.[4]J.Engwerda,“OntheexistenceofapositivedefinitesolutiontothematrixequationX+ATX−1A=I,”LAA,194,pp.91-108,1993.[5]V.B.Kolmanovskii,S.-I.NiculescuandK.Gu,“Delayeffectsonstabil-ity:Asurvey,”Proc.38thIEEEConf.Dec.Control,Phoenix,AZ,pp.1993-1998,199).[6]A.C.M.RanandM.C.B.Reurings,“OnthenonlinearmatrixequationX+A∗F(X)A=Q:solutionsandperturbationtheory,”LinearAlgebraanditsApplications,vol.346,pp.15-26,2002.[7]E.I.Verriest,“Robuststability,adjoints,andLQcontrolofscale-delaysystems,”Proc.38-thConf.Dec.Control,Phoenix,AZ,pp.209-214,1999.[8]E.I.Verriest,“Robuststabilityandstabilization:fromlineartonon-linear,”Proceedingsofthe2ndIFACWorkshoponLinearTimeDelaySystems,Ancona,Italy,pp.184-195,September2000. RICCATISTABILITY53[9]E.I.VerriestandA.F.Ivanov,“Robuststabilizationofsystemswithde-layedfeedback,”ProceedingsoftheSecondInternationalSymposiumonImplicitandRobustSystems,Warszawa,Poland,pp.190-193,July1991. Problem1.12StateandfirstorderrepresentationsJanC.Willems1DepartmentofElectricalEngineering-SCD(SISTA)UniversityofLeuvenKasteelparkArenberg10B-3001Leuven-HeverleeBelgiumJan.Willems@esat.kuleuven.ac.be1DESCRIPTIONOFTHEPROBLEMWeconjecturethatthesolutionsetofasystemoflinearconstantcoefficientPDEsisMarkovianifandonlyifitisthesolutionsetofasystemoffirstorderPDEs.Ananalogousconjectureregardingstatesystemsisalsomade.NotationFirst,weintroduceournotationforthesolutionsetsoflinearPDEsinthenrealindependentvariablesx=(x,...,x).LetD0denote,asusual,theset1nnofrealdistributionsonRn,andLwthelinearsubspacesof(D0)wconsistingnnofthesolutionsofasystemoflinearconstantcoefficientPDEsinthewreal-valueddependentvariablesw=col(w1,...,ww).Moreprecisely,eachelementB∈LwisdefinedbyapolynomialmatrixR∈R•×w[ξ,ξ,...,ξ],n12nwithwcolumns,butanynumberofrows,suchthat0w∂∂∂B={w∈(Dn)|R(,,...,)w=0}.∂x1∂x2∂xnWerefertoelementsofLwaslineardifferentialn-Dsystems.TheabovenPDEiscalledakernelrepresentationofB∈Lw.NotethateachB∈Lwhasnnmanykernelrepresentations.Foranin-depthstudyofLw,see[1]and[2].n1ThisresearchissupportedbytheBelgianFederalGovernmentundertheDWTCprogramInteruniversityAttractionPoles,PhaseV,2002-2006,DynamicalSystemsandControl:Computation,IdentificationandModelling. STATEANDFIRSTORDERREPRESENTATIONS55Next,weintroduceaclassofspecialthree-waypartitionsofRn.DenotebyPthefollowingsetofpartitionsofRn:[(S,S,S)∈P]:⇔[(S,S,SaredisjointsubsetsofRn)−0+−0+∧(S∪S∪S=Rn)∧(SandSareopen,andSisclosed)].−0+−+0Finally,wedefineconcatenationofmapsonRn.Letf,f:Rn→F,and−+letπ=(S,S,S)∈P.Definethemapf∧f:Rn→F,calledthe−0+−π+concatenationof(f−,f+)alongπ,byf−(x)forx∈S−(f−∧πf+)(x):=f+(x)forx∈S0∪S+MarkoviansystemsDefineB∈LwtobeMarkovian:⇔n[(w,w∈B∩C∞(Rn,Rw))∧(π=(S,S,S)∈P)−+−0+∧(w−|S0=w+|S0)]⇒[(w−∧πw+∈B].ThinkofS−asthe“past”,S0asthe“present”,andS+asthe“future.”MarkovianmeansthatiftwosolutionsofthePDEagreeonthepresent,thentheirpastsandfuturesarecompatible,inthesensethatthepast(andpresent)ofone,concatenatedwiththe(presentand)futureoftheother,isalsoasolution.Inthelanguageofprobability:thepastandthefutureareindependentgiventhepresent.Wecometoourfirstconjecture:B∈LwisMarkoviannifandonlyifithasakernelrepresentationthatisfirstorder.Thus,itisconjecturedthataMarkoviansystemadmitsakernelrepresen-tationoftheform∂∂∂R0w+R1w+R2w+···Rnw=0.∂x1∂x2∂xnOberst[2]hasproventhatthereisaone-to-onerelationbetweenLwandthensubmodulesofRw[ξ,ξ,...,ξ],eachB∈Lwbeingidentifiablewithitssetofannihilators12nnw>∂∂∂NB:={n∈R[ξ1,ξ2,...,ξn]|n(,,...,)B=0}.∂x1∂x2∂xnMarkovianityishenceconjecturedtocorrespondexactlytothoseB∈LwnforwhichthesubmoduleNBhasasetoffirstordergenerators. 56PROBLEM1.12StatesystemsInthissectionweconsidersystemswithtwokindofvariables:wreal-valuedmanifestvariables,w=col(w1,...,ww),andzreal-valuedstatevariables,z=col(z1,...,zz).TheirjointbehaviorisagainassumedtobethesolutionsetofasystemoflinearconstantcoefficientPDEs.Thusweconsiderbe-haviorsinLw+z,whenceeachelementB∈Lw+zisdescribedintermsoftwonnpolynomialmatrices(R,M)∈R•×(w+z)[ξ,ξ,...,ξ]by12nB={(w,z)∈(D0)w×(D0)z|nn∂∂∂∂∂∂R(,,...,)w+M(,,...,)z=0}.∂x1∂x2∂xn∂x1∂x2∂xnDefineB∈Lw+ztobeastatesystemwithstatez:⇔n[((w,z),(w,z)∈B∩C∞(Rn,Rw+z))∧(π=(S,S,S)∈P)−−++−0+∧(z−|S0=z+|S0)]⇒[(w−,z−)∧π(w+,z+)∈B].ThinkagainofS−asthe“past”,S0asthe“present”,S−+asthe“future”.Statemeansthatifthestatecomponentsoftwosolutionsagreeonthepresent,thentheirpastsandfuturesarecompatible,inthesensethatthepastofonesolution(involvingboththemanifestandthestatevariables),concatenatedwiththepresentandfutureoftheothersolution,isalsoasolu-tion.Inthelanguageofprobability:thepresentstate“splits”thepastandthepresentplusfutureofthemanifestandthestatetrajectorycombined.Wecometooursecondconjecture:B∈Lw+zisastatesystemnifandonlyifithasakernelrepresentationthatisfirstorderinthestatevariableszandzero-thorderinthemanifestvariablesw.I.e.,itisconjecturedthatastatesystemadmitsakernelrepresentationoftheform∂∂∂R0w+M0z+M1z+M2z+···Mnz=0.∂x1∂x2∂xn2MOTIVATIONANDHISTORYOFTHEPROBLEMTheseopenproblemsaimatunderstandingstateandstateconstructionforn-Dsystems.Maxwell’sequationsconstituteanexampleofaMarkoviansystem.Thediffusionequationandthewaveequationarenon-examples. STATEANDFIRSTORDERREPRESENTATIONS573AVAILABLERESULTSItisstraightforwardtoprovethe“if”-partofbothconjectures.Theconjec-turesaretrueforn=1,i.e.,intheODEcase,see[3].BIBLIOGRAPHY[1]H.K.PillaiandS.Shankar,“Abehavioralapproachtocontrolofdis-tributedsystems,”SIAMJournalonControlandOptimization,vol.37,pp.388-408,1999.[2]U.Oberst,“Multidimensionalconstantlinearsystems,”ActaApplican-daeMathematicae,vol.20,pp.1-175,1990.[3]P.RapisardaandJ.C.Willems,“Statemapsforlinearsystems,”SIAMJournalonControlandOptimization,vol.35,pp.1053-1091,1997. Problem1.13ProjectionofstatespacerealizationsAntoineVandendorpeandPaulVanDoorenDepartmentofMathematicalEngineeringUniversit´ecatholiquedeLouvainB-1348Louvain-la-NeuveBelgium1DESCRIPTIONOFTHEPROBLEMWeconsidertwom×pstrictlypropertransferfunctionsT(s)=C(sI−A)−1B,Tˆ(s)=Cˆ(sI−Aˆ)−1B,ˆ(1)nkofrespectiveMcMillandegreesnandkkifandonlyifthereexisttworegularpencils,Mr−sNrandMl−sNlsuchthatthematricesL,L,R,ˆR,QˆlandQrofthefollowingequationsA−sIn0BRNrR0Aˆ−sIkBˆRNˆr=Rˆ(Mr−sNr),(9)C−Cˆ0Qr0AT−sI0CTLNLnl0AˆT−sI−CˆT−LNˆ=−Lˆ(M−sN),(10)klllBTBˆT0Ql0satisfythefollowingconditions:RNr1.NTLT−NTLˆTQT(M−Ns)RNˆr=0,lllQr PROJECTIONOFSTATESPACEREALIZATIONS612.dimIm(RNˆr)=dimIm(LNˆl)=k.Moreover,suchmatricesalwaysexistprovided2k≤2n−m−p.Theconditionsgivenbyourconjectureareatleastsufficient.Indeed,fromequations(10),and(9)andtheregularityassumptionofMr−sNrandMl−sNl,itfollowsthatCRN=CˆRNˆ,NTLTB=NTLˆTB.ˆ(11)rrllThen,fromcondition1,NTLTRN=NTLˆTRNˆ,NTLTARN=NTLˆTAˆRNˆ.(12)lrlrlrlrFinally,conditions1and2implythatthematricesRNˆrandLNˆlarerightinvertible.DefiningZ,V∈Cn×kbyZ=LN(LNˆ)−r,V=RN(RNˆ)−r,(13)llrrwecaneasilyverifyequations(1)and(2).Anotherjustificationisthat(bylookingcarefullyattheproofoftheorem1)Conjecture3istruefortheSISOcase.WenowpresentthelinkwiththeKrylovtechniques.Equations(9)and(10)giveusthefollowingSylvesterequations:ARN−RM+BQ=0,ATLN−LM+CTQ=0.(14)rrrlllTheseSylvesterequationscorrespondtogeneralizedleftandrighteigenspa-cesofthesystemzeromatrix(8).Moreprecisely,Im(RNr)andIm(LNl)canbeexpressedasgeneralizedKrylovspacesoftheform(4)and(5).ThechoiceofmatricesMl,Nl,Mr,Nr,Ql,andQrcorrespondrespectivelytoleftandrighttangentialinterpolationconditionsattheeigenvaluesσiof(Mr−sNr)andγjof(Ml−sNl),thataresatisfiedbetweenT(s)andTˆ(s)(see[5]).TheseeigenspacescorrespondtodisjointpartsofthespectrumofM−NssuchthattheproductNTLTRN=NTLˆTRNˆisinvertible(seelrlr[5]formoredetails).Inotherwords,ourconjectureisthatanyprojectedreduced-ordertransferfunctioncanbeobtainedbyimposingsomeinterpolationconditionsorsomemodalapproximationconditionswithrespecttotheoriginaltransferfunc-tion.Moreover,asolutionalwaysexistsprovided2k≤2n−m−p(i.e.,forallTˆ(s)ofsufficientlysmalldegreek).Ifthisturnsouttobetrue,wecouldhopetofindtheinterpolationconditionsthatyield,e.g.,theoptimalHankelnormoroptimalH∞normreduced-ordermodelsusingcheapinterpolationtechniques.4AVAILABLERESULTSIndependently,Halevirecentlyprovedin[6]newresultsconcerningthegen-eralframeworkofmodelorderreductionviaprojection.Theunknowns 62PROBLEM1.13ZandVhave2nkparameters(ordegreesoffreedom),while(2)imposesm+p(2k+m+p)kconstraints.Heshowsthatthecasek=n−corresponds2toafinitenumberofsolutions.Moreover,fortheparticularcasem=pandk=n−m,heshowsthatanypairofprojectingmatricesZ,Vsatisfying(2)canbeseenasgeneralizedeigenspacesofacertainmatrixpencil.ThematrixpencilusedbyHalevicanbelinkedtothesystemzeromatrixoftheerrortransferfunctiondefinedinequation(8).MatricesZandVsatisfying(2)arealsothektrailingrowsofS−1,respec-tivelycolumnsofSwhichtransformthesystem(A,B,C)tothesystem(S−1AS,S−1B,CS):∗∗∗S−1AS−sIS−1Bn=∗Aˆ−sIBˆ.(15)kCS0∗Cˆ0TheexistenceofprojectingmatricesZ,Vsatisfying(1and2)isthereforerelatedtotheabovesubmatrixproblem.AsquarematrixAˆissaidtobeembeddedinasquarematrixAwhenthereexistsachangeofcoordinatesSsuchthatAˆ−sIisasubmatrixofS−1(A−sI)S.Necessaryandsufficientknconditionsfortheembeddingofsuchmonicpencilsaregivenin[9],[8].Asformonicpencils,wesaythatthepencilMˆ−NsˆisembeddedinthepencilM−NswhenthereexistinvertiblematricesLe,RisuchthatMˆ−Nsˆisasub-matrixofLe(M−Ns)Ri.Findingnecessaryandsufficientcondi-tionsfortheembeddingofsuchgeneralpencilsisstillanopenproblem[7].Nevertheless,oneobtainsfrom[9],[8],[7]necessaryconditionson(C,ˆA,ˆBˆ)Aˆ−sIkBˆA−sInBand(C,A,B)fortobeembeddedin.TheseCˆ0C0obviouslygivenecessaryconditionsfortheexistenceofprojectingmatricesZ,Vsatisfying(1and2).Wehopetobeabletoshednewlightontheneces-saryandsufficientconditionsfortheembeddingproblemviatheconnectionsdevelopedinthispaper.BIBLIOGRAPHY[1]A.C.Antoulas,“Lecturesontheapproximationoflarge-scaledynamicalsystems,”SIAMBookSeries:AdvancesinDesignandControl,2002.[2]J.A.Ball,I.GohbergandL.Rodman,Interpolationofrationalmatrixfunctions,Birkh¨auserVerlag,Basel,1990.[3]K.Gallivan,A.Vandendorpe,andP.VanDooren,“Modelreductionviatangentialinterpolation,”MTNS2002(15thSymp.ontheMathemati-calTheoryofNetworksandSystems),UniversityofNotreDame,SouthBend,Indiana,USA,August2002. PROJECTIONOFSTATESPACEREALIZATIONS63[4]K.Gallivan,A.Vandendorpe,andP.VanDooren,“ModelReductionviatruncation:aninterpolationpointofview,”LinearAlgebraAppl.,submitted.[5]K.Gallivan,A.Vandendorpe,andP.VanDooren,“Modelre-ductionofMIMOsystemsviatangentialinterpolation,”Inter-nalreport,Universit´ecatholiquedeLouvain,2002.Availableathttp://www.auto.ucl.ac.be/vandendorpe/.[6]Y.Halevi,“Onmodelorderreductionviaprojection,”15thIFACWorldCongressonautomaticcontrol,Barcelona,July2002.[7]J.J.Loiseau,S.Mondi´e,I.ZaballaandP.Zagalak,“AssigningtheKro-neckerinvariantsofamatrixpencilbyroworcolumncompletions,”LinearAlgebraAppl.,278,pp.327-336,1998.[8]E.MarquesdeS´a,“Imbeddingconditionsforλ-matrices,”LinearAlgebraAppl.,24,pp.33-50,1979.[9]R.C.Thompson,“InterlacingInequalitiesforInvariantFactors,”LinearAlgebraAppl.,24,pp.1-31,1979.[10]K.Zhou,J.C.Doyle,andK.Glover,Robustandoptimalcontrol,UpperSaddleRiver,PrenticeHall,Inc,N.J.:1996. PART2StochasticSystems Problem2.1OnerrorofestimationandminimumofcostforwidebandnoisedrivensystemsAgamirzaE.BashirovDepartmentofMathematicsEasternMediterraneanUniversityMersin10Turkeyagamirza.bashirov@emu.edu.tr1DESCRIPTIONOFTHEPROBLEMThesuggestedopenproblemconcernstheerrorofestimationandthemini-mumofthecostinthefilteringandoptimalcontrolproblemsforapartiallyobservablelinearsystemcorruptedbywidebandnoiseprocesses.Recentresultsallowtoconstructawidebandnoiseprocessinacertaininte-gralformonthebasisofitsautocovariancefunctionanddesigntheoptimalfilterandtheoptimalcontrolforapartiallyobservablelinearsystemcor-ruptedbysuchwidebandnoiseprocesses.Moreover,explicitformulaefortheerrorofestimationandfortheminimumofthecosthavebeenobtained.But,theinformationaboutwidebandnoisecontainedinitsautocovariancefunctionisincomplete.Hence,everyautocovariancefunctiongeneratesin-finitelymanywidebandnoiseprocessesrepresentedintheintegralform.Consequently,theerrorofestimationandtheminimumofthecostmen-tionedaboveareforasamplewidebandnoiseprocesscorrespondingtothegivenautocovariancefunction.Thefollowingproblemarises:givenanautocovariancefunction,whataretheleastupperandgreatestlowerboundsoftherespectiveerrorofestimationandtherespectiveminimumofthecost?Whatarethedistributionsoftheerrorofestimationandtheminimumofthecost?Whataretheparametersofthewidebandnoiseprocessproducingtheaverageerrorandtheaverageminimumofthecost? 68PROBLEM2.12MOTIVATIONANDHISTORYOFTHEPROBLEMModernstochasticoptimalcontrolandfilteringtheoriesusewhitenoisedrivensystems.ResultssuchastheseparationprincipleandtheKalman-Bucyfilteringarebasedonthewhitenoisemodel.Infact,whitenoise,beingamathematicalidealization,givesonlyanapproximatedescriptionofrealnoise.Insomefieldstheparametersofrealnoiseareneartotheparametersofwhitenoiseand,so,themathematicalmethodsofcontrolandfilteringforwhitenoisedrivensystemscanbesatisfactorilyappliedtothem.Butinmanyfieldswhitenoiseisacrudeapproximationtorealnoise.Consequently,thetheoreticaloptimalcontrolsandthetheoreticaloptimalfiltersforwhitenoisedrivensystemsbecomenotoptimaland,indeed,mightbequitefarfrombeingoptimal.Itbecomesimportanttodevelopthecontrolandes-timationtheoriesforthesystemsdrivenbynoisemodelsthatdescriberealnoisemoreadequately.Suchanoisemodelisthewidebandnoisemodel.TheimportanceofwidebandnoiseprocesseswasmentionedbyFlemingandRishel[1].AnapproachtowidebandnoisebasedonapproximationsbywhitenoisewasusedinKushner[2].Anotherapproachtowidebandnoisebasedonrepresentationinacertainintegralformwassuggestedin[3]anditsapplicationstospaceengineeringandgravimetrywasdiscussedin[4,5].Filtering,smoothing,andpredictionresultsforwidebandnoisedrivenlinearsystemsareobtainedin[3,6].Theproofsin[3,6]aregiventhroughthedualityprincipleand,technically,theyareroutine,makingfur-therdevelopmentsinthetheorydifficult.Amorehandletechniquebasedonthereductionofawidebandnoisedrivensystemtoawhitenoisedrivensystemwasdevelopedin[7,8,9].Thistechniqueallowstofindtheexplicitformulaefortheoptimalfilterandfortheoptimalcontrol,aswellasfortheerrorofestimationandfortheminimumofthecostinthefilteringandoptimalcontrolproblemsforawidebandnoisedrivenlinearsystem.Inparticulartheopenproblemdescribedherewasoriginallyformulatedin[9].Acompletediscussionofthesubjectcanbefoundintherecentbook[10].3AVAILABLERESULTSANDDISCUSSIONTherandomprocessϕwiththepropertycov(ϕ(t+s),ϕ(t))=λ(t,s)if0≤s<εandcov(ϕ(t+s),ϕ(t))=0ifs≥ε,whereε>0isasmallvalueandλisanonzerofunction,iscalledawidebandnoiseprocessanditissaidtobestationary(inwidesense)ifthefunctionλ(calledtheautocovatiancefunctionofϕ)dependsonlyons(seeFlemingandRishel[8]).Startingfromtheautocovariancefunctionλ,onecanconstructtherespectivewidebandnoiseprocessϕintheintegralformZ0ϕ(t)=φ(θ)w(t+θ)dθ,t≥0,(1)−min(t,ε) WIDEBANDNOISECONTROLANDFILTERING69wherewisawhitenoiseprocesswithcov(w(t),w(s))=δ(t−s),δistheDirac’sdelta-function,ε>0andφisasolutionoftheequationZ−sφ(θ)φ(θ+s)dθ=λ(s),0≤s≤ε.(2)−εThesolutionϕof(2)iscalledarelaxingfunction.Sincein(2)φhasonlyonevariabletheprocessϕfrom(1)isstationaryinwidesense(exceptsmalltimeinterval[0,ε]).Thefollowingtheoremfrom[8,9]iscrucialfortheproposedproblem.Theorem:Letε>0andletλbeacontinuousreal-valuedfunctionon[0,ε].Definethefunctionλ0astheevenextensionofλtothereallinevanishingoutsideof[−ε,ε]andassumethatλ0isapositivedefinitefunctionwithF(λ)1/2∈L(−∞,∞)whereF(λ)istheFouriertransformationofλ.0200Thenthereexistsaninfinitenumberofsolutionsoftheequation(2)inthespaceL2(−ε,0)ifλisanonzerofunctiona.e.on[−ε,0].Thenonuniquenessofthesolutionofequation(2)demonstratesthatthecovariancefunctionλdoesnotprovidecompleteinformationaboutthere-spectivewidebandnoiseprocessϕ.Therefore,forgivenλ,asamplesolutionφof(2)generatestherandomprocessϕintheform(1)thatcouldbecon-sideredasalessormoreadequatemodelofrealnoise.Inordertomakeareasonabledecisionabouttherelaxingfunction,oneofthewaysisstudyingthedistributionsoftheerrorofestimationandtheminimumofthecostinfilteringandcontrolproblems,findingtheaverageerrorandtheaverageminimumandidentifyingtherelaxingfunctionφ¯producingtheseaveragevalues.Forthis,theexplicitformulaefrom[7,8,9](theyarenotgivenherebecauseofthelength)canbeusedtoinvestigatetheproblemanalyticallyornumerically.Also,theproofofthetheoremfrom[8,9]canbeusefulforconstructiondifferentsolutionsofequation(2).Finally,notethatinapartiallyobservablesystemboththestate(signal)andtheobservationsmaybedisturbedbywidebandnoiseprocesses.Hence,thesuggestedproblemconcernsboththesecasesandtheircombinationaswell.BIBLIOGRAPHY[1]W.H.FlemingandR.W.Rishel,DeterministicandStochasticOptimalControl,NewYork,SpringerVerlag,1975,p.126.[2]H.J.Kushner,WeakConvergenceMethodsandSingularlyPerturbedStochasticControlandFilteringProblems,Boston,Birkh¨auser,1990.[3]A.E.Bashirov,“Onlinearfilteringunderdependentwidebandnoises”,Stochastics,23,pp.413-437,1988. 70PROBLEM2.1[4]A.E.Bashirov,L.V.EppelbaumandL.R.Mishne,“ImprovingE¨otv¨oscorrectionsbywidebandnoiseKalmanfiltering,”Geophys.J.Int.,108,pp.193-127,1992.[5]A.E.Bashirov,“Controlandfilteringforwidebandnoisedrivenlinearsystems”,JournalonGuidance,ControlandDynamics,16,pp.983-985,1993.[6]A.E.Bashirov,H.EtikanandN.S¸emi,“Filtering,smoothingandpre-dictionofwidebandnoisedrivensystems”,J.FranklinInst.,Eng.Appl.Math.,334B,pp.667-683,1997.[7]A.E.Bashirov,“Onlinearsystemsdisturbedbywidebandnoise”,Pro-ceedingsofthe14thInternationalConferenceonMathematicalTheoryofNetworksandSystems,Perpignan,France,June19-23,7p.,2000.[8]A.E.Bashirov,“Controlandfilteringoflinearsystemsdrivenbywidebandnoise,”1stIFACSymposiumonSystemsStructureandControl,Prague,CzechRepublic,August29-31,6p.,2001.[9]A.E.BashirovandS.U˘gural,“Analyzingwidebandnoisewithappli-cationtocontrolandfiltering”,IEEETrans.AutomaticControl,47,pp.323-327,2002.[10]A.E.Bashirov,PartiallyObservableLinearSystemsUnderDependentNoises,Basel,Birkh¨auser,2003. Problem2.2OnthestabilityofrandommatricesGiuseppeCalafiore∗,FabrizioDabbene∗∗∗Dip.diAutomaticaeInformatica∗∗IEIIT-CNRPolitecnicodiTorinoC.soDucadegliAbruzzi,24Torino,Italy{giuseppe.calafiore,fabrizio.dabbene}@polito.it1INTRODUCTIONANDMOTIVATIONInthetheoryoflinearsystems,theproblemofassessingwhethertheomoge-neoussystem˙x=Ax,A∈Rn,nisasymptoticallystableisawellunderstood(andfundamental)one.Ofcourse,thesystem(andweshallsayalsothema-trixA)isstableifandonlyifReλi<0,i=1,...,n,beingλitheeigenvaluesofA.Evolvingfromthisbasicnotion,muchresearchefforthasbeendevotedinrecentyearstothestudyofrobuststabilityofasystem.Withoutenteringinthedetailsofmorethanthirtyyearsoffruitfulresearch,wecouldcondensetheessenceoftherobuststabilityproblemasfollows:givenaboundedset∆andastablematrixA∈Rn,n,statewhetherA=A+∆isstablefor∆all∆∈∆.Sincetheabovedeterministicproblemmaybecomputationallyhardinsomecases,arecentlineofstudyproposestointroduceaprobabilitydistributionover∆,andthentoassesstheprobabilityofstabilityoftherandommatrixA+∆.Actually,intheprobabilisticapproachtorobuststability,thisprobabilityisnotanalyticallycomputedbutonlyestimatedbymeansofrandomizedalgorithms,whichmakestheproblemfeasiblefromacomputationalpointofview,see,forinstance,[3]andthereferencestherein.Leavingaparttherandomizedapproach,whichcircumventstheproblemofanalyticalcomputations,thereisacleardisparitybetweentheabundanceofresultsavailableforthedeterministicproblem(bothpositiveandnegativeresults,intheformofcomputational“hardness,”[2])andtheirdeficiencyintheprobabilisticone.Inthislattercase,almostnoanalyticalresultisknownamongcontrolresearchers. 72PROBLEM2.2Theobjectiveofthisnoteistoencourageresearchonrandommatricesinthecontrolcommunity.Theonewhoadventuresinthisfieldwillencounterunexpectedandexcitingconnectionsamongdifferentfieldsofscienceandbeautifulbranchesofmathematics.Inthenextsection,weresumesomeoftheknownresultsonrandomma-trices,andstateasimplenew(tothebestofourknowledge)closedformresultontheprobabilityofstabilityofacertainclassofrandommatrices.Then,insection3weproposethreeopenproblemsrelatedtotheanalyticalassessmentoftheprobabilityofstabilityofrandommatrices.Theproblemsarepresentedinwhatwebelieveistheirorderofdifficulty.2AVAILABLERESULTSNotation:ArealrandommatrixXisamatrixwhoseelementsarerealrandomvariables.Theprobabilitydensity(pdf)ofX,fX(X)isdefinedasthejointpdfofitselements.ThenotationX∼YmeansthatX,Yarerandomquantitieswiththesamepdf.TheGaussiandensitywithmeanµandvarianceσ2isdenotedasN(µ,σ2).ForamatrixX,ρ(X)denotesthespectralradius,andkXktheFrobeniusnorm.ThemultivariateGammafunctionisdefinedasΓ(x)=πn(n−1)/4QnΓ(x−(i−1)/2),whereΓ(·)isni=1thestandardGammafunction.Inthisnote,weconsidertheclassofrandommatrices(aclassofrandommatricesisoftencalledan“ensemble”inthephysicsliterature)whosedensityisinvariantunderorthogonalsimilarity.ForarandommatrixXinthisclass,wehavethatX∼UXUT,foranyfixedorthogonalmatrixU.Forsymmetricorthogonalinvariantrandommatrices,itcanbeprovedthatthepdfofXis.afunctionofonlyitseigenvaluesΛ=diag(λ1,...,λn),i.e.,fX(X)=gX(Λ).(1)Orthogonalinvariantrandommatricesmayseemspecialized,butweprovidebelowsomenotableexamples:1.Gn:Gaussianmatrices.Itistheclassofn×nrealrandommatri-ceswithindependentidenticallydistributed(iid)elementsdrawnfromN(0,1).2.Wn:Whishartmatrices.Symmetricn×nrandommatricesoftheformXXT,whereXisG.n3.GOEn:GaussianOrthogonalEnsemble.Symmetricn×nrandommatricesoftheform(X+XT)/2,whereXisG.n4.Sn:Symmetricorthogonalinvariantensemble.Genericsymmetricn×nrandommatriceswhosedensitysatisfies(1).WnandGOEnarespecialcasesofthese. ONTHESTABILITYOFRANDOMMATRICES73ρ5.USn:Symmetricn×nrandommatricesfromSn,whichareuniformovertheset{X∈Rn,n:ρ(X)≤1}.F6.USn:Symmetricn×nrandommatricesfromSn,whichareuniformovertheset{X∈Rn,n:kXk≤1}.Whishartmatriceshavealonghistory,andarewellstudiedinthestatisticsliterature,see[1]foranearlyreference.TheGaussianOrthogonalEnsembleisafundamentalmodelusedtostudythetheoryofenergylevelsinnuclearphysics,andithasbeenoriginallyintroducedbyWigner[9,8].Athoroughaccountofitsstatisticalpropertiesispresentedin[7].AfundamentalresultfortheSnensembleisthatthejointpdfoftheeigenval-uesofrandommatricesbelongingtoSnisknownanalytically.Inparticular,ifλ1≥λ2≥...≥λnaretheeigenvaluesofarandommatrixXbelongingtoSn,thentheirpdffΛ(Λ)is2πn/2YfΛ(Λ)=gX(Λ)(λi−λj).(2)Γn(n/2)1≤i0,betweenanytwosystemsinS?Incasetheanswerisaffirmative,anaturalfollow-upquestionfromthedif-ferentialgeometricpointofviewwouldbewhetheritispossibletoconstructafiniteatlasofchartsforthemanifoldS,suchthatthechartsassubsetsofEuclideanspacearebounded(i.e.,containedinanopenballinEuclideanspace),whilethedistortionofeachchartremainsfinite.2MOTIVATIONANDBACKGROUNDOFTHEPROBLEMAnimportantandwell-studiedprobleminlinearsystemsidentificationistheconstructionofparametrizationsforvariousclassesoflinearsystems.Intheliteratureagreatnumberofparametrizationsforlinearsystemshavebeenproposedandused.Fromthegeometricpointofviewthequestionariseswhetheronecanqualifyvariousparametrizationsasgoodorbad.Aparametrizationisawayto(locally)describeageometricobject.Intuitively,aparametrizationisbetterthemoreitreflectsthe(local)structureofthegeometricobject.Animportantconsiderationinthisrespectisthescaleoftheparametrization,orratherthespectrumofscales,see[4].Toexplainthis,considerthetangentspaceofadifferentialmanifoldofsystems,suchasS.ThedifferentiablemanifoldcanbesuppliedwithaRiemanniangeometry,forexample,bysmoothlyembeddingthedifferentiablemanifoldinanap-propriateHilbertspace:thenthetangentspacestothemanifoldarelinearsubspacesoftheHilbertspace,whichinducesaninnerproductoneachofthetangentspacesandaRiemannianmetricstructureonthemanifold.IfsuchaRiemannianmetricisdefined,thenanysufficientlysmoothparametrizationwillhaveanassociatedRiemannianmetrictensor.Inlocalcoordinates(i.e.,intermsoftheparametersused)itisrepresentedbyasymmetric,positivedefinitematrixateachpoint.Theeigenvaluesofthismatrixreflectthelo-calscalesoftheparametrization:thescaleofanyinfinitesimalmovementstartingfromagivenpoint,willvarybetweenthelargestandthesmallesteigenvalueoftheRiemannianmetrictensoratthepointinvolved.OverasetofpointsthescalewillclearlyvarybetweenthelargesteigenvaluetobefoundinthespectraofthecorrespondingsetofRiemannianmetricmatricesandthesmallesteigenvaluetobefoundinthatsamesetofspectra.Follow-ingMilnor(see[12]),whoconsideredthequestionoffindinggoodchartsfortheearth,wedefinethedistortionofaparametrization,whichwewillcalltheMilnordistortion,asthequotientofthelargestscaleandthesmallestscaleoftheparametrization.NotethatthisconceptofMilnordistortionisageneralizationoftheconceptoftheconditionnumberofamatrix.Howeveritis(ingeneral)notthemaximumoftheconditionnumbersofthesetofRiemannianmetricmatrices. 78PROBLEM2.3Indeed,thelargesteigenvalueandthesmallesteigenvaluethatenterintothedefinitionoftheMilnordistortiondonothavetocorrespondtotheRiemannianmetrictensoratthesamepoint.Ifonehasanatlasofoverlappingcharts,onecancalculatetheMilnordis-tortionineachofthechartsandconsiderthelargestdistortioninanyofthechartsoftheatlas.Onecouldnowbetemptedtodefinethisnumberasthedistortionoftheatlasandlookforatlaseswithrelativelysmalldistortion.However,inthiscase,theproblemshowsupthatitisalwayspossibletotakealargenumberofsmallcharts,eachonedisplayingverylittledistortion(i.e.,distortionclosetoone),whilesuchanatlasmaystillnotbedesirableasitmayrequireahugenumberofcharts.ThedifficultyhereistotradeoffthenumberofchartsinanatlasagainsttheMilnordistortionineachofthosecharts.Atthispoint,wehavenoclearnaturalcandidateforthistrade-off.ButatleastforatlaseswithanequalfinitenumberofchartstheconceptofmaximalMilnordistortioncouldbeusedtocomparetheatlases.3AVAILABLERESULTSIntryingtoapplytheseideastothequestionofparametrizationoflin-earsystems,theproblemarisesthatmanyparametrizationsturnouttohaveinfactaninfiniteMilnordistortion.ConsiderforexamplethecaseofrealSISOdiscrete-timestrictlyproperstablesystemsoforderone.(Seealso[9]and[13,section4.7].)Thissetcanbedescribedbytworealpa-rameters,e.g.,bywritingtheassociatedtransferfunctionintotheformh(z)=b/(z−a).Here,theparameteradenotesthepoleofthesystemandtheparameterbisassociatedwiththegain.TheRiemannianmetrictensorinducedbytheH2normofthisparametrizationcanbecomputedb2(1+a2)/(1−a2)3ab/(1−a2)2as222,see[9].Thereforeittendstoab/(1−a)1/(1−a)infinitywhenaapproachesthestabilityboundary|a|=1,whencetheMil-nordistortionofthisparametrizationbecomesinfinity.Inthisexamplethegeometryisthatofaflatdoubleinfinite-sheetedRiemannsurface.Locally,itisisometricwithEuclideanspaceandthereforeonecanconstructchartsthathavetheidentitymatrixastheirRiemannianmetrictensor(see[13]).However,inthiscase,thismeansthatclosetothestabilityboundarythedistancesbetweenpointsbecomearbitrarilylarge.Therefore,althoughitispossibletoconstructchartswithoptimalMilnordistortion,thiscanonlybedoneatthepriceofhavingtoworkwithinfinitelylarge(i.e.,unbounded)charts.Ifonewantstoworkwithchartsinwhichthedistancesremainboundedthenonewillneedinfinitelymanyofthemonsuchoccasions.InthecaseofstochasticGaussiantime-invariantlineardynamicalsystemswithoutobservedinputs,theclassofstableminimum-phasesystemsplaysanimportantrole.Forsuchstochasticsystemsthe(asymptotic)Fisherinforma-tionmatrixiswell-defined.Thismatrixisdependentontheparametrization FISHERGEOMETRYFORLINEARSYSTEMS79usedandadmitstheinterpretationofaRiemannianmetrictensor(see[15]).ThereisanextensiveliteratureonthecomputationofFisherinformation,especiallyforARandARMAsystems.See,e.g.,[6,7,11].Muchofthisinterestderivesfromthemanyapplicationsinpracticalsettings:itcanbeusedtoestablishlocalparameteridentifiability,itisusedforparameteres-timationinthemethodofscoring,anditisalsoknowntodeterminethelocalconvergencepropertiesofthepopularGauss-Newtonmethodforleast-squaresidentificationoflinearsystemsbasedonthemaximumlikelihoodprinciple(see[10]).InthecaseofstableARsystems,theFishermetrictensorcan,forinstance,becalculatedusingtheparametrizationwithSchurparameters.Fromtheformulasin[14]itfollowsthattheFisherinformationforscalarARsystemsoforderonedrivenbyzeromeanGaussianwhitenoiseofunitvarianceisequalto1/(1−γ2).Hereγisrequiredtorangebetween−1and1(to11imposestability)andtobenonzero(toimposeminimality).AlthoughthisagainimpliesaninfiniteMilnordistortion,thesituationhereisstructurallydifferentfromthesituationinthepreviouscase:thelengthofthecurveofsystemsobtainedbylettingγ1rangefrom0to1isfinite!Indeed,theR1(Fisher)lengthofthiscurveiscomputedas√1dγ=π/2.01−γ211LettheinnergeometryofaconnectedRiemannianmanifoldofsystemsbedefinedbytheshortestpathdistance:d(Σ1,Σ2)istheRiemannianlengthoftheshortestcurveconnectingthetwosystemsΣ1andΣ2.Then,inthissimplecase,theFishergeometryhasthepropertythatthecorrespondinginnergeometryhasauniformupperbound.Therefore,thisexampleprovidesaninstanceofasubsetofthemanifoldSforwhichtheanswertothequestionraisedisaffirmative.Asamatteroffact,ifonenowreparametrizesthesetofsystemsasin[17]byθdefinedthroughγ1=sin(θ),thentheresultingFisherinformationquantitybecomesequalto1everywhere.Thus,itisboundedandtheMilnordistor-tionofthisreparametrizationisfinite.Butatthesametimetheparameterchartitselfremainsbounded!Hence,alsothe“follow-upquestion”oftheprevioussectionisansweredaffirmativehere.IfoneconsidersSISOstableminimum-phasesystemsoforder1,itcanbeshownlikewisethatalsoheretheFisherdistancebetweentwosystemsisuniformlyboundedandthatafiniteatlaswithboundedchartsandfiniteMilnordistortioncanbedesigned.Whetherthisalsooccursforlargerstate-spacedimensionsisstillunknown(tothebestoftheauthors’knowledge)andthisispreciselytheopenproblempresentedabove.Toconclude,wenotethattheroleplayedbythecovariancematrixΩofthedrivingwhitenoiseisratherlimited.Itiswellknownthatifthesystemequationsandthecovariancematrixareparametrizedindependentlyofeachother,thentheFisherinformationmatrixattainsablock-diagonalstructure(see,e.g.,[18,Ch.7].ThecovariancematrixΩthenappearsasaweightingmatrixfortheblockoftheFisherinformationmatrixassociatedwiththe 80PROBLEM2.3parametersinvolvedinthesystemequations.Therefore,ifΩisknown,orratherifanupperboundonΩisknown(whichislikelytobethecaseinanypracticalsituation!),itsrolewithrespecttothequestionsraisedcanbelargelydisregarded.ThisallowstorestrictattentiontothesituationwhereΩisfixedtotheidentitymatrixIm.BIBLIOGRAPHY[1]S.-I.Amari,Differential-GeometricalMethodsinStatistics,LectureNotesinStatistics28,SpringerVerlag,Berlin,1985.[2]S.-I.Amari,“DifferentialgeometryofaparametricfamilyofinvertiblelinearsystemsRiemannianmetric,dualaffineconnections,anddiver-gence,”MathematicalSystemsTheory,20,53–82,1987[3]C.AtkinsonandA.F.S.Mitchell,“Rao’sdistancemeasure.Sankhya¯:TheIndianJournalofStatistics,SeriesA,43(3),345–365,1981.[4]R.W.BrockettandP.S.Krishnaprasad“Ascalingtheoryforlinearsystems,”IEEETrans.Aut.Contr.,AC-25,197–206,1980.[5]J.M.C.Clark,“Theconsistentselectionofparametrizationsinsys-temidentification,”Proc.JointAutomaticControlConference,576–580.PurdueUniversity,Lafayette,Indiana,1976.[6]B.Friedlander,“OntheComputationoftheCramer-RaoBoundforARMAParameterEstimation,”IEEETransactionsonAcoustics,SpeechandSignalProcessing,ASSP-32(4),721–727.[7]E.J.GodolphinandJ.M.Unwin,“EvaluationofthecovariancematrixforthemaximumlikelihoodestimatorofaGaussianautoregressive-movingaverageprocess,”Biometrika,70(1),279–284,1983.[8]E.J.HannanandM.Deistler,TheStatisticalTheoryofLinearSystems.JohnWiley&Sons,NewYork,1988.[9]B.Hanzon,Identifiability,RecursiveIdentificationandSpacesofLinearDynamicalSystems,CWITracts63and64,CentrumvoorWiskundeenInformatica(CWI),Amsterdam,1989.[10]B.HanzonandR.L.M.Peeters,“OntheRiemannianInterpreta-tionoftheGauss-NewtonAlgorithm,”In:M.K´arn´yandK.Warwick(eds.),MutualImpactofComputingPowerandControlTheory,111–121.PlenumPress,NewYork,1993.[11]A.KleinandG.M´elard,“OnAlgorithmsforComputingtheCovari-anceMatrixofEstimatesinAutoregressiveMovingAverageProcesses,”ComputationalStatisticsQuarterly,1,1–9,1989. FISHERGEOMETRYFORLINEARSYSTEMS81[12]J.Milnor,“Aproblemincartography,”AmericanMath.Monthly,76,1101–1112,1969.[13]R.L.M.Peeters,SystemIdentificationBasedonRiemannianGeome-try:TheoryandAlgorithms.TinbergenInstituteResearchSeries,vol.64,ThesisPublishers,Amsterdam,1994.[14]R.L.M.PeetersandB.Hanzon,“SymboliccomputationofFisherin-formationmatricesforparametrizedstate-spacesystems,”Automatica,35,1059–1071,1999.[15]C.R.Rao,“Informationandaccuracyattainableintheestimationofstatisticalparameters,”Bull.CalcuttaMath.Soc.,37,81–91,1945.[16]N.Ravishanker,E.L.MelnickandC.-L.Tsai,“DifferentialgeometryofARMAmodels,”JournalofTimeSeriesAnalysis,11,259–274,.[17]A.L.Rijkeboer,“FisheroptimalapproximationofanAR(n)-processbyanAR(n-1)-process,”In:J.W.Nieuwenhuis,C.PraagmanandH.L.Trentelmaneds.,Proceedingsofthe2ndEuropeanControlConferenceECC’93,1225–1229,Groningen,1993.[18]T.S¨oderstr¨omandP.Stoica,SystemIdentification,Prentice-Hall,NewYork,1989. Problem2.4OntheconvergenceofnormalformsforanalyticcontrolsystemsWeiKangDepartmentofMathematics,NavalPostgraduateSchoolMonterey,CA93943USAwkang@nps.navy.milArthurJ.KrenerDepartmentofMathematics,UniversityofCaliforniaDavis,CA95616USAajkrener@ucdavis.edu1BACKGROUNDAfruitfultechniqueforthelocalanalysisofadynamicalsystemconsistsofusingalocalchangeofcoordinatestotransformthesystemtoasimplerform,whichiscalledanormalform.Thequalitativebehavioroftheoriginalsystemisequivalenttothatofitsnormalformwhichmaybeeasiertoanalyze.Abifurcationofaparameterizeddynamicsoccurswhenachangeintheparameterleadstoachangeinitsqualitativeproperties.Therefore,normalformsareusefulinanalyzingwhenandhowabifurcationwilloccur.Inhisdissertation,Poincar´estudiedtheproblemoflinearizingadynamicalsystemaroundanequilibriumpoint,lineardynamicsbeingthesimplestnormalform.Poincar´e’sideaistosimplifythelinearpartofasystemfirst,usingalinearchangeofcoordinates.Thenthequadratictermsinthesystemaresimplified,usingaquadraticchangeofcoordinates,thenthecubicterms,andsoon.Forsystemsthatarenotlinearizable,thePoincar´e-Dulactheoremprovidesthenormalform.GivenaC∞dynamicalsysteminitsTaylorexpansionaroundx=0,x˙=f(x)=Fx+f[2](x)+f[3](x)+···(1) CONVERGENCEOFNORMALFORMS83wherex∈0,ν>0ifC|m·λ−λk|≥|m|νForeigenvaluesinthePoincar´edomain,thereareatmostafinitenumberofresonances.Foreigenvaluesoftype(C,ν),therearenoresonancesandas|m|→∞therateatwhichresonancesareapproachediscontrolled.Aformalchangeofcoordinatesisaformalpowerseriesz=Tx+θ[2](x)+θ[3](x)+···(2)whereTisinvertible.IfT=I,thenitiscalledanearidentitychangeofcoordinates.Ifthepowerseriesconvergeslocally,thenitdefinesarealanalyticchangeofcoordinates.Theorem1:(Poincar´e-Dulac)Ifthesystem(1)isC∞thenthereexistsaformalchangeofcoordinates(2)transformingittoz˙=Az+w(z)whereAisinJordanformandw(z)consistssolelyofresonantterms.(IfsomeoftheeigenvaluesofFarecomplexthenthechangeofcoordinateswillalsobecomplex).Inthisnormalform,w(z)neednotbeunique.Ifthesystem(1)isrealanalyticanditseigenvalueslieinthePoincar´edo-main(2),thenw(z)isapolynomialvectorfieldandthechangeofcoordinates(2)isrealanalytic.Theorem2:(Siegel)Ifthesystem(1)isrealanalyticanditseigenvaluesareoftype(C,ν)forsomeC>0,ν>0,thenw(z)=0andthechangeofcoordinates(2)isrealanalytic.Asispointedoutin[1],evenincaseswheretheformalseriesaredivergent,themethodofnormalformsturnsouttobeapowerfuldeviceinthestudyofnonlineardynamicalsystems.Afewlowdegreetermsinthenormalformoftengivesignificantinformationonthelocalbehaviorofthedynamics. 84PROBLEM2.42THEOPENPROBLEMIn[3],[4],[5],[10],and[8],Poincar´e’sideaisappliedtononlinearcontrolsys-tems.Anormalformisderivedfornonlinearcontrolsystemsunderchangeofstatecoordinatesandinvertiblestatefeedback.ConsideraC∞controlsystemx˙=f(x,u)=Fx+Gu+f[2](x,u)+f[3](x,u)+···(3)wherex∈2.Also,g00(y)=m−j1g0(y)=g(y)andg00(y)isexpressedintermsofg0(y),i=0,1,2,···,(m−11mm−i1)andhm−j(y),j=0,1,2,···,(m−2),m≥2.Assumingthathm−j(y),αm−j(y),βm−j−1(y),j=1,2,···,(m−2),m>2,areknown,f00(y)andg00(y)canbecomputed.Withoutlossofgenerality,mm−1onecanassumematrixAtobediagonalandG=[1,1,···,1,1]tbyapplyingachangeofcoordinateto(5)involvingVandermondematrix[10].Onecanthenproceedtosolve(15)forhm(y)intermsofαm(y)andsubstitutethesameinto(16)tosetupalinearsystemofequationsintheunknowncoefficientsofpolynomialsαm(y)andβm−1(y).Form=2,ithasbeenshownin[9]thatthecorrespondingsystemoflinearn(n−1)equationscanbereducedtoasystemof()equationsinnvariables2whoserankis(n−1).Itisconjecturedthatasimilarreductionofthelinearsystemofequations,inthearbitraryordercase,shouldalsobepossible.Formulationofthepropertiesandsolution,ifitexists,ofthelinearsystemofequationsinvolvingthecoefficientsofthepolynomialsαm(.),βm−1(.)andhm(.),m>2willconstitutethesolutiontotheopenproblem. 96PROBLEM3.2BIBLIOGRAPHY[1]A.J.KrenerandW.Kang,“Extendednormalformsofquadraticsys-tems,”Proc.29thConf.DecisionandControl,pp.2091-2095,1990.[2]P.Brunovsky,“AClassificationoflinearcontrollablesystems,”Kyber-neticacislo,vol.3,pp.173-188,1970.[3]R.Devanathan,“Linearizationconditionthroughstatefeedback,”IEEETransactionsonAutomaticControl,vol.46,no.8,pp.1257-1260,2001.[4]V.I.Arnold,GeometricalMethodsintheTheoryofOrdinaryDifferentialEquations,Springer-Verlag,NewYork,pp.177-188,1983.[5]J.GuckenheimerandP.Holmes,NonlinearOscillations,DynamicalSys-temsandBifurcationofVectorFields,Springer-Verlag,NewYork,1983.[6]A.D.Bruno,LocalMethodsinNonlinearDifferentialEquations,Springer-Verlag,Berlin,1989.[7]GiampaoloCicogna,“Ontheconvergenceofnormalizingtransformationsinthepresenceofsymmetries,”DepartmentodiFisica,UniversitadiPisa,P.zaTorricelli2,I-56126,Pisa,Italy.[8]A.J.Krener,S.Karahan,andM.Hubbard,“Approximatenormalformsofnonlinearsystems,”Proc.27thConf.DecisionandControl,pp.1223-1229,1988.[9]R.Devanathan,“Solutionofsecondorderlinearization,”FifteenthInter-nationalSymposiumonMathematicalTheoryofNetworksandSystems,UniversityofNotreDame,NotreDame,IN,USA(2002).[10]B.C.Kuo,AutomaticControlSystems,FourthEd.,Prentice-Hall,pp.227-240,1982. Problem3.3BasesforLiealgebrasandacontinuousCBHformulaMatthiasKawski1DepartmentofMathematicsandStatisticsArizonaStateUniversityTempe,AZ85287-1804USAkawski@asu.edu1DESCRIPTIONOFTHEPROBLEMManytime-varyinglinearsystems˙x=F(t,x)naturallysplitintotime-invariantgeometriccomponentsandtime-dependentparameters.Aspecialcasearenonlinearcontrolsystemsthatareaffineinthecontrolu,andspec-ifiedbyanalyticvectorfieldsonamanifoldMnXmx˙=f0(x)+ukfk(x).(1)k=1Itisnaturaltosearchforsolutionformulasforx(t)=x(t,u)thatseparatethetime-dependentcontributionsofthecontrolsufromtheinvariant,ge-ometricroleofthevectorfieldsfk.Ideally,onemaybeabletoaprioriobtainclosed-formexpressionsfortheflowsofcertainvectorfields.Thequadraturesofthecontrolmightbedoneinreal-time,ortheintegralsofthecontrolsmaybeconsiderednewvariablesfortheoreticalpurposessuchaspath-planningortracking.Tomakethisschemework,oneneedssimpleformulasforassemblingthesepiecestoobtainthesolutioncurvex(t,u).Suchformulasarenontrivialsinceingeneralthevectorfieldsfkdonotcommute:alreadyinthecaseoflinearsystems,exp(sA)·exp(tB)6=exp(sA+tB)(formatricesAandB).Thusthedesiredformulasnotonlyinvolvetheflowsofthesystemvectorfieldsfibutalsotheflowsoftheiriteratedcommutators[fi,fj],[[fi,fj],fk],andsoon.UsingHall-ViennotbasesHforthefreeLiealgebrageneratedbyminde-terminatesX1,...Xm,Sussmann[22]gaveageneralformulaasadirected1SupportedinpartbyNSF-grantDMS00-72369 98PROBLEM3.3infiniteproductofexponentialsY→x(T,u)=exp(ξH(T,u)·fH).(2)H∈HHerethevectorfieldfHistheimageoftheformalbracketHunderthecanonicalLiealgebrahomomorphismthatmapsXitofi.Usingthechrono-RTlogicalproduct(U∗V)(t)=U(s)V0(s)ds,theiteratedintegralsξare0RHTdefinedrecursivelybyξXk(T,u)=0uk(t)dtandξHK=ξH∗ξK(3)ifH,K,HKareHallwordsandtheleftfactorofKisnotequaltoH[9,22].(Inthecaseofrepeatedleftfactors,theformulacontainsanadditionalfactorial.)Analternativetosuchinfiniteexponentialproduct(inLiegrouplanguage,“coordinatesofthe2ndkind”)isasingleexponentialofaninfiniteLieseries(“coordinatesofthe1stkind”).Xx(T,u)=exp(ζB(T,u)·fB)(4)B∈BItisstraightforwardtoobtainexplicitformulasforζBforsomespanningsetsBofthefreeLiealgebra[22],butmuchpreferableareseriesthatusebasesB,andwhich,inaddition,yieldassimpleformulasforζBas(3)doesforξH.Problem1:ConstructbasesB={Bk:k≥0}forthefreeLiealgebraonafinitenumberofgeneratorsX1,...XmsuchthatthecorrespondingiteratedintegralfunctionalsζBdefinedby(4)havesimpleformulas(similarto(3)),suitableforcontrolapplications(bothanalysisanddesign).Theformulae(4)and(2)arisefromthe“freecontrolsystem”onthefreeassociativealgebraonmgenerators.Itsuniversalitymeansthatitssolutionsmaptosolutionsofspecificsystems(1)onMnviatheevaluationhomomor-phismXi7→fi.However,theresultingformulascontainmanyredundanttermssincethevectorfieldsfBarenotlinearlyindependent.Problem2:ProvideanalgorithmthatgeneratesforanyfinitecollectionofanalyticvectorfieldsF={f,...,f}onMnabasisforL(f,...,f)1m1mtogetherwitheffectiveformulasforassociatediteratedintegralfunctionals.Withoutlossofgenerality,onemayassumethatthecollectionFsatisfiestheLiealgebrarankcondition,i.e.,L(f1,...,fm)(p)=TpMataspecifiedinitialpointp.ItmakessensetofirstconsiderspecialclassesofsystemsF,e.g.,whicharesuchthatL(f1,...,fm)isfinite,nilpotent,solvable,etc.Thewordssimpleandeffectivearenotusedinatechnicalsenseinproblems1and2(asinformalstudiesofcomputationalcomplexity)butinsteadrefertocomparisonwiththeelegantformula(3),whichhasprovenconvenientfortheoreticalstudies,numericalcomputation,andpracticalimplementations. BASESFORLIEALGEBRAS992MOTIVATIONANDHISTORYOFTHEPROBLEMSeriesexpansionsofsolutiontodifferentialequationshavealonghistory.PmElementaryPicarditerationoftheuniversalcontrolsystemS˙=i=1Xiuionthefreeassociativealgebraover(X1,...,Xm)yieldstheChenFliessseries[5,11,21].OthermajortoolsareVolterraseries,andtheMagnusexpansion[14],whichgroupsthetermsinadifferentwaythantheFliessseries.ThemaindrawbackoftheFliessseriesisthat(unlikeitsexponentialproductexpansion(2))nofinitetruncationistheexactsolutionofanyap-proximatingsystem.Akeyinnovationisthechronologicalcalculusof1970sAgrachevandGamkrelidze[1].However,itisgenerallynotformulatedusingexplicitbases.Theseriesandproductexpansionshavemanifoldusesincontrolbeyondsim-plecomputationofintegralcurvesandanalysisofreachablesets(whichin-cludescontrollabilityandoptimality).Theseincludestate-spacerealizationsofsystemsgivenininput-outputoperatorform[8,20],outputtracking,andpath-planning.Forthelatter,expressthetargetorreferencetrajectoryintermsoftheξorζ,nowconsideredascoordinatesofasuitablyliftedsystem(e.g.,freenilpotent)andinverttherestrictionofthemapu7→{ξB:B∈BN}oru7→{ζB:B∈BN}(forsomefinitesubbasisBN)toafinitelyparame-terizedfamilyofcontrolsu,e.g.,piecewisepolynomial[7]ortrigonometricpolynomial[12,17].TheCampbell-Baker-Hausdorffformula[18]isaclassictooltocombineproductsofexponentialsintoasingleexponentialeaeb=eH(a,b)whereH(a,b)=a+b+1[a,b]+1[a,[a,b]]−1[b,[a,b]+....Ithasbeenexten-21212sivelyusedfordesigningpiecewiseconstantcontrolvariationsthatgeneratehighordertangentvectorstoreachablesets,e.g.,forderivingconditionsforoptimality.However,repeateduseoftheformulaquicklyleadstounwieldlyexpressions.Theexpansion(2)isthenaturalcontinuousanalogueoftheCBHformula,andtheproblemistofindthemostusefulform.Theusesoftheseexpansions(2)and(4)extendfarbeyondcontrol,astheyapplytoanydynamicalsystemsthatsplitintodifferentinteractingcom-ponents.Inparticular,closelyrelatedtechniqueshaverecentlyfoundmuchattentioninnumericalanalysis.ThisstartedwithasearchforRunge-Kutta-likeintegrationschemessuchthattheapproximatesolutionsinherentlysat-isfyalgebraicconstraints(e.g.,conservationlaws)imposedonthedynamicalsystem[3].Muchefforthasbeendevotedtooptimizesuchschemes,inpar-ticularminimizingthenumberofcostlyfunctionevaluations[16].Forarecentsurveysee,[6].Clearly,theform(4)ismostattractiveasitrequirestheevaluationofonlyasingle(computationallycostly)exponential.Thegeneralareaofnoncommutingformalpowerseriesadmitsbothdynam-icalsystems/analyticandpurelyalgebraic/combinatorialapproaches.Alge-braically,underlyingtheexpansions(2)and(4)istheChenseries[2],whichiswell-knowntobeanexponentialLieseries,compare[18],thusguarantee- 100PROBLEM3.3ingtheexistenceofthealternativeexpansionsXXY→!!w⊗w=expζB⊗B=exp(ξB⊗B)(5)w∈Z∗B∈BB∈BThefirstbasesforfreeLiealgebrasbuildonHall’sworkinthe1930soncom-mutatorgroups.Whileseveralotherbases(Lyndon,Sirsov)havebeenpro-posedinthesequel,Viennot[23]showedthattheyareallspecialcasesofgen-eralizedHallbases.Underlyingtheirconstructionisauniquefactorizationprinciple,whichinturniscloselyrelatedtoPoincar-Birckhoff-Wittbases(oftheuniversalenvelopingalgebraofaLiealgebra)andLazardelimination.FormulasforthedualPBWbasesξBhavebeengivenbySch¨utzenberger,Sussmann[22],Grossman,andMelanconandReutenauer[15].Foranintro-ductorysurvey,see[11],while[15]elucidatestheunderlyingHopfalgebrastructure,and[18]istheprincipaltechnicalreferenceforcombinatoricsoffreeLiealgebras.3AVAILABLERELATEDRESULTSThedirectexpansionofthelogarithmintoaformalpowerseriesmaybesimplifiedusingsymmetrization[18,22],butthisstilldoesnotyieldwell-defined“coordinates”withrespecttoabasis.Explicitbutquiteunattractiveformulasforthefirst14coefficientsζHinthecaseofm=2andaHall-basisarecalculatedin[10].Thiscalculationcanbeautomatedinacomputeralgebrasystemfortermsofconsiderablyhigherorder,butnoapparentalgebraicstructureisdiscernible.Theseresultssufficeforsomenumericalpurposes,buttheydonotprovidemuchstructuralinsight.Severalnewalgebraicstructuresintroducedin[19]leadtosystematicfor-mulasforζBusingspanningsetsBthataresmallerthanthosein[22],butarenotbases.TheseformulascanberefinedtoapplytoHall-bases,butatthecostofloosingtheirsimplestructure.Furtherrecentinsightsintotheunderlyingalgebraicstructuresmaybefoundin[4,13].Theintroductorysurvey[11]laysoutinelementarytermsthecloseconnec-tionsbetweenLazardelimination,Hall-sets,chronologicalproducts,andtheparticularlyattractiveformula(3).TheseintimateconnectionssuggestthattoobtainsimilarlyattractiveexpressionsforζBonemayhavetostartfromtheverybeginningbybuildingbasesforfreeLiealgebrasthatdonotrelyonrecursiveuseofLazardelimination.WhileitisdesirablethatanysuchnewbasesstillrestricttobasesofthehomogeneoussubspacesofthefreeLiealgebra,wesuggestconsiderbalancingthesimplicityofthebasisfortheLiealgebraandstructuralsimplicityoftheformulasforthedualobjectsζB.Inparticular,considerbaseswhoseelementsarenotnecessarilyLiemonomialsbutpossiblynontriviallinearcombinationsofiteratedLiebracketsofthegenerators. BASESFORLIEALGEBRAS101BIBLIOGRAPHY[1]A.AgrachevandR.Gamkrelidze,“Chronologicalalgebrasandnonsta-tionaryvectorfields,”JournalSovietMath.,17,pp.1650–1675,1979.[2]K.T.Chen,“Integrationofpaths,geometricinvariantsandageneral-izedBaker-Hausdorffformula,”AnnalsofMathematics,65,pp.163–178,1957.[3]P.CrouchandR.Grossman,“Theexplicitcomputationofintegrational-gorithmsandfirstintegralsforordinarydifferentialequationswithpoly-nomialcoefficientsusingtrees,”Proc.Int.SymposiumonSymbolicandAlgebraicComputation,pp.89-94,ACMPress,1992.[4]A.Dzhumadil’daev,“Non-associativealgebraswithoutunit,”Comm.Al-gebra,2002.[5]M.Fliess,“Fonctionellescausalesnonlin´eairesetindetermin´eesnoncom-mutatives,”Bull.Soc.Math.France,109,pp.3–40,1981.[6]A.Iserles,“Expansionsthatgrowontrees,”NoticesAMS,49,pp.430-440,2002.[7]G.Jacob,“Motionplanningbypiecewiseconstantorpolynomialinputs,”Proc.NOLCOS,Bordeaux,1992.[8]B.Jakubczyk,“Convergenceofpowerseriesalongvectorfieldsandtheircommutators;aCartan-K¨ahlertypetheorem,”Ann.Polon.Math.,74,pp.117-132,2000.[9]M.KawskiandH.J.Sussmann“Noncommutativepowerseriesandfor-malLie-algebraictechniquesinnonlinearcontroltheory,”In:Operators,Systems,andLinearAlgebra,U.Helmke,D.Pr¨atzel-WoltersandE.Zerz,eds.Teubner,pp.111–128,1997.[10]M.Kawski,“CalculatingthelogarithmoftheChenFliessseries,”Proc.MTNS,Perpignan,2000.[11]M.Kawski,“Thecombinatoricsofnonlinearcontrollabilityandnon-commutingflows,”LectureNotesseriesoftheAbdusSalamICTP,2001.[12]G.LafferriereandH.Sussmann,“Motionplanningforcontrollablesys-temswithoutdrift,”Proc.Int.Conf.Robot.Automat.,pp.1148–1153,1991.[13]J.-L.LodayandT.Pirashvili,“UniversalenvelopingalgebrasofLeibnizalgebrasand(co)homology,”Math.Annalen,196,pp.139–158,1993.[14]W.Magnus,“Ontheexponentialsolutionofdifferentialequationsforalinearoperator,”Comm.PureAppl.Math.,VII,pp.649–673,1954. 102PROBLEM3.3[15]G.Melan¸conandC.Reutenauer,“Lyndonwords,freealgebrasandshuffles,”CanadianJ.Math.,XLI,pp.577–591,1989.[16]H.Munthe-KaasandB.Owren,“ComputationsinafreeLiealgebra,”,-RoyalSoc.LondonPhilos.Trans.Ser.AMath.Phys.Eng.Sci.,357,pp.957–982,1999.[17]R.MurrayandS.Sastry,“Nonholonomicpathplanning:steeringwithsinusoids,”IEEETrans.Aut.Control,38,pp.700–716,1993.[18]C.Reutenauer,FreeLiealgebras,Oxford:ClarendonPress,1993.[19]E.Rocha,“OncomputataionofthelogarithmoftheChen-Fliessseriesfornonlinearsystems,”Proc.NCN,Sheffield,2001.[20]E.SontagandY.Wang,“Ordersofinput/outputdifferentialequationsandstatespacedimensions,”SIAMJ.ControlandOptimization,33,pp.1102-1126,1995.[21]H.Sussmann,“Liebracketsandlocalcontrollability:Asufficientcondi-tionforscalar-inputsystems,”SIAMJ.Cntrl.&Opt.,21,pp.686–713,1983.[22]H.Sussmann,“AproductexpansionoftheChenseries,”TheoryandApplicationsofNonlinearControlSystems,C.I.ByrnesandA.Lindquist,eds.,Elsevier,North-Holland,pp.323–335,1986.[23]G.Viennot,“Alg`ebresdeLieLibresetMono¨ıdesLibres,”LectureNotesMath.,692,Springer,Berlin,1978. Problem3.4AnextendedgradientconjectureLuizCarlosMartinsJr.UniversidadePaulista-UNIP15091-450,S.J.doRioPreto,SPBrazillcmartinsjr@bol.com.brGeraldoNunesSilvaDepartamentodeComputacaoeEstatisticaUniversidadeEstadualPaulista-UNESP15054-000,S.J.doRioPreto,SPBrazilgsilva@dcce.ibilce.unesp.br1DESCRIPTIONOFTHEPROBLEMLetf:Rn→RbealocallyLipschitzfunction,i.e.,forallx∈Rthereis>0andaconstantKdependingonsuchthat|f(x1)−f(x2)|≤Kkx1−x2k,∀x1,x2∈x+B.HereBdenotestheopenunitballofRn.Letv∈Rn.Thegeneralizeddirectionalderivativeoffatx,inthedirectionv,denotedbyf0(x;v),isdefinedasfollows:0f(y+sv)−f(y)f(x;v)=limsup.y→xss→0+Herey∈Rn,s∈(0,+∞).Thegeneralizedgradientoffatx,denotedby∂f(x),isthesubsetofRngivenby{ξ∈Rn:f0(x;v)≥hξ,vi,∀v∈R}.Forthepropertiesandbasiccalculusofthegeneralizedgradient,standardreferencesare[1]and[2].Theproblemweproposehereisregardingthefollowingdifferentialinclusionx˙(t)∈∂f(x(t))a.e.t∈[0,β),(1) 104PROBLEM3.4whereβisapositivescalar.Asolutionof(1)isanabsolutelycontinuousfunctionx:[0,β)→Rnthat,togetherwith˙x,itsderivativewithrespecttot,satisfies(1).Notethat˙xmayfailtoexistonasetA⊂[0,∞)ofzeroLebesguemeasure.TakeStobetheset[0,∞)A.Wesaythatx˙d:=lim,t→βkx˙kSwhenthelimitexists,isatangentialdirectionofxat0∈Rn.Thenotationt→βmeansthatthelimitistakenfort∈S.SWearenowinapositiontoproposeourproblem.Conjecture:Supposethatf(0)=0andletxbeasolutionof(1)suchthatx(t)→0,ast→β.Thenthereexistsauniquetangentialdirection.2MOTIVATIONANDHISTORYOFTHEPROBLEMThisproblemhasbeenstated,forthefirsttime,inthesmoothcase,thatis,inthesituationwherefisarealanalyticfunctiononanopenneighborhoodU⊂Rnofapointx,andxisamaximalcurveof(1)with5f,the00gradientoff,replacingthegeneralizedgradientoffandx(t)→x0,ast→β.Underthisconditions,R.Thomaskedwhetherthetangentofx(t)atx0waswell-defined.Thiswaslaternamedtheconjectureofthegradient,see,forexample,[4,5,6].Wenowshowthatthiswasanaturalquestiontoask.Assumingthatfisananalyticfunctionasaboveandthatx0=0andf(0)=0,Lojasiewiczprovedin[8,p.92]thatthereexists0<θ<1suchthatθ|∇f(x)|>|f(x)|,forx∈U0.ThisresultisknownasLojasiewiczInequalityandisthemaintoolintheproofofthenextstatedresult.Foranaccountonthissee,forexample,[7]and[9].Theorem(Lojasiewicz):LetA=f−1(0)∩U.Thenβ=+∞andifx(t)0tendstowardA,thenx(t)tendstoauniquepointofA.(Asimpleproofofthistheoremisprovidedin[3]).Since,fromthetheoremabove,weseethatamaximaltrajectoryxlivesinthewholeinterval[0,∞)andapproximatesauniquepointintheinverseimageof0byf,itisnaturaltoaskifthetangentofx(t)inthelimitpointisalsounique.ThiswaspreciselywhatR.Thomconjecturedandbecamethewell-knowngradientconjecture.Inthiswork,weproposeanextensionofthisconjecturetothenonsmoothcase. ANEXTENDEDGRADIENTCONJECTURE1053KNOWNRESULTSANDREMARKSThegradientconjecture,asitisknownintheregularcase,isequivalenttofactthattheintegralcurvesof∇fhavetangentinallpointsofω(x).Partialresultsontheconjectureofthegradientwasgivenin[3],[11],and[9].Thefirstproofofthegeneralregularcasewasgivenin[4]andasimplermodifiedproofhasappearedin[6].Actually,ithasbeenprovedastrongerresultthatstatesthattheradialprojectionofx(t)fromx(0)intothesphereSn−1hasfinitelength.TheargumentsoftheproofrelyontheLojasiewiczInequality.Thenewconjectureofthegradienteisstatedinthenonsmoothsettingandiscalledtheextendedgradientconjecture.Asfarasweknow,noresulthasappearedinthisdirection.Wereckonthatasimpleextensionofthestandardtechniquesusedtoprovetheregularcaseisnotenough.Itwillbenecessarytocomeupwithnewideastoprovethisconjectureifithappenstobetrue.BIBLIOGRAPHY[1]F.H.Clarke,OptimizationandNonsmoothAnalysis,WileyInter-science,NewYork1983;reprintedasvol.5ofClassicsinAppliedMath-ematics,SIAM,Philadelphia,PA,1990;Russiantranslation,Nauka,Moscow,1988.[2]F.H.Clarke,Yu.S.Ledyaev,R.J.Stern,P.R.Wolenski,NonsmoothAnalysisandControlTheory,GTM178,NewYork,Springer-Verlag,1998.[3]F.Ichikawa,“Thom’sconjectureonsingulatiriesofgradientvectorfields,”KodaiMath.J.15,pp.134-140,1992.[4]K.Kurdyka,T.Mostowski,TheGradientConjectureofR.Thom,preprint,1996(revised1999).http://www.lama.univ-savoie.fr/sitelama/Membres/pagesweb/KURDIKA/index.html.[5]K.Kurdyka,“OnthegradientconjectureofR.Thom,”SeminaridiGeometria1998-1999,Universit`adiBologna,IstitutodiGeometria,Di-partamentodiMatematica,pp.143-151,2000.[6]K.Kurdyka,T.Mostowski,A.Parusinki,“Proofofthegradientcon-jectureofR.Thom,”AnnalsofMathematics,152(2000),pp.763-792.[7]S.Lojasiewicz,“Unepropri´et´etopologiquedessous-ensemblesanaly-tiquesr´eels,”ColloquesInternationauxduC.N.R.S.#117,les´equationsauxd´eriv´eespartielles,Paris25-30juin(1962),pp.87-89.[8]S.Lojasiewicz,Ensemblessemi-analytiques,IHESpreprint,1965. 106PROBLEM3.4[9]R.Moussu,“Surladynamiquedesgradients.Existencedevari´et´esin-variants,”Math.Ann.,307(1997),pp.445-460.[10]F.Sanz,“Nonoscillatingsolutionsofanalyticgradientvectorfields,”Ann.Inst.Fourier,Grenoble48(4)1998,pp.1045-1067.[11]H.XingLin,Surlastructuredeschampsdegradientsdefonctionsanalytiquesr´eelles,Ph.D.Th`ese,Universit´eParisVII,1992. Problem3.5OptimaltransactioncostsfromaStackelbergperspectiveGeertJanOlsderFacultyofInformationTechnologyandSystemsDelftUniversityofTechnologyP.O.Box5031,2600GADelftTheNetherlandsg.j.olsder@its.tudelft.nl1DESCRIPTIONOFTHEPROBLEMTheproblemtobeconsideredisx˙=f(x,u),x(0)=x0,(1)ZTZTmaxJF=max{q(x(T))+g(x,u)dt−γ(u(t))dt},(2)uu00ZTmaxJL=maxγ(u(t))dt,(3)γ(·)γ(·)0withf,gandqbeinggivenfunctions,thestatex∈Rn,thecontrolu∈R,andγ(·)isascalarfunctionwhichmapsRintoR.TheproblemconcernsadynamicgameprobleminwhichuisthedecisionvariableofoneplayercalledtheFollower,andthefunctionγisuptothechoiceoftheotherplayercalledtheLeader.AnessentialfeatureoftheproblemisthattheLeader’sprofit(3)isadirectlossfortheFollowerin(2).TheLeaderlivesasaparasiteontheFollower.Inthenextsection,amoreconcretemotivationwillbegiven.Thefunctionγmustbechosensubjecttotheconstraintsγ(0)=0,γ(·)≥0andifatallpossibleitmustbenondecreasingwithrespectto|u|,andpossiblyalsoγ(u)=γ(−u).Bymeansofthenotationintroducedandthenamesoftheplayersitshouldbeclearthattheproblemformulatedisa(specialkindof)Stackelberggame[2].TheLeaderannouncesthefunctionγthatthusbecomesknowntotheFollowerwhosubsequentlychoosesu.Thustheoptimaluisafunctionofγ(·). 108PROBLEM3.52MOTIVATIONANDHISTORYOFTHEPROBLEMForn=1,i.e.,x∈R,whichweassumehenceforth,aninterpretationofthismodelisthatx(t)representstheFollower’swealthattimet.ThisFollowerisaninvestorandwhowouldliketomaximizeZTg(x,u)dt+q(x(T)).(4)0Thetermq(x(T))inthiscriterionisafunctionofthewealthoftheinvestoratRTthefinaltimeTandthetermg(x,u)dtrepresentstheconsumptionduring0thetimeinterval[0,T].Thedecisionvariableu(t)denotesthetransactionswiththebankattimet(e.g.,sellingorbuyingstocks).Tobemoreprecise,u(t)denotesatransactiondensity,i.e.,duringthetimeinterval[t,t+dt]thenumberoftransactionsequalsu(t)dt.Foru=0,notransactionstakeplaceandthebankdoesnotearnmoney(becauseγ(0)=0).Transactionscostmoneyandweassumethatthebank(i.e.,theLeader)wantstomaximizethesetransactioncostsasindicatedby(3).Thesecostsaresubtractedfrom(4)andhencetheultimatecriterionoftheFollowerisgivenby(2).Therestrictionsposedonγ(nondecreasingwithrespectto|u|andγ(0)=0)nowhaveaclearmeaning.Thehigherthenumberoftransactions(eitherbuyingorselling,onebeingrelatedtoapositiveu,theotheronetoanegativeu),thehigherthecosts.Equation(1)issupposedtotellhowthewealthxevolvesintime.Usually,suchmodelsarerepresentedbystochasticdifferentialequations,butduetothecomplexityoftheproblem,werestrictourselvestoalessrealisticdeterministicdifferentalfunction.3AVAILABLERESULTSANDBACKGROUNDProblemswithtransactioncostshavebeenstudiedbefore,seee.g.,[1,3,4],butneverfromthepointofviewofthebanktomaximizethesecosts.Theproblemasstatedisadifficultone,see[7]forsomefirstsolutionattempts.Theprincipaldifficultyisthatcomposedfunctionsareinvolved,i.e.,onefunctionistheargumentofanother[6].Hence,wewillalsoconsiderthefollowingstaticproblem,whichissimplerthanthetime-dependentone:max(q(u)−γ(u)),maxγ(u),uγ(·)subjecttoγ(·)≥0andγ(0)=0andpossiblyalsoγ(u)nondecreasingwithrespectto|u|.Withthesameinterpretationasbefore,theinvestorissecuredofaminimumvalueq(0)byplayingu=0.Therefore,hewillonlytakeu-valuesintoconsiderationforwhichq(u)≥q(0).Thisstaticproblemisaspecialcaseoftheso-calledinverseStackelbergproblemasitwasintroducedin[5]andasolutionmethodisknown,seechapter7of[2]. OPTIMALTRANSACTIONCOSTS109Tostartwith,inaconventionalStackelberggame,therearetwoplayers,calledLeaderandFollowerrespectively,eachhavingtheirowncostfunctionJL(uL,uF),JF(uL,uF),whereuF,uL∈R.Eachplayerwantstochoosehisowndecisionvariableinsuchawayastomaximizehisowncriterion.Withoutgivinganequilibriumconcept,theproblemasstatedsofarisnotwelldefined.Suchanequilib-riumconceptcould,forinstance,beonenamedafterNashorPareto.IntheStackelbergequilibriumconcept,oneplayer,theLeader,announceshisdecisionuL,whichissubsequentlyknowntotheotherplayer,theFollower.Withthisknowledge,theFollowerchooseshisuF.Hence,uFbecomesafunctionofuL,writtenasuF=lF(uL),whichisdeterminedthroughtherelationminJF(uL,uF)=JF(uL,lF(uL)),uFprovidedthatthisminimumexistsandisasingletonforeachpossiblechoiceuLoftheLeader.ThefunctionlF(·)issometimescalledareactionfunction.BeforetheLeaderannounceshisdecisionuL,hewillrealizehowtheFollowerwillreactandhencetheLeaderchoosesuLsuchastominimizeJL(uL,lF(uL)).InaninverseStackelberggame,theLeaderdoesnotannouncehischoiceuLaheadoftime,asabove,butinsteadafunctionγL(uF).Think(asanothermotivatingexample)oftheLeaderbeingthegovernmentandoftheFollowerasacitizen.ThegovernmentstateshowmuchincometaxthecitizenhastopayandthistaxwilldependontheincomeuFofthecitizen.Itisuptothecitizenastohowmuchmoneytoearn(byworkingharderornot)andthushecanchooseuF.TheincometaxthegovernmentwillreceiveequalsγL(uF),wherethe”rulefortaxation”γL(·)wasmadeknownaheadoftime.BIBLIOGRAPHY[1]M.Akian,J.L.Menaldi,andA.Sulem,“Onaninvestment-consumptionmodelwithtransactioncosts,”SIAMJ.ControlandOptim.,vol.34pp.329-364,1996.[2]T.Ba¸sarandG.J.Olsder,DynamicNoncooperativeGameTheory,SIAM,Philadelphia,1999.[3]P.Bernhard,“Arobustcontrolapproachtooptionpricingincludingtransactioncosts,”In:AnnalsoftheISDG7,A.Nowak,ed.,Birkh¨auser,2002. 110PROBLEM3.5[4]E.R.GrannanandG.H.Swindle,“Minimizingtransactioncostsofoptionhedgingstrategies,”Mathematicalfinance,vol.6,no.4,341–364,1996.[5]Y.-C.Ho,P.B.LuhandG.J.Olsder,“Acontrol-theoreticviewonincentives,”Automatica,vol.18,pp.167-179,1982.[6]M.Kuczma,FunctionalEquationsinaSingleVariable,PolishScientificPublishers,1968.[7]G.J.Olsder,“Differentialgame-theoreticthoughtsonoptionpricingandtransactioncosts,”InternationalGameTheoryReview,vol.2,pp.209-228,2000. Problem3.6Doescheapcontrolsolveasingularnonlinearquadraticproblem?YuriV.OrlovElectronicsDepartmentCICESEResearchCenterEnsenada,BC22860Mexicoyorlov@cicese.mx1DESCRIPTIONOFTHEPROBLEMAstandardcontrolsynthesisforaffinesystemsx˙=f(x)+g(x)u,x∈Rn,u∈Rm(1)underdegenerateperfomancecriterionZ∞J(u)=xT(t)Px(t)dt,P=PT>0(2)0dependingonthestatevectorx(t)only,replacesthissingularoptimizationproblembyitsregularizationthroughε-approximationZ∞J(u)=[xT(t)Px(t)+εuT(t)Ru(t)]dt,ε>0,R=RT>0(3)ε0ofthiscriterionwithsmall(cheap)penaltyonthecontrolinputu.Here-after,functionsf,gareassumedsufficientlysmooth,andallquantitiesin(1)through(3)areassumedtohavecompatibledimensions.Theoptimalcontrolsynthesiscorrespondingto(2)isthenobtainedasalimitasε→0oftheoptimalcontrollawu0correspondingto(3).Sinceonlyεparticularapproximationistakenwhileotherapproximationsarepossibleaswellthereisnoguaranteethattheoriginalperfomancecriterionisminimizedbythecontrollawobtainedviathisprocedure.AnopenproblemthatariseshereistoprovethatinfJ(u)=liminfJε(u)(4)uε→0uorpresentacounterexampleofsystem(1)wherethelimitingrelation(4)isnotsatisfied. 112PROBLEM3.62MOTIVATIONANDHISTORYOFTHEPROBLEMTheaboveproblemiswell-understoodinthelinearcasewhensystem(1)isspecifiedasfollows:x˙=Ax+Bu,x∈Rn,u∈Rm.(5)Underthestabilizabilityanddetectabilityconditionsthelinearsystem(5)drivenbythecheapcontrolu0exhibitsaninitialfasttransientfollowedbyaεslowmotiononasingulararc(see,e.g.,[3,section6]andreferencestherein).Inthelimitε→0,asingularperturbationanalysisrevealsthatthestablefastmodesdecayinstantaneouslyasiftheywouldbedrivenbytheimpulsivecomponentofthecontrollerminimizingthedegenerateperformancecriterion(2).Thisfeature,however,doesnotadmitastraightforwardextensiontothesysteminquestionbecauseincontrasttothelinearsystem(5),aninstanta-neousimpulseresponseoftheaffinesystem(1),generallyspeaking,dependsonanapproximationoftheimpulse[2].Thus,itmighthappenthattheoriginalperformance(2)isnotminimizedthroughtheε-approximation(3)ofthiscriterion.3AVAILABLERESULTSAdistribution-orientedvariationalanalysis[1]ofthesingularnonlinearquad-raticproblem(1),(2),admittingbothintegrableandimpulsiveinputs,re-vealsthattheinfimumofthedegeneratecriterion(2)istypicallyattainedbyacontrollerwithimpulsivebehaviorattheinitialtimemoment.Inthatcase,aninstantaneousimpulseresponseoftheclosed-loopsystemdoesnotdependonanapproximationoftheimpulseifandonlyiftheaffinesystem(1)satisfiestheFrobeniuscondition,i.e.,thedistributionspannedbythecolumnsofg(x)isinvolutive(see[2]fordetails).Motivatedbythesearguments,theauthorsuspectsthatthelimitingrelation(4)holdswheneversystem(1)satisfiestheFrobeniuscondition,andacoun-terexampleofsystem(1),violating(4),isindeedpossibleiftheFrobeniusconditionisnotimposedonthesystemanymore.BIBLIOGRAPHY[1]Y.Orlov,“Necessaryoptimalityconditionsofgeneralizedcontrolactions1,2,”AutomationandRemoteControl,vol.44,no.7,pp.868-877;vol.44,no.8,pp.998-1105,1984.[2]Y.Orlov,“Instantaneousimpulseresponseofnonlinearsystems,”IEEETrans.Aut.Control,vol.45,no.5,pp.999-1001,2000. SINGULARNONLINEARQUADRATICPROBLEM113[3]V.R.Saksena,J.O’Reilly,andP.V.Kokotovic,“Singularperturbationsandtime-scalemethodsincontroltheory:Survey1976-1983,”Automat-ica,vol.20,no.3,pp.273-293,1984. Problem3.7Delta-SigmamodulatorsynthesisAndersRantzerDept.ofAutomaticControlLundInstituteofTechnologyP.O.Box118SE-22100LUND,Swedenrantzer@control.lth.se1DESCRIPTIONOFTHEPROBLEMDelta-Sigmamodulatorsareamongthekeycomponentsinmodernelectron-ics.Theirmainpurposeistoprovidecheapconversionfromanalogtodigitalsignals.Inthefigurebelow,theanalogsignalrwithvaluesintheinterval[−1,1]issupposedtobeapproximatedbythedigitalsignaldthattakesonlytwovalues,−1and1.Onecannotexpectgoodapproximationatallfrequencies.Hence,thedynamicsystemDshouldbechosentominimizetheerrorfinagivenfrequencyrange[ω1,ω2].ThereisarichliteratureonDelta-Sigmamodulators.See[2,1]andref-erencestherein.Thepurposeofthisnoteistoreachabroadaudiencebyfocusingonthecentralmathematicalproblem.rfd-e-Dynamic--systemD6−1Tomakeapreciseproblemformulation,weneedtointroducesomenotation:Notation:Thesignalspace`[0,∞]isthesetofallsequences{f(k)}∞suchk=0thatf(k)∈[−1,1]fork=0,1,2,....AmapD:`[0,∞]→`[0,∞]iscalledacausaldynamicsystemifforeveryu,v∈`[0,∞]suchthatu(k)=v(k)fork≤Titholdsthat[D(u)](k)=[D(v)](k)fork≤T.Definealsothe DELTA-SIGMAMODULATORSYNTHESIS115function(1ifx≥0sgn(x)=−1elseProblem:Givenr∈`[0,∞]andacausaldynamicsystemD,defined,f∈`[0,∞]by(
d(k+1)=sgn[D(f)](k),d(0)=0f(k)=r(k)−d(k)andfindacausaldynamicsystemDsuchthatregardlessofthereferenceinputr,theerrorsignalfbecomessmallinaprespecifiedfrequencyinterval[ω1,ω2].Theproblemformulationisintentionallyleftvagueonthelastline.Thesizeoffcanbemeasuredinmanydifferentways.Oneoptionistorequireauniformboundon1XT−ikωlimsupef(k)T→∞Tk=0forallω∈[ω1,ω2]andallreferencesignalsr∈`[0,∞].AnotheroptionistoallowDtobestochasticsystemandputaboundonthespectraldensityoffinthefrequencyinterval.Thiswouldbeconsistentwiththewide-spreadpracticetoaddastochastic“ditheringsignal”beforethenonlinearityinordertoavoidundesiredperiodicorbits.2AVAILABLERESULTSThesimplestandbestunderstoodcaseiswhere(x(k+1)=x(k)+f(k)f(k)=r(k)−sgn(x(k))Inthiscase,itiseasytoseethatthesetx∈[−2,2]isinvariant,sowithXTXTF(z)=z−kf(k)X(z)=z−kx(k)TTk=0k=0 116PROBLEM3.7itholdsthatZω0Zω01iω21iωiω2|FT(e)|dω=|(e−1)XT(e)|dωT0T0Zω01iω2=2(1−cosω)|XT(e)|dωT0Zπ1iω2≤2(1−cosω0)|XT(e)|dωT0XTπ2=2(1−cosω0)x(k)Tk=0≤8π(1−cosω0)whichclearlyboundstheerrorfatlowfrequencies.Manymodificationsusinghigherorderdynamicshavebeensuggestedinordertofurtherreducetheerror.However,thereisstillastrongdemandforimprovementsandabetterunderstandingofthenonlineardynamics.Thefollowingtworeferencesaresuggestedasentriestotheliteratureon∆-Σ-modulators:BIBLIOGRAPHY[1]JamesA.Cherry,ContinuousTimeDelta-SigmaModulatorsforHigh-SpeedA/DConversion:Theory,Practice&FundamentalPerformanceLimits,Kluwer,1999.[2]S.R.Norsworthy,R.Schreier,andG.C.Temes,Delta-SigmaDataCon-verters,IEEEPress,NewYork,1997. Problem3.8Determiningofvariousasymptoticsofsolutionsofnonlineartime-optimalproblemsviarightidealsinthemomentalgebraG.M.SklyarSzczecinUniversityWielkopolskastr.15,70-451Szczecin,Poland;KharkovNationalUniversitySvobodasqr.4,61077KharkovUkrainesklar@sus.univ.szczecin.pl,sklyar@univer.kharkov.uaS.Yu.IgnatovichKharkovNationalUniversitySvobodasqr.4,61077Kharkov,Ukrainebob@online.kharkiv.com1MOTIVATIONANDHISTORYOFTHEPROBLEMThetime-optimalcontrolproblemisoneofthemostnaturalandatthesametimedifficultproblemsintheoptimalcontroltheory.Forlinearsystems,themaximumprincipleallowstoindicateaclassofopti-malcontrols.However,theexplicitformofthesolutioncanbegivenonlyinanumberofparticularcases[1-3].Atthesametime[4],anarbitrarylineartime-optimalproblemwithanalyticcoefficientscanbeapproximated(inaneighborhoodoftheorigin)byacertainlinearproblemoftheformx˙=−tqiu,i=1,...,n,q<···0isthen×ncovariancematrixandXisann×pmatrix.Actually,itisnontrivialtoprovethatthiscubicmatrixdifferentialequationisindeedaPSAintheabovesenseandthus,generically,convergestoadominanteigenspacebasis.Another,moregeneralexampleofaPSAflowisthatintroducedby[12,13]and[17]:X˙=CXN−XNX0CX(2)HereN=N0>0denotesanarbitrarydiagonalk×kmatrixwithdistincteigenvalues.ThissystemisactuallyajointgeneralizationofOja’sflow(1)andofBrockett’s[1]gradientflowonorthogonalmatricesX˙=[C,XNX0]X(3)Forsymmetricmatrixdiagonalisation,seealso[6].In[19],Oja’sflowwasre-derivedbyfirstproposingthegradientflowX˙=(C(I−XX0)+(I−XX0)C)X(4)andthenomittingthefirsttermC(I−XX0)XbecauseC(I−XX0)X=CX(I−X0X)→0,aconsequenceofbothtermsin(4)forcingXtotheinvariantmanifold{X:X0X=I}.Interestingly,ithasrecentlybeenreal-ized[8]that(4)hascertaincomputationaladvantagescomparedwith(1),however,arigorousconvergencetheoryismissing.Ofcourse,thesethreesystemsarejustprominentexamplesfromabiggerlistofpotentialPSA 124PROBLEM3.9flows.Oneopenprobleminmostofthecurrentresearchisalackofafullconvergencetheory,establishingpointwiseconvergencetotheequilibria.Inparticular,asolutiontothefollowingthreeproblemswouldbehighlydesir-able.Thefirstproblemaddressesthequalitativeanalysisoftheflows.Problem1.Developacompletephaseportraitanalysisof(1),(2)and(4).Inparticular,provethattheflowsarePSA,determinetheequilibriapoints,theirlocalstabilitypropertiesandthestableandunstablemanifoldsfortheequilibriumpoints.Theprevioussystemsareusefulforprincipalcomponentanalysis,buttheycannotbeusedimmediatelyforminorcomponentanalysis.Ofcourse,onepossibleapproachmightbetoapplyanyoftheaboveflowswithCreplacedbyC−1.Oftenthisisnotreasonablethough,asinmostapplicationsthecovariancematrixCisimplementedbyrecursiveestimatesandonedoesnotwanttoinverttheserecursiveestimatesonline.Anotheralternativecouldbetoputanegativesigninfrontoftheequations.Butthisdoesnotworkeither,astheminorcomponentequilibriumpointremainsunstable.Intheliterature,therefore,otherapproachestominorcomponentanalysishavebeenproposed[2,3,5],butwithoutacompleteconvergencetheory.1Moreover,aguidinggeometricprinciplethatallowsforthesystematiccon-structionofminorcomponentflowsismissing.Thekeyideahereseemstobeanappropriateconceptofdualitybetweenprincipalandminorcompo-nentanalysis.Conjecture1.Principalcomponentflowsaredualtominorcomponentflows,viaaninvolutioninmatrixspaceRn×p,thatestablishesabijectivecorrespondencebetweensolutionsofPSAflowsandMSAflows,respectively.IfaPSAflowisactuallyagradientflowforacostfunctionf,asisthecasefor(1),(2)and(4),thenthecorrespondingdualMSAflowisagradientflowfortheLegendredualcostfunctionf∗off.Whenimplementingthesedifferentialequationsonacomputer,suitabledis-cretizationsaretobefound.SinceweareworkinginunconstrainedEu-clideanmatrixspaceRn×p,weconsiderEulerstepdiscretizations.Thus,e.g.,forsystem(1)considerX=X−s(I−XX0)CX,(5)t+1tttttwithsuitablysmallstepsizes.SuchEulerdiscretizationschemesarewidelyusedintheliterature,butusuallywithoutexplicitstep-sizeselectionsthatguarantee,forgenericinitialconditions,convergencetothepdominantor-thonormaleigenvectorsofA.Afurtherchallengeistoobtainstep-sizese-lectionsthatachievequadraticconvergencerates(e.g.,viaaNewton-typeapproach).1Itisremarkedthattheconvergenceproofin[5]appearsflawed;theyarguethatdvecQbecause=G(t)vecQforsomematrixG(t)<0thenQ→0.However,counter-dtexamplesareknown[15]whereG(t)isstrictlynegativedefinite(withconstanteigenvalues)yetQdiverges. DYNAMICSOFPRINCIPALANDMINORCOMPONENTFLOWS125Problem2.Developasystematicconvergencetheoryfordiscretisationsoftheflows,byspecifyingstep-sizeselectionsthatimplyglobalaswellaslocalquadraticconvergencetotheequilibria.2MOTIVATIONANDHISTORYEigenvaluecomputationsareubiquitousinMathematicsandEngineeringSciences.Inapplications,thematriceswhoseeigenvectorsaretobefoundareoftendefinedinarecursiveway,thusdemandingrecursivecomputa-tionalmethodsforeigendecomposition.Subspacetrackingalgorithmsarewidelyusedinneuralnetworks,regressionanalysis,andsignalprocessingapplicationsforthispurpose.Subspacetrackingalgorithmscanbestudiedbyreplacingthestochastic,recursivealgorithmthroughanaveragingproce-durebyanonlinearordinarydifferentialequation.Similarly,newsubspacetrackingalgorithmscanbedevelopedbystartingwithasuitableordinarydif-ferentialequationandthenconvertingittoastochasticapproximationalgo-rithm[7].Therefore,understandingthedynamicsofsuchflowsisparamounttothecontinuingdevelopmentofrecursiveeigendecompositiontechniques.ThestartingpointformostofthecurrentworkinprincipalcomponentanalysisandsubspacetrackinghasbeenOja’ssystemfromneuralnetworktheory.UsingasimpleHebbianlawforasingleperceptronwithalinearactivationfunction,Oja[9,10]proposedtoupdatetheweightsaccordingtoX=X−s(I−XX0)uu0X.(6)t+1tttttttHereXtdenotesthen×pweightmatrixanduttheinputvectoroftheperceptron,respectively.ByapplyingtheODEmethodtothissystem,Ojaarrivesatthedifferentialequation(1).HereC=E(uu0)isthecovariancettmatrix.Similarly,theotherflows,(2)and(4),haveanalogousinterpreta-tions.In[9,11]itisshownforp=1that(1)isaPSAflow,i.e.,itconvergesforgenericinitialconditionstoanormaliseddominanteigenvectorofC.In[11]thesystem(1)wasstudiedforarbitraryvaluesofpanditwasconjecturedthat(1)isaPSAflow.Thisconjecturewasfirstprovenin[18],assumingpositivedefinitenessofC.Moreover,in[18,4],explicitinitialconditionsintermsofintersectiondimensionsforthedominanteigenspacewiththeinitalsubspaceweregiven,suchthattheflowconvergestoabasismatrixofthep-dimensionaldominanteigenspace.ThisisreminiscentofSchuberttypeconditionsinGrassmannmanifolds.AlthoughtheOjaflowservesasaprincipalsubspacemethod,itisnotusefulforprincipalcomponentanalysisbecauseitdoesnotconvergeingeneraltoabasisofeigenvectors.Flowsforprincipalcomponentanalysissuchas(2)havebeenfirststudiedin[14,12,13,17].However,pointwiseconvergencetotheequilibriapointswasnotestablished.In[16]aLyapunovfunctionfortheOjaflow(1)wasgiven,butwithoutrecognizingthat(1)isactuallyagradient 126PROBLEM3.9flow.Therehavebeenconfusingremarksintheliteratureclaimingthat(1)cannotbeagradientsystemasthelinearizationisnotasymmetricmatrix.However,thisisduetoamisunderstandingoftheconceptofagradient.In[20]itisshownthat(2),andinparticular(1),isactuallyagradientflowforthecostfunctionf(X)=1/4tr(AXNX0)2−1/2tr(A2XD2X0)andasuitableRiemannianmetriconRn×p.Moreover,startingfromanyinitialconditioninRn×p,pointwiseconvergenceofthesolutionstoabasisofkindependenteigenvectorsofAisshowntogetherwithacompletephaseportraitanalysisoftheflow.Firststepstowardaphaseportraitanalysisof(4)aremadein[8].3AVAILABLERESULTSIn[12,13,17]theequilibriumpointsof(2)werecomputedtogetherwithalocalstabilityanalysis.Pointwiseconvergenceofthesystemtotheequilibriaisestablishedin[20]usinganearlyresultbyLojasiewiczonrealanalyticgradientflows.Thustheseresultstogetherimplythat(2),andhence(1),isaPSA.Ananalogousresultfor(4)isforthcoming;see[8]forfirststepsinthisdirection.SufficientconditionsforinitialmatricesintheOjaflow(1)toconvergetoadominantsubspacebasisaregivenin[18,4],butnotfortheother,unstableequilibria,norforsystem(2).Acompletecharacterizationofthestable/unstablemanifoldsiscurrentlyunknown.BIBLIOGRAPHY[1]R.W.Brockett,“Dynamicalsystemsthatsortlists,diagonalisema-trices,andsolvelinearprogrammingproblems,”LinearAlgebraAppl.,146:79–91,1991.[2]T.Chen,“ModifiedOja’salgorithmsforprincipalsubspaceandminorsubspaceextraction,”NeuralProcessingLetters,5:105–110,April1997.[3]T.Chen,S.Amari,andQ.Lin,“Aunifiedalgorithmforprincipalandminorcomponentextraction,”NeuralNetworks,11:385–390,1998.[4]T.Chen,Y.Hua,andW.-Y.Yan,“GlobalconvergenceofOja’ssub-spacealgorithmforprincipalcomponentextraction,”IEEETransac-tionsonNeuralNetworks,9(1):58–67,1998.[5]S.C.Douglas,S.-Y.Kung,andS.Amari,“Aself-stabilizedminorsub-spacerule,”IEEESignalProcessingLetters,5(12):328–330,December1998.[6]U.HelmkeandJ.B.Moore,OptimizationandDynamicalSystems,Springer-Verlag,1994. DYNAMICSOFPRINCIPALANDMINORCOMPONENTFLOWS127[7]H.J.KushnerandG.G.Yin,StochasticApproximationAlgorithmsandApplications,Springer,1997.[8]J.H.Manton,I.M.Y.Mareels,andS.Attallah,“AnanalysisofthefastsubspacetrackingalgorithmNOja,”In:IEEEConferenceonAcoustics,SpeechandSignalProcessing,Orlando,Florida,May2002.[9]E.Oja,“Asimplifiedneuronmodelasaprincipalcomponentanalyzer,”JournalofMathematicalBiology,15:267–273,1982.[10]E.Oja,“Neuralnetworks,principalcomponents,andsubspaces,”In-ternationalJournalofNeuralSystems,1:61–68,1989.[11]E.OjaandJ.Karhunen,“Onstochasticapproximationoftheeigenvec-torsandeigenvaluesoftheexpectationofarandommatrix,”JournalofMathematicalAnalysisandApplications,106:69–84,1985.[12]E.Oja,H.Ogawa,andJ.Wangviwattana,“Principalcomponentanal-ysisbyhomogeneousneuralnetworks,PartI:Theweightedsubspacecriterion,”IEICETransactionsonInformationandSystems,3:366–375,1992.[13]E.Oja,H.Ogawa,andJ.Wangviwattana,“Principalcomponentanal-ysisbyhomogeneousneuralnetworks,PartII:Analysisandextensionsofthelearningalgorithms,”IEICETransactionsonInformationandSystems,3:376–382,1992.[14]T.D.Sanger,“Optimalunsupervisedlearninginasingle-layerlinearfeedforwardnetwork,”NeuralNetworks,2:459–473,1989.[15]J.L.Willems,StabilityTheoryofDynamicalSystems,StudiesinDy-namicalSystems,London,Nelson,1970.[16]J.L.WyattandI.M.Elfadel,“Time-domainsolutionsofOja’sequa-tions,”NeuralComputation,7:915–922,1995.[17]L.Xu,“Leastmeansquareerrorrecognitionprincipleforselforganizingneuralnets,”NeuralNetworks,6:627–648,1993.[18]W.-Y.Yan,U.Helmke,andJ.B.Moore,“GlobalanalysisofOja’sflowforneuralnetworks,”IEEETransactionsonNeuralNetworks,5(5):674–683,September1994.[19]B.Yang,“Projectionapproximationsubspacetracking”,IEEETrans-actionsonSignalProcessing,43(1):95–107,January1995.[20]S.Yoshizawa,U.Helmke,andK.Starkov,“Convergenceanalysisforprincipalcomponentflows,”InternationalJournalofAppliedMathe-maticsandComputerScience,11:223–236,2001. PART4DiscreteEvent,HybridSystems Problem4.1L2-inducedgainsofswitchedlinearsystemsJo˜aoP.Hespanha1Dept.ofElectricalandComputerEngineeringUniversityofCalifornia,SantaBarbaraUSAhespanha@ece.ucsb.edu1SWITCHEDLINEARSYSTEMSInthe1999collectionofOpenProblemsinMathematicalSystemsandCon-trolTheory,weproposedtheproblemofcomputinginput-outputgainsofswitchedlinearsystems.Recentdevelopmentsprovidednewinsightsintothisproblemleadingtonewquestions.Aswitchedlinearsystemisdefinedbyaparameterizedfamilyofrealizations{(Ap,Bp,Cp,Dp):p∈P},togetherwithafamilyofpiecewiseconstantswitchingsignalsS:={σ:[0,∞)→P}.Hereweconsiderswitchedsys-temsforwhichallthematricesAp,p∈PareHurwitz.Thecorrespondingswitchedsystemisrepresentedbyx˙=Aσx+Bσu,y=Cσx+Dσu,σ∈S(1)andbyasolutionto(1),wemeanapair(x,σ)forwhichσ∈Sandxisasolutiontothetime-varyingsystemx˙=Aσ(t)x+Bσ(t)u,y=Cσ(t)x+Dσ(t)u,t≥0.(2)GivenasetofswitchingsignalsS,wedefinetheL2-inducedgainof(1)byinf{γ≥0:kyk2≤γkuk2,∀u∈L2,x(0)=0,σ∈S},whereyiscomputedalongsolutionsto(1).TheL2-inducedgainof(1)canbeviewedasa“worstcase”energyamplificationgainfortheswitchedsystem,overallpossibleinputsandswitchingsignalsandisanimportanttooltostudytheperformanceofswitchedsystems,aswellasthestabilityofinterconnectionsofswitchedsystems.1ThismaterialisbaseduponworksupportedbytheNationalScienceFoundationunderGrantNo.ECS-0093762. 132PROBLEM4.12PROBLEMDESCRIPTIONWeareinterestedhereinfamiliesofswitchingsignalsforwhichconsecutivediscontinuitiesareseparatedbynolessthanapositiveconstantcalledthedwell-time.ForagivenτD>0,wedenotebyS[τD]thesetofpiecewiseconstantswitchingsignalswithintervalbetweenconsecutivediscontinuitiesnosmallerthanτD.Thegeneralproblemthatweproposeisthecomputationofthefunctiong:[0,∞)→[0,∞]thatmapseachdwell-timeτDwiththeL2-inducedgainof(1),forthesetofdwell-timeswitchingsignalsS:=S[τD].Untilrecently?littlemorewasknownaboutgotherthanthefollowing:1.gismonotonedecreasing2.gisboundedbelowbyg:=supkC(sI−A)−1B+Dk,staticpppp∞p∈PwherekTk∞:=sup<[s]≥0kT(s)kdenotestheH∞-normofatransfermatrixT.WerecallthatkTk∞isnumericallyequaltotheL2-inducedgainofanylineartime-invariantsystemwithtransfermatrixT.Item1isatrivialconsequenceofthefactthatgiventwodwell-timesτD1≤τD2,wehavethatS[τD1]⊃S[τD2].Item2isaconsequenceofthefactthateverysetS[τD],τD>0containsalltheconstantswitchingsignalsσ=p,p∈P.Itwasshownin[2]thatthelower-boundgstaticisstrictandingeneralthereisagapbetweengstaticandgslow:=limg[τD].τD→∞Thismeansthatevenswitchingarbitrarilyseldom,onemaynotbeabletorecovertheL2-inducedgainsofthe“unswitchedsystems.”In[2]aprocedurewasgiventocomputegslow.Oppositetowhathadbeenconjectured,gslowisrealizationdependentandcannotbedeterminedjustfromthetransferfunctionsofthesystemsbeingswitched.Thefunctiongthuslooksroughlyliketheonesshowninfigure4.1.1,where(a)correspondstoasetofrealizationsthatremainsstableforarbitrarilyfastswitchingand(b)toasetthatcanexhibitunstablebehaviorforsufficientlyfastswitching[3].In(b),thescalarτmindenotesthesmallestdwell-timeforwhichinstabilitycanoccurforsomeswitchingsignalinS[τmin].Severalimportantbasicquestionsremainopen:1.Underwhatconditionsisgbounded?Thisisreallyastabilityproblemwhosegeneralsolutionhasbeeneludingresearchersforawhilenow(cf.,thesurveypaper[3]andreferencestherein).2.Incasegisunbounded(case(b)infigure4.1.1),howtocomputethepositionoftheverticalasymptote?Or,equivalently,whatisthesmallestdwell-timeτminforwhichonecanhaveinstability? L2-INDUCEDGAINSOFSWITCHEDLINEARSYSTEMS133g(τD)g(τD)gslowgslowgstaticgstaticτDτminτD(a)(b)Figure4.1.1L2-inducedgainversusthedwell-time.3.Isgaconvexfunction?Isitsmooth(orevencontinuous)?Evenifdirectcomputationofgprovestobedifficult,answerstothepreviousquestionsmayprovideindirectmethodstocomputetightboundsforit.Theyalsoprovideabetterunderstandingofthetrade-offbetweenswitchingspeedandinducedgain.Asfarasweknow,currentlyonlyverycoarseupper-boundsforgareavailable.Theseareobtainedbycomputingaconservativeupper-boundτupperforτminandthenanupper-boundforgthatisvalidforeverydwell-timelargerthanτupper(cf.,e.g.,[4,5]).Theseboundsdonotreallyaddressthetrade-offmentionedabove.BIBLIOGRAPHY[1]J.P.HespanhaandA.S.Morse,“Input-outputgainsofswitchedlin-earsystems,”In:OpenProblemsinMathematicalSystemsTheoryandControl,V.D.Blondel,E.D.Sontag,M.Vidyasagar,andJ.C.Willems,eds.,London:Springer-Verlag,1999.[2]J.P.Hespanha,“Computationofroot-mean-squaregainsofswitchedlinearsystems,”presentedattheFifthHybridSystems:ComputationandControlWorkshop,Mar.2002.[3]D.LiberzonandA.S.Morse,“Basicproblemsinstabilityanddesignofswitchedsystems,”IEEEContr.Syst.Mag.,vol.19,pp.59–70,Oct.1999.[4]J.P.HespanhaandA.S.Morse,“Stabilityofswitchedsystemswithaveragedwell-time,”In:Proc.ofthe38thConf.onDecisionandContr.,pp.2655–2660,Dec.1999.[5]G.Zhai,B.Hu,K.Yasuda,andA.N.Michel,“Disturbanceattenuationpropertiesoftime-controlledswitchedsystems,”submittedtopublica-tion,2001. Problem4.2ThestatepartitioningproblemofquantisedsystemsJanLunzeInstituteofAutomationandComputerControlRuhr-UniversityBochumD-44780BochumGermanyLunze@esr.ruhr-uni-bochum.de1DESCRIPTIONOFTHEPROBLEMConsideracontinuoussystemwhosestatecanonlybeaccessedthroughaquantizer.Thequantizerisdefinedbyapartitionofthestatespace.Thesystemgeneratesaneventifthesystemtrajectorycrossestheboundarybetweenadjacentpartitions.Theproblemconcernsthepredictionoftheeventsequencegeneratedbythesystemforagiveninitialevent.Astheinitialeventdoesnotdefinetheinitialsystemstateunambiguouslybutonlyrestrictstheinitialstatetoapartitionboundary,whenpredictingthesystembehaviorthebundleofallstatetrajectorieshavetobeconsideredthatstartonthispartitionboundary.Thequestiontobeansweredis:underwhatconditionsonthevectorfieldofthesystemandthestatepartitionistheeventsequenceunique?Inmoredetail,considerthecontinuous-variablesystemx˙=f(x(t)),x(0)=x0(1)withthestatex∈X⊆Rn.ThevectorfieldfsatisfiesaLipschitzconditionsothateqn.(1)has,forallx0∈X,auniquesolution.ThestatespaceXispartitionedintoNdisjointsetsQx(i)(i=1,2,...,N)thatsatisfytheconditions[NX=Qx(i)andQx(i)∩Qx(j)=∅fori6=j.i=1ThesetQ={Qx(i):i=1,2,...,N} THESTATEPARTITIONINGPROBLEMOFQUANTISEDSYSTEMS135iscalledastatequantization.Thequantizedstateisdenotedby[x]anddefinedby[x]=i⇔x∈Qx(i).(2)Thechangeofthequantizedstateiscalledanevent,wheretheeventeijoccursattimet¯iftherelations[x(t¯+δt)]=iand[x(t¯−δt)]=jholdforsmallδt>0.Hence,attimet¯thestatexisontheboundarybetweenthestatepartitionsQx(i)andQx(j)x(t¯)∈δQx(i)∩Qx(j),whereδQxdenotesthehullofQx.Thesystem(1)togetherwiththequan-tizationQiscalledthequantizedsystem.Forgiveninitialstatex0thesystem(1)generates,forthetimeinterval[0,T],auniquestatetrajectoryx(x0,t)and,hence,auniqueeventsequenceE=(e0,e1,...,eH)=Quant(x(x0,t)),whichformallycanberepresentedastheresultoftheoperatorQuantappliedtothestatetrajectory.Histhenumberofeventsgeneratedbythesystemwithinthetimeinterval[0,T].Thefollowingconsiderationsconcernonlythoseinitialeventse0forwhichthequantizedsystemgeneratesaneventsequencewithH>1.Ifinsteadoftheinitialstatex0onlytheinitialevente0=eijisgiven,theinitialstateisonlyknowntolieontheboundaryδQx(i)∩Qx(j)betweenthestatepartitionsQx(i)andQx(j).Consequently,thebundleoftrajectoriesstartinginalltheseinitialstateshavetobeconsidered.ThesetrajectoriesyieldthesetE(e0)={E=Quant(x(x0,t))forx0∈δQx(i)∩Qx(j)}ofeventsequences.IfthesetEhasmorethanoneelement,thequantizedsystemisnondeterministicinthesensethattheknowledgeoftheinitialevente0isnotsufficienttopredictthefutureeventsequenceunambiguously.Ontheotherhand,thequantizedsystemiscalledtobedeterministicifthesetE(e0)isasingletonforallpossibleinitialeventse0.Inordertodefinetheeventsprecisely,thestatepartitionshouldsatisfythefollowingassumptions:A1.Thetrajectoriesdonotlieinthehypersurfacesthatrepresentthepar-titionboundaries.A2.Thesystemcannotgenerateaninfinitenumberofeventsinafinitetimeinterval.A3.Nofix-pointofthevectorfieldflieonapartitionboundary.Theseassumptionscanbesatisfiedbyappropriatelydefiningthestatepar-titionsforthegivenvectorfieldf. 136PROBLEM4.2Statepartitioningproblem.Findconditionsunderwhichthequantisedsystemisdeterministic.Thisproblemcanbereformulatedintwoversions:Problem:Forgivenvectorfieldf,findapartitionofthestatespacesuchthatthequantizedsystemisdeterministic.ProblemB:ForgivenvectorfieldfandastatequantizationQ,testwhetherthequantizedsystemisdeterministic.Bothformulationshavetheirengineeringrelevance.WhereproblemAcon-cernsthepracticalsituationinwhichastatepartitionhastobeselected,problemBreferstothetestofthedeterminismofthesystemforgivenpartition.Theproblemstatedsofaris,possibly,toogeneralintworespects.First,theproblemfortestingthedeterminismofthesystemshouldbeassimpleaspossible.ForagivenpartitionconsistingofNdisjointsets,ProblemBcanbesolvedbyconsideringalltrajectorybundlesthatstartonallpartitionboundaries.Here,thecharacterizationofclassesofvectorfieldsfandparti-tioningmethodsisinterestingforwhichthecomplexityofthetestisconstantorgrowsonlylinearlywithN.Second,forproblemAitisinterestingtofindpartitionsthatcanbedistinguishedwithonlyafewmeasurements.Forexample,rectangularpartitionsareinterestingfromapracticalviewpointwhichresultfromseparatequantizationsofallnstatevariablesxi.Nonautonomoussystems.Theproblemcanbeextendedtononautonomousquantizedsystemsx˙=f(x(t),u(t)),x(0)=x0(3)y(t)=g(x(t),u(t))(4)withinputu∈U⊆Rmandoutputy∈Y⊆Rr.ThefunctionsfandgsatisfyaLipschitzconditionsothateqns.(3),(4)have,forallx0∈Xandu(t),auniquesolution.TheoutputquantizerisdefinedbyapartitionoftheoutputspaceYintothesetsQy(i)wherethequantizedoutput[y]isdefinedanalogouslytoequation(2).TheeventsequenceEisnowdefinedintermsoftheeventsthattheoutputsignalygenerates.Thesystemisconsideredwiththequantizedinput[u].AninjectorassociateswitheachinputauniqueelementofthefinitediscretesetU={u1,u2,...,uM}suchthatu(t)=uiif[u(t)]=i.Again,achangeofthequantizedinputvalueiscalledan(input)event.Itisassumedthattheinputandoutputeventsoccursynchronously.Thisassumptionfixesthetimeinstancesinwhichtheinputchangesitsvalue.Itismotivatedbythefactthatinclosed-loop THESTATEPARTITIONINGPROBLEMOFQUANTISEDSYSTEMS137systemsasupervisordefinesthequantizedinputinthesametimeinstantinwhichanoutputeventoccurs.Herethestatepartitioningproblemincludesalsotodefineanoutputparti-tionandaninputsetUsuchthatthequantizedsystemisdeterministicforallinputsequences.2MOTIVATIONANDHISTORYOFTHEPROBLEMTheproblemresultsfromhybridsystems,whosesimplestformisaconti-nuous-variablesystemwithdiscreteinputs.Manytechnologicalsystemsthatarecontrolledbyprogrammablelogiccontrollers(PLC)haveacontinuousstatespaceandarecontrolledbydiscreteinputs.Thecontrastoftheconti-nousstateandthediscreteinputdoesnotmatterbecausemanysystemsaredesignedinsuchawaythatanyaccessibleinputresultsinanunambiguousstateoroutputevent.Forexample,the(simplified)statespaceofalifthasthestatevariables“vehicleposition,”and“doorposition”bothofwhicharequantizedwherethevehiclepositionreferstothefloorinwhichitstopsandthetwodiscretedoorpositionarecalled“open”or“closed.”Fortheperformanceofthissystem,onlytheeventsareimportant,whichrefertothebeginningandtheendofthepresenceofthevehicleorthedoorinoneofthesepositions.AsthePLCcanonlyswitchonoroff,themotorsofthevehicleorthedooranditisprogrammedsothatthenextcommandisgivenonlyafterthenextoutputeventhasoccurred,everynewinputeventisfollowedbyexactlyoneoutputevent(unlessthesystemisfaulty).So,theliftisacontinuous-variablesystem(3),(4)withquantizedinputandoutput,thatisdeterministic.Inthiscase,thesolutiontothestatepartitioningproblemissimple.Thedeterminismofthequantizedsystemresultsfromthefactthatthesystemtrajectoriesareparalleltothecoordinateaxesofthestatespaceforallaccessibleinputsandthequantizationreferstoseparateintervalsofbothstatevariables.So,theendpointofanymovementinitiatedbyaPLCcommandisapointinthestatespaceandeverytrajectoryoftheclosed-loopsystemresultsinpreciselyoneoutputevent.Inamoregeneralsetting,continuous-variablesystemsaredealtwithasquantizedsystemsforprocesssupervisiontasks.Thenthesystemisnotdesignedtobehavelikeadiscrete-eventsystembuthasacontinuousstatespace.Thequantizersareintroduceddeliberatelytoreducetheinformationtobeprocessed.Forexample,alarmmessagesshowthatacertainsignalhasexceededathreshold.Thestatepartitioningproblemasksfortheachoiceofdiscretesensorssuchthatthesystembehaviorisdeterministic.Asthethirdmotivationforthestatepartitioningproblem,hybridsystemstheoryconcernsdynamicalsystemswithcontinuous-variableanddiscrete-eventsubsystems.Theinterfacesbetweenbothpartsarethequantizerand 138PROBLEM4.2theinjectorintroducedabovethattransformthediscreteoutputsignalofthediscretesubsystemintoareal-valuedinputsignalofthecontinuoussub-systemandviceversa.Theproblemoccursunderwhatconditiontheoverallhybridsystemhasadeterministicinput-outputbehaviorifonlythediscreteinputsandoutputsofthediscretesubsystemareconsidered.Themainsourceofnondeterminismresultsfromthequantizationofthesignalspaceofthecontinuoussubsystem,whichagainleadstothestatepartitioningproblem.Inallthesesituations,thediscretebehaviorofacontinuoussystemisconsid-ered.Intheliteratureonfaultdiagnosisandverificationofdiscretecontrolalgorithmsthehybridnatureoftheclosed-loopsystemisremovedbyusingadiscrete-eventrepresentationofthequantizedsystem.Asinmanypracticalsituationsthequantizerscanbechosen,thestatepartitioningproblemasksforguidelinesofthisselection.Foradeterministicdiscretebehavior,ade-terministicmodelcanbeusedtodescribethequantisedsystem.If,however,thediscretebehaviorisnon-deterministic,anondeterministicmodellikeanondeterministicorstochasticautomatonoraPetrinethastobeused.Severalwaysfordeterminingsuchmodelsforagivenquantizedsystemhavebeenelaboratedrecently([3],[4],[6],[7],[8],[9]).3AVAILABLERESULTSThefirstresultonthestatepartitioningproblemconcernsdiscrete-timesystems(ratherthancontinuous-timesystems)withquantizedstatespace.Reference[4]givesanecessaryandsufficientconditionforthedeterminismofthediscretebehaviorforlinearautonomoussystemswithastatespacepartitionthatregularlydecomposeseachstatevariableintointervalsofthesamesize.In[5]ithasbeenshownhowstatepartitionscanbegeneratedbymappingagiveninitialsetQx(1)bythemodel(1)thatisusedwithreversedtimeaxis.Fortheproblemstatedhere,onlypreliminaryresultsareavailable.Ifthesystemtrajectoriesare,likeintheliftexample,paralleltothecoordinateaxesofthestatespaceandthequantizationboundariesdefinerectangularcellswhoseaxesareparalleltothecoordinateaxes,thediscretebehaviorisdeterministic.Thissituationisencounteredif,forexample,thestatevariablesaredecoupledandcontrolledbyseparateinputs.Hence,themodelcanbedecomposedintox˙i=fi(xi,ui)yi=gi(xi),whichcorrespondsagaintothesimpleliftexample.Anotherexampleisanundampedoscillatorwithastatepartitionthatdecomposesthestatespaceintothetwohalf-planes.Thenthefix-pointliesonthepartitionboundary(and,thus,violatesassumptionA3).However,theoscillatorgenerates,for THESTATEPARTITIONINGPROBLEMOFQUANTISEDSYSTEMS139eachinitialstate,aunique(alternating)eventsequence.Resultsonsymbolicdynamicsarecloselyrelatedtotheproblemstatedhere(cf.[1],[2]).Abundleoftrajectories(orflows)isconsidered,whichgenerateasymbolicoutputifsomepartitionboundaryiscrossed.ThepartitioniscalledMarkovianifalltrajectoriesofthebundlecrossthesamepartitionboundaryand,hence,generatethesamesymbol.Intheterminologyusedthere,theproblemposedhereasksthequestionhowtofindMarkovianpartitions.BIBLIOGRAPHY[1]A.Lasota,M.C.Mackey,Chaos,FractalsandNoise,Springer-Verlag,NewYork,1994.[2]D.Lind,B.Marcus,AnIntroductiontoSymbolicDynamicsandCoding,CambridgeUniversityPress,1995.[3]J.Lunze,“APetrinetapproachtoqualitativemodellingofcontinuousdynamicalsystems,”SystemsAnalysis,Modelling,Simulation,9,pp.885-903,1992.[4]J.Lunze,“Qualitativemodellingoflineardynamicalsystemswithquan-tizedstatemeasurements,”Automatica,30,pp.417-431,1994.[5]J.Lunze,B.Nixdorf,J.Schr¨oder,“Onthenondeterminismofdiscrete-eventrepresentationsofcontinuous-variablessystems,”Automatica,35,pp.395-408,1999.[6]H.A.Preisig,M.J.H.Pijpers,M.Weiss,“Adiscretemodellingprocedureforcontinuousprocessesbasedonstate-discretisation,”2nMATHMOD,Vienna,pp.189-194,1997.[7]J.Raisch,S.O’Young,“Atotallyorderedsetofdiscreteabstractionsforagivenhybridorcontinuoussystem,”pp.342-360In:P.Antsaklis,W.Kohn,A.Nerode,S.Sastry,eds.,HybridSystemsII,Springer-Verlag,Berlin,1995.[8]J.Schr¨oder,Modelling,StateObservationandDiagnosisofQuantisedSystems,Springer-Verlag,Heidelberg,2002.[9]O.Stursberg,S.Kowalewski,S.Engell,“Generatingtimeddiscretemodels,”2nMATHMOD,Vienna,pp.203-207,1997. Problem4.3FeedbackcontrolinflowshopsS.P.SethiSchoolofManagementTheUniversityofTexasatDallasBox830688,MailStationJO4.7Richardson,TX75083USAsethi@utdallas.eduQ.ZhangDepartmentofMathematicsUniversityofGeorgiaAthens,GA30602USAqingz@math.uga.edu1DESCRIPTIONOFTHEPROBLEMConsideramanufacturingsystemproducingasinglefinishedproductusingmmachinesintandemthataresubjecttobreakdownandrepair.Wearegivenafinite-stateMarkovchainα(·)=(α1(·),...,αm(·))onaprobabilityspace(Ω,F,P),whereαi(t),i=1,...,m,isthecapacityofthei-thmachineattimet.Weuseui(t)todenotetheinputratetothei-thmachine,i=1,...,m,andxi(t)todenotethenumberofpartsinthebufferbetweenthei-thand(i+1)-thmachines,i=1,...,m−1.Finally,thesurplusisdenotedbyxm(t).Thedynamicsofthesystemcanthenbewrittenasfollows:x˙(t)=Au(t)+Bz,x(0)=x,(1)wherezistherateofdemandand1−10···0001−1···00A=....andB=........···....000···1−1 FEEDBACKCONTROLINFLOWSHOPS141Sincethenumberofpartsintheinternalbufferscannotbenegative,weimposethestateconstraintsxi(t)≥0,i=1,...,m−1.Toformulatetheproblemprecisely,letS=[0,∞)m−1×(−∞,∞)⊂Rmdenotethestateconstraintdomain.Forα=(α1,...,αm)≥0,letU(α)={u=(u1,...,um):0≤ui≤αi,i=1,...,m},andforx∈S,letU(x,α)={u:u∈U(α);xi=0⇒ui−ui+1≥0,i=1,...,m−1}.LetM={α1,...,αp}foragivenintegerp≥1,whereαj=(αj,...,αj)1mjwithαidenotingthepossiblecapacitystatesofthei-thmachine,i=1,...,m.Lettheσ-algebraFt=σ{α(s):0≤s≤t}.Definition1:Acontrolu(·)isadmissiblewithrespecttotheinitialstatex∈Sandα∈Mif:(i)u(·)is{Ft}-adapted,(ii)u(t)∈U(α(t))forallt≥0,and(iii)thecorrespondingstateprocessx(t)=(x1(t),...,xm(t))∈Sforallt≥0.LetA(x,α)denotethesetofadmissiblecontrols.Definition2:Afunctionu(x,α)iscalledafeedbackcontrol,if(i)foranygiveninitialx,theequation(1)hasauniquesolution;and(ii)u(·)={u(t)=u(x(t),α(t)),t≥0}∈A(x,α).Theproblemistofindanadmissiblecontrolu(·)thatminimizesZ∞J(x,α,u(·))=Ee−ρtG(x(t),u(t))dt,(2)0whereG(x,u)definesthecostofsurplusxandproductionu,αistheinitialvalueofα(t),andρ>0isthediscountrate.WeassumethatG(x,u)≥0isjointlyconvexandlocallyLipschitz.Thevaluefunctionisthendefinedasv(x,α)=infJ(x,α,u(·)).(3)u(·)∈A(x,α)Theoptimalcontrolofthisproblemwasconsideredin[1]usingHJBequa-tionswithdirectionalderivatives.Itisshownthatthereexistsauniqueoptimalcontrol.Inaddition,averificationtheoremassociatedwiththeHJBequationsisobtained.However,theseHJBequationsaredifficulttosolvenumerically,especiallywhenthestatespaceofMislarge.Inthiscase,itisdesirabletoderiveanapproximatesolutioninstead.Weconsiderthecasewhenα(·)jumpsrapidly.Inparticular,weassumeα(t)=αε(t)∈M,t≥0,tobeaMarkovchainwiththegeneratorQε=1Qe+Q,bεwhereQe=(˜qij)andQb=(ˆqij)aregeneratormatricesandQeisweaklyir-reducible.Hereεisasmallparameter.WeusePεtodenoteourcontrolproblem.Asεgetssmallerandsmaller,oneexpectsthatPεapproachestoa 142PROBLEM4.3limitingproblem.Toobtainsuchlimitingproblem,letν=(ν1,...,νp)de-notetheequilibriumdistributionofQe.Weconsidertheclassofdeterministiccontrolsdefinedbelow.Definition3:Forx∈S,letA0(x)denotethesetofthefollowingmeasur-ablecontrolsU(·)=(u1(·),...,up(·))=((u1(·),...,u1(·)),...,(up(·),...,up(·)))1m1mjjsuchthat0≤ui(t)≤αiforallt≥0,i=1,...,nandj=1,...,p,andthecorrespondingsolutionsx(·)ofthesystemXpx˙(t)=Aνjuj(t)+Bz,x(0)=xj=1satisfyx(t)∈Sforallt≥0.TheobjectiveofthelimitingproblemistochooseacontrolU(·)∈A0(x)thatminimizesZ∞XpJ0(x,U(·))=e−ρtνjG(x(t),uj(t))dt.0j=1WeuseP0todenotethelimitingproblemandv0(x)thecorrespondingvaluefunction.2MOTIVATIONANDHISTORYOFTHEPROBLEMItisshownin[1]thatthevaluefunctionvε(x,α)convergestov0(x)asε→0.Thelimitingproblemismucheasiertosolve.Thegoalistousethesolutionofthelimitingproblemtoconstructacontrolfortheoriginalproblemthatisnearlyoptimal.3AVAILABLERESULTSTheideaistouseanoptimal(oranearoptimal)controltoconstructacontrolfortheoriginalproblemPε.ThemaindifficultyishowtoconstructanadmissiblecontrolforPεinawaythatstillguaranteestheasymptoticoptimalityasεgoestozero.Partialresultswereobtainedusinga“lifting”and“modification”approach.Thiswasappliedtoopen-loopcontrols;see[1].Constructionofanasymptoticoptimalfeedbackcontrolremainsopen.Aresolutionofthisproblemwouldperhapsalsoapplytoamorecomplicatedjobshopconsideredin[1]. FEEDBACKCONTROLINFLOWSHOPS143BIBLIOGRAPHY[1]S.P.SethiandQ.Zhang,HierarchicalDecisionMakinginStochasticManufacturingSystems,Birkh¨auser,Boston,1994. Problem4.4DecentralizedcontrolwithcommunicationbetweencontrollersJanH.vanSchuppenCWIP.O.Box94079,1090GBAmsterdamTheNetherlandsJ.H.van.Schuppen@cwi.nl1DESCRIPTIONOFTHEPROBLEMProblem1:DecentralizedcontrolwithcommunicationbetweencontrollersConsideracontrolsystemwithinputsfromrdifferentcontrollers.Eachcontrollerhaspartialobservationsofthesystemandthepartialobservationsofeachpairofcontrollersisdifferent.Thecontrollersareallowedtoexchangeonlineinformationontheirpartialobservations,stateestimates,orinputvalues,butthereareconstraintsonthecommunicationchannelsbetweeneachtupleofcontrollers.Inaddition,thereisspecifiedacontrolobjective.Theproblemistosynthesizercontrollersandacommunicationprotocolforeachdirectedtupleofcontrollers,suchthatwhenthecontrollersallusetheirreceivedcommunicationsthecontrolobjectiveismetaswellaspossible.Theproblemcanbeconsideredforadiscrete-eventsystemintheformofagenerator,foratimeddiscrete-eventsystem,forahybridsystem,forafinite-dimensionallinearsystem,forafinite-dimensionalGaussiansystem,etc.Ineachcase,thecommunicationconstrainthastobechosenandaformulationhastobeproposedonhowtointegratethereceivedcommunicationsintothecontroller.Remarksonproblem(1)Theconstraintsonthecommunicationchannelsbetweencontrollersareessentialtotheproblem.Withoutit,everycontrollercommunicatesallhis/herpartialobservationstoallothercontrollersandoneobtainsacontrolproblemwithacentralizedcontroller,albeitonewhereeachcontrollercarriesoutthesamecontrolcomputations. DECENTRALIZEDCONTROLWITHCOMMUNICATION145(2)Thecomplexityoftheproblemislarge,forcontrolofdiscrete-eventsys-temsitislikelytobeundecidable.Therefore,theproblemformulationhastoberestricted.Notethattheproblemisanalogoustohumancommunicationingroups,firms,andorganizationsandthatthecommunicationproblemsinsuchorganizationsareeffectivelysolvedonadailybasis.Yetthereisscopeforafundamentalstudyofthisproblemalsoforengineeringcontrolsystems.Theapproachtotheproblemisbestfocusedontheformulationandanalysisofsimplecontrollawsandontheformulationofnecessaryconditions.(3)Thebasicunderlyingproblemseemstobe:whatinformationofacon-trollerissoessentialinregardtothecontrolpurposethatithastobecommunicatedtoothercontrollers?Asystemtheoreticapproachissuitableforthis.(4)Theproblemwillalsobeusefulforthedevelopmentofhierarchicalmod-els.Theinformationtobecommunicatedhastobedealtwithatagloballevel,theinformationthatdoesnotneedtobecommunicatedcanbetreatedatthelocallevel.Toassistthereaderwiththeunderstandingoftheproblem,thespecialcasesfordiscrete-eventsystemsandforfinite-dimensionallinearsystemsarestatedbelow.Problem2:Decentralizedcontrolofadiscrete-eventsystemwithcommunicationbetweensupervisorsConsideradiscrete-eventsystemintheformofageneratorandr∈Z+supervisors:G=(Q,E,f,q0),Q,thestateset,q0∈Q,theinitialstate,E,theeventset,f:Q×E→Q,thetransitionfunction,L(G)={s∈E∗|f(q,s)isdefined},0∀k∈Zr={1,2,...,r},apartition,E=Ec,k∪Euc,k,Ecp,k={Ee⊆E|Euc,k⊆Ee},∀k∈Z,apartition,E=E∪E,p:E∗→E∗,∀k∈Z,ro,kuo,kko,kraneventisenabledifitisenabledbyallsupervisors,{vk:pk(L(G))→Ecp,k,∀k∈Zr},thesetofsupervisorsbasedonpartialobservations,Lr,La⊆L(G),requiredandadmissiblelanguage,respectively.Theproblemorbetter,avariantofit,istodetermineasetofsubsetsoftheeventsetthatrepresenttheeventstobecommunicatedbyeachsupervisor 146PROBLEM4.4totheothersupervisorsandasetofsupervisors,∀(i,j)∈Zr×Zr,Eo,i,j⊆Eo,i,pi,j:E→Eo,i,j,thesetofsupervisorsbasedonpartialobservationsandoncommunications,{vk(pk(s),{pj,k(s),∀j∈Z+{k}})7→Ecp,k,∀k∈Zr};issuchthattheclosed-looplanguage,L(v1∧...∧vr/G),satisfiesLr⊆L(v1∧...∧vr/G)⊆La,andthecontrolledsystemisnonblocking.Problem3:Decentralizedcontrolofafinite-dimensionallinearsystemwithcommunicationbetweencontrollersConsiderafinite-dimensionallinearsystemwithr∈Z+inputsignalsandroutputsignals,Xrx˙(t)=Ax(t)+Bkuk(t),x(t0)=x0,k=1Xryj(t)=Cjx(t)+Dj,kuk(t),∀j∈Zr={1,2,...,r},k=1ys,j(t)=Cj(vs,j(t))x(t),whereyj,srepresentsthecommunicationsignalfromControllerstoCon-trollerj,wherevs,jisthecontrolinputofControllersforthecommunicationtoControllerj,andwherethedimensionsofthestate,theinputsignals,theoutputsignals,andofthematriceshavebeenomitted.Theithcontrollerobservesoutputyiandprovidestothesysteminputui.SupposethatCon-troller2communicatessomecomponentsofhisobservedoutputsignaltoController1.Canthesystemthenbestabilized?Howmuchcanaquadraticcostbeloweredbydoingso?Theproblembecomesdifferentifthecom-municationsfromController2toController1arenotcontinuousbutarespacedperiodicallyintime.Howshouldtheperiodbechosenforstabilityorforacostminimization?Theperiodwillhavetotakeaccountofthefeedbackachievabletimeconstantsofthesystem.Afurtherrestrictiononthecommunicationchannelistoimposethatmessagescancarryatmostafinitenumberofbits.Thenquantizationisrequired.Forarecentworkonquantizationinthecontextofcontrolsee,[17].2MOTIVATIONTheproblemismotivatedbycontrolofnetworks:forexample,ofcommuni-cationnetworks,oftelephonenetworks,oftrafficnetworks,firmsconsistingofmanydivisions,etc.Controloftrafficontheinternetisaconcreteexam-ple.Insuchnetworks,therearelocalcontrollersatthenodesofthenetwork, DECENTRALIZEDCONTROLWITHCOMMUNICATION147eachhavinglocalinformationaboutthestateofthenetworkbutnoglobalinformation.Decentralizedcontrolisusedbecauseitistechnologicallydemandingandeconomicallyexpensivetoconveyallobservedinformationstoothercon-trollers.Yetitisoftenpossibletocommunicateinformationatacost.Thisviewpointhasnotbeenconsideredmuchincontroltheory.Inthetrade-off,theeconomiccostsofcommunicationhavetobecomparedwiththegainsforthecontrolobjectives.Thiswasalreadyremarkedoninthecontextofteamtheoryalongtimeago.Butthishasnotbeenusedincontroltheorytillrecently.Thecurrenttechnologicaldevelopmentsmakethecommunica-tionrelativelycheapandthereforethetrade-offhasshiftedtowardtheuseofmorecommunication.3HISTORYOFTHEPROBLEMThedecentralizedcontrolproblemwithcommunicationbetweensupervisorswasformulatedbytheauthorofthispaperaround1995.Theplanforthisproblemisolder,though,buttherearenowrittenrecords.WithKaiC.Wonganecesaryandsufficientconditionwasderived(see[20])forthecaseoftwocontrollerswithasymmetriccommunication.Theaspectoftheproblemthatasksfortheminimalinformationtobecommunicatedwasnotsolvedinthatpaper.Subsequentresearchhasbeencarriedoutbymanyresearchersincontrolofdiscrete-eventsystems,includingGeorgeBarrett,ReneBoel,RamiDebouk,StephaneLafortune,LaurieRicker,KarenRudie,DemosTeneketzis;see[1,2,3,4,5,11,12,13,14,15,16,19].Besidesthecontrolproblem,thecorrespondingproblemforfailurediagnosishasalsobeenanalyzed;see[6,7,8,9].Theproblemforfailurediagnosisissimplerthanthatforcontrolduetothefactthatthereisnorelationofthediagnosingviatheinputtothefutureobservations.Theproblemfortimeddiscrete-eventsystemshasbeenformulatedalsobecauseincommunicationnetworkstimedelaysduetocommunicationneedtobetakenintoaccount.Therearerelationsoftheproblemwithteamtheory;see[10].Therearealsorelationswiththeasymptoticagreementproblemindistributedesti-mation;see[18].TherearealsorelationsoftheproblemtographmodelsandBayesianbeliefnetworkswherecomputationsforlargescalesystemsarecarriedoutinadecentralizedway.4APPROACHSuggestionsfollowforthesolutionoftheproblem.Approachesare:(1)Ex-plorationofsimplealgorithms.(2)Developmentoffundamentalpropertiesofcontrollaws. 148PROBLEM4.4AnexampleofasimplealgorithmistheIEEE802.11protocolforwirelesscommunication.Theprotocolprescribesstationswhentheycantransmitandwhennot.Allstationsareincompetitionwitheachotherfortheavail-ablebroadcastingtimeonaparticularfrequency.Theprotocoldoesnothaveatheoreticalanalysisandwasnotdesignedviaacontrolsynthesispro-cedure.Yetitisabeautifulexampleofadecentralizedcontrollawwithcommunicationbetweensupervisors.Thealternatingbitprotocolisanotherexample.Inarecentpaper,S.Morsehasanalyzedanotheralgorithmfordecentralizedcontrolwithcommunicationbasedonamodelforaschooloffishes.Amorefundamentalstudywillhavetobedirectedatstructuralproperties.DecentralizedcontroltheoryisbasedontheconceptofNashequilibriumfromgametheoryandontheconceptofperson-by-personoptimalityfromteamtheory.Thecomputationofanequilibriumisdifficultbecauseitisthesolutionofafixpointequationinfunctionspace.However,propertiesofthecontrollawmaybederivedfromtheequilibriumequation,asisroutinelydoneforoptimalcontrolproblems.Considerthentheproblemforaparticularcontroller:itregardsasthecom-binedsystemtheplantwiththeothercontrollersbeingfixed.Thecontrollerthenfacestheproblemofdesigningacontrollawforthecombinedsystem.However,duetocommunicationwithothersupervisors,itcaninadditionselectcomponentsofthestatevectorofthecombinedsystemforitsownobservationprocess.Aquestiontheniswhichcomponentstoselect.Thisapproachleadstoasetofequations,which,combinedwiththoseforothercontrollers,havetobesolved.Specialcasesofwhichthesolutionmaypointtogeneralizationsarethecaseoftwocontrollerswithasymmetriccommunicationandthecaseofthreecontrollers.Forlargernumberofcontrollersgraphtheorymaybeexploitedbutitislikelythatsimplealgorithmswillcarrytheday.Constraintscanbeformulatedintermsofinformation-likequantitiesasin-formationrate,butthisseemsmostappropriatefordecentralizedcontrolofstochasticsystems.Constraintscanalsobebasedoncomplexitytheoryasdevelopedincomputerscience,wherecomputationsarecounted.Thiscasecanbeextendedtocountingbitsofinformation.BIBLIOGRAPHY[1]G.Barrett,Modeling,AnalysisandControlofCentralizedandDe-centralizedLogicalDiscrete-EventSystems,Ph.D.thesis,UniversityofMichigan,AnnArbor,2000.[2]G.BarrettandS.Lafortune,“Anovelframeworkfordecentralizedsu-pervisorycontrolwithcommunication,”In:Proc.1998IEEESystems,Man,andCyberneticsConference,NewYork,IEEEPress,1998. DECENTRALIZEDCONTROLWITHCOMMUNICATION149[3]G.BarrettandS.Lafortune,“Onthesynthesisofcommunicatingcon-trollerswithdecentralizedinformationstructuresfordiscrete-eventsys-tems,”In:ProceedingsIEEEConferenceonDecisionandControl,pp.3281–3286,NewYork,IEEEPress,1998.[4]G.BarrettandS.Lafortune,“Someissuesconcerningdecentralizedsupervisorycontrolwithcommunicatio,”In:Proceedings38thIEEEConferenceonDecisionandControl,pp.2230–2236,NewYork,IEEEPress,1999.[5]G.BarrettandS.Lafortune,“Decentralizedsupervisorycontrolwithcommunicatingcontrollers,”IEEETrans.AutomaticControl,45:1620–1638,2000.[6]R.K.BoelandJ.H.vanSchuppen,“Decentralizedfailurediagnosisfordiscrete-eventsystemswithcostlycommunicationbetweendiagnosers,”In:ProceedingsofInternationalWorkshoponDiscreteEventSystems(WODES2002),pp.175–181,LosAlamitos,IEEEComputerSociety,2002.[7]R.Debouk,Failurediagnosisofdecentralizeddiscrete-eventsystems,Ph.D.thesis,UniversityofMichigan,AnnArbor,2000.[8]R.Debouk,S.Lafortune,andD.Teneketzis,“Coordinateddecentral-izedprotocolsforfailurediagnosisofdiscrete-eventsystems,”DiscreteEventDynamicsSystems,10:33–86,2000.[9]R.Debouk,S.Lafortune,andD.Teneketzis,“Coordinateddecentral-izedprotocolsforfailurediagnosisofdiscreteeventsystems,”ReportCGR-97-17,CollegeofEngineering,UniversityofMichcigan,AnnAr-bor,1998.[10]R.Radner,“Allocationofascarceresourceunderuncertainty:Anex-ampleofateam,”In:C.B.McGuireandR.Radner,eds,Decisionandorganization,pp.217–236.North-Holland,Amsterdam,1972.[11]S.RickerandK.Rudie,“Knowmeansno:Incorporatingknowledgeintodecentralizeddiscrete-eventcontrol,”In:Proc.1997AmericanControlConference,1997.[12]S.L.Ricker,KnowledgeandCommunicationinDecentralizedDiscrete-EventControl,Ph.D.thesis,Queen’sUniversity,DepartmentofCom-putingandInformationScience,August1999.[13]S.L.RickerandG.Barrett,“Decentralizedsupervisorycontrolwithsingle-bitcommunications,”In:ProceedingsofAmericanControlCon-ference(ACC01),pp.965–966,2001. 150PROBLEM4.4[14]S.L.RickerandK.Rudie,“Incorporatingcommunicationandknowl-edgeintodecentralizeddiscrete-eventsystems,”In:Proceedings38thIEEEConferenceonDecisionandControl,pp.1326–1332,NewYork,IEEEPress,1999.[15]S.L.RickerandJ.H.vanSchuppen,“Asynchronouscommunicationintimeddiscreteeventsystems,”In:ProceedingsoftheAmericanControlConference(ACC2001),pp.305–306,2001.[16]S.L.RickerandJ.H.vanSchuppen,“Decentralizedfailurediagnosiswithasynchronuouscommunicationbetweensupervisors,”In:Proceed-ingsoftheEuropeanControlConference(ECC2001),pp.1002–1006,2001.[17]S.C.Tatikonda,Controlundercommunicationconstraints,Ph.D.the-sis,DepartmentofElectricalEngineeringandComputerScience,MIT,Cambridge,MA,2000.[18]D.TeneketzisandP.Varaiya,“Consensusindistributedestimationwithinconsistentbeliefs,”Systems&ControlLett.,4:217–221,1984.[19]J.H.vanSchuppen,“Decentralizedsupervisorycontrolwithinforma-tionstructures,”In:ProceedingsInternationalWorkshoponDiscreteEventSystems(WODES98),pp.36–41,London,IEE,1998.[20]K.C.WongandJ.H.vanSchuppen,“Decentralizedsupervisorycontrolofdiscrete-eventsystemswithcommunication,”In:ProceedingsInter-nationalWorkshoponDiscreteEventSystems1996(WODES96),pp.284–289,London,IEE,1996. PART5DistributedParameterSystems Problem5.1InfinitedimensionalbacksteppingfornonlinearparabolicPDEsAndrasBaloghandMiroslavKrsticDepartmentofMAEUniversityofCaliforniaatSanDiegoLaJolla,CA92093–0411USAabalogh@ucsd.eduandkrstic@ucsd.edu1INTRODUCTIONThisnoteexploresanapproachtoglobalstabilizationofboundarycon-trollednonlinearPDEsbyatechniqueinspiredbyfinitedimensionalback-stepping/feedbacklinearization.Solutionoftheproblempresentedhereinwouldbeofenormoussignificancebecausethesearetheonlytrulyconstruc-tiveandsystematictechniquesinfinitedimension.WeconsidernonlinearparabolicPDEsoftheformut(x,t)=εuxx(x,t)+f(u(x,t))(1)forx∈(0,1),t>0,withboundaryconditionsu(0,t)=0,(2)u(1,t)=α1(u),(3)initialconditionu(x,0)=u0(x),x∈[0,1],andundertheassumptionε>0,f∈C∞(R).1(4)Thetaskistoderiveanonlinear(feedback)functionalα1:C([0,1])→Rthatstabilizesthetrivialsolutionu(x,t)≡0inanappropriateway.Anin-finitedimensionalversionofbacksteppingwasintroducedin[2]thatsolves1Thesmoothnessrequirementisexplainedafterformula(18). 154PROBLEM5.1thisproblemforf(u)=λuwithλ>0arbitrarilylarge.Superlinearnonlin-earitiescanimplyfinitetimeblow–upfortheuncontrolledcase[6,7,9,10].However,numericalresultsinaseriesofpapersbyBoskovicandKrstic[3,4,5]showpromisefortheapplicabilityoftheinfinitedimensionalback-steppingtononlinearproblems,atleastforfinite–griddiscretizations.Inthisnote,wepresenttheopenproblemofconvergenceofnonlinearback-steppingschemesasthediscretizationgridbecomesinfinitelyrefined.Notethatthisproblemisdifferentfromthequestionofcontrollability[1,8].2BACKSTEPPINGTRANSFORMATIONWelookforacoordinatetransformationoftheformw=u−α(u),(5)whereα:C([0,1])→C([0,1])isanonlinearoperatortobefound,thattransformssystem(1)–(3)intotheexponentiallystablesystemwt(x,t)=εwxx(x,t),x∈(0,1),t>0,(6)withboundaryconditionsw(0,t)=0,(7)w(1,t)=0.(8)Oncetransformation(5)isfound,itisrealizedthroughthestabilizingbound-aryfeedbackcontrol(3)withα1(u)=α(u)|x=1.Inordertofind(5)inaconstructiveway,wefirstdiscretizeinspace(1)–(3),thenwedevelopastabilizingcoordinatetransformationforthesemi–discretizedsystem.Themainquestionofshowingthatthediscretizationconvergestoaninfinitedimensionaltransformationisopeninthecaseoffunctionsf(u)thatarenonlinear.Wedefineun=u(ih,t)fori,j=0,1,...,n+1,n=1,2,...whereh=i1/(n+1),andthefinitedifferencediscretizationoftherestofthefunctionsisdefinedthesameway.Thediscretizedversionofcoordinatetransformation(5)nowhastheformwn=(I−αn)(un)n=1,2,...(9)whereαnisann–vectorvaluedfunctionofunandTwn=wn,wn,...,wn,(10)01n+1Tun=un,un,...,un.(11)01n+1Thediscretizedformofsystem(1)–(3)isun=0,(12)0un−2un+unu˙n=εi+1ii−1+f(un),i=1,...,n,(13)ih2iun=αn(un,un,...,un).(14)n+1n12n BACKSTEPPINGFORNONLINEARPDES155withtheconventionofαn=0.Thediscretizedformofsystem(6)–(8)is0wn=0,(15)0wn−2wn+wnni+1ii−1w˙i=ε2,i=1,2,...,n,(16)hwn=0.(17)n+1Combining(16),(9)and(13),andsolvingforαn,weobtainthefinalformioftherecursiveformulaforthetransformation:h2αn=−f(un)+2αn−αniii−1i−2εXi−1n2
∂αi−1nnnhn+uj+1−2uj+uj−1+fuj(18)∂ujεj=1fori=1,2,...,n.Thisrecursiveformulacontainsthefunctionsf(u)(whichisnonlinearingeneral,)anditinvolvesdifferentiation.Asaresult,asn→∞,eventuallyinfinitesmoothnessofthefunctionfisrequired.Afewvaluesofαn:iαn=0,(19)0h2αn=−f(un),(20)11εh2h2h2h2αn=−f(un)−2f(un)−f0(un)un−2un+f(un),(21)2211211εεεεh2h2h2αn=−f(un)−2f(un)−3f(un)3321εεεh2h2−2f0(un)un−2un+f(un)1211εε2!h2h2h2+−f00(un)un−2un+f(un)−f0(un)·12111εεεh2·un−2un+f(un)211εh2h2h2−f0(un)+f0(un)un−2un+f(un)(22)21322εεε3OPENPROBLEMUsingtheabovebacksteppingapproach,theproblemoffindingthecoordi-natetransformation(5)andthecorrespondingstabilizingboundarycontrol(3)requirestwosteps. 156PROBLEM5.11.Findassumptionsonthenonlinearfunctionfthatensurestheconver-genceofthediscretizedcoordinatetransformation(18)toa(nonlinear)operatorαinordertoobtainthefeedbackboundarycontrollaw(5).2.EstablishtheboundedinvertibilityofoperatorI−α(seeequation(5))inappropriatefunctionspaces.4KNOWNLINEARRESULTForthelinearcasef(u)=λuwehavethefollowingresult[2].Theorem1:Foranyλ∈Randε,c>0thereexistsafunctionk1∈L∞(0,1)suchthatforanyu
0∈L∞(0,1)theuniqueclassicalsolutionu(x,t)∈C1(0,∞);C2(0,1)ofsystem(1)–(3)withboundaryfeedbackcontrolZ1α1(u)=k1(ξ)u(ξ,t)dξ(23)0isexponentiallystableintheL2(0,1)andmaximumnormswithdecayratec.Theprecisestatementsofstabilitypropertiesarethefollowing:thereexistsapositiveconstantM2suchthatforallt>0ku(t)k≤Mkuke−ct(24)0andmax|u(t,x)|≤Msup|u(x)|e−ct.(25)0x∈[0,1]x∈[0,1]Inthislinearcase,thetransformationisaboundedlinearoperatorα:RxL1→L1intheformofα(u)=0k(x,ξ)u(ξ)dξwithintegralkernelk∈R1L∞([0,∞]×[0,∞]).Theboundarycontrolisα1(u)=0k(1,ξ)u(ξ)dξ.TheexplicitformofαiisXiαn=knun,i=1,...,n,(26)ii,jjj=1where!j+1ni(c+λ)ki,i−j=−j+12ε(n+1)[j/2]!j−2l+1X1j−li−l(c+λ)−(i−j)(27)ll−1j−2lε(n+1)2l=1fori=1,...,n,j=1,...,i.2Mgrowswithc,λand1/ε. BACKSTEPPINGFORNONLINEARPDES1575NUMERICALRESULTSInthenonlinearcase,weneedatleasttheuniformboundednessofsequences{αn(u)}n⊂Rasn→∞forallufromsomereasonablefunctionspace.ii=1WeusedMathematicaandMuPADtocalculateαn(u)symbolicallyusingntherecursiverelationship(18)andthentoevaluateitforseveraldifferentfunctionsu(x)andfordifferentnonlinearfunctionsf(u).Sincewefoundnoqualitativedifferencebetweenresultscorrespondingtofunctionsu(x)ofthesamesize,wepresenthereonlytheresultsforfunctionsoftheformu(x)=psin(πx)withdifferentvaluesofp.Thesymboliccalculationbe-comesextremelydemandingcomputationallyforincreasingvaluesofn.Wewereabletoevaluateαnforvaluesupton=9orn=10dependingonthencomplexityofthenonlinearfunctionf(u).Theresultsarecollectedbelowintwotables.2
f(u)u→∞1.Inthecaseoff(u)=uln1+u,wehavesuperlinearity−−−−→Ru∞du∞,butthecondition<∞,whichisnecessaryforfinitetimebf(u)blowup(see,e.g.,[9])isnotsatisfiedforanyb>0.Also,thezerosolutionofequation(1)islocallystable.Thevaluep=1.5correspondstoaninitialvalueforwhichtheopen–loopsolutionconvergestozero.Asthecorrespondingcolumninthetablebelowshows,thecontroloperatorαnconvergestoafinitevalue.Forp=2theuncontrollednsolutionof(1)doesnotconvergetozero,butstillαnconvergestoanfinitevalue.Forlargervaluesofp,theconvergenceisnotobviousfromthecalculations,buttheconcavityofthefunctiongraphs(decreasingratesofchangeinthevaluesofαn)suggestthatwehaveconvergencenforincreasingvaluesofnwithadecreasingrateofconvergenceasthesizeoftheinitialfunctionisincreased.
αnforf(u)=uln1+u2nnp=1.5p=2p=5p=101−4.4−8.0−40.7−115.32−4.5−11.0−97.4−356.23−4.4−11.6−141.1−615.14−4.3−12.3−178.4−867.15−4.3−12.6−209.0−1099.16−4.2−12.8−233.4−1301.57−4.2−13.0−252.5−1472.68−4.2−13.1−267.6−1615.49−4.2−13.2−279.5−1733.62.Forf(u)=u2solutionscorrespondingtolargeinitialdataexhibitfinitetimeblow–up.Infact,allofthepresentpvaluescorrespondtoinitialfunctionsthatresultinfinitetimeblow–up.However,forp=1.5 158PROBLEM5.1andp=2,thecontrolvaluesseemtoconvergeasthetablebelowshows.Forlargervalues(p=5andp=10),numericalcalculationssuggestfastdivergence.αnforf(u)=u2nnp=1.5p=2p=5p=101−5.6−10.0−62.5−250.02−7.2−16.1−221.3−1687.03−7.6−18.6−402.0−4974.24−8.0−21.1−637.3−11202.15−8.2−22.6−926.7−22798.36−8.3−23.8−1244.8−41999.67−8.3−24.6−1578.1−70862.28−8.4−25.3−1915.4−111498.49−8.4−25.8−2247.4−165709.210−8.5−26.1−2567.5−234811.7BIBLIOGRAPHY[1]S.AnitaandV.Barbu,“Nullcontrollabilityofnonlinearconvectiveheatequations,”ESAIMCOCV,vol.5,pp.157–173,2000.[2]A.BaloghandM.Krstic,“Infinitedimensionalbackstepping–stylefeed-backtransformationsforaheatequationwithanarbitrarylevelofin-stability,”EuropeanJournalofControl,2002.[3]D.BoskovicandM.Krstic,“Nonlinearstabilizationofathermalcon-vectionloopbystatefeedback,”Automatica,vol.37,pp.2033-2040,2001.[4]D.BoskovicandM.Krstic,“Backsteppingcontrolofchemicaltubularreactors,”ComputersandChemicalEngineering,vol.26,pp.1077-1085,2002.[5]D.BoskovicandM.Krstic,“Stabilizationofasolidpropellantrocketinstabilitybystatefeedback,”Int.J.RobustandNonlinearControl,inpress.[6]L.A.CaffarrelliandA.Friedman,“Blow–upofsolutionsofnonlinearheatequations,”J.Math.Anal.Appl.,129,pp.409–419,1988.[7]M.Chipot,M.Fila,andP.Quittner,“Stationarysolutions,blowupandconvergencetostationarysolutionsforsemilinearparabolicequationswithnonlinearboundaryconditions,”ActaMath.Univ.Comenianae,vol.LX,no.1,pp.35–103,1991. BACKSTEPPINGFORNONLINEARPDES159[8]E.Fernandez–Cara,“Nullcontrollabilityofthesemilinearheatequa-tion,”ESAIMCOCV,vol.2,pp.87–103,1997.[9]A.A.Lacey,“Mathematicalanalysisofthermalrunawayforspatiallyinhomogeneousreactions,”SIAMJ.Appl.Math.,vol.43,no.6,pp.1350–1366,1983.[10]S.Seo,“Blowupofsolutionstoheatequationswithnonlocalboundaryconditions,”KobeJ.Math.,vol.13,pp.123–132,1996. Problem5.2ThedynamicalLamesystemwithboundarycontrol:onthestructureofreachablesetsM.I.Belishev1Dept.oftheSteklovMathematicalInstitute(POMI)Fontanka27St.Petersburg191011Russiabelishev@pdmi.ras.ru1MOTIVATIONThequestionsposedbelowcomefromdynamicalinverseproblemsforthehyperbolicsystemswithboundarycontrol.ThesequestionsariseintheframeworkoftheBC–method,whichisanapproachtoinverseproblemsbasedontheirrelationstotheboundarycontroltheory[1],[2].2GEOMETRYLetΩ⊂R3beaboundeddomainwiththesmooth(enough)boundaryΓ;λ,µ,ρsmoothfunctions(Lameparameters)satisfyingρ>0,µ>0,3λ+2µ>0inΩ.¯TheparametersdeterminetwometricsinΩ¯|dx|2dl2=,α=p,sαc2αwhere11λ+2µ2µ2cp:=,cs:=ρρarethevelocitiesofp−(pressure)ands−(shear)waves;letdistαbethecorrespondingdistances.1SupportedbytheRFBRgrantNo.02-01-00260. THEDYNAMICALLAMESYSTEMWITHBOUNDARYCONTROL161Thedistantfunctions(eikonals)τα(x):=distα(x,Γ),x∈Ω¯determinethesubdomainsΩT:={x∈Ω|τ(x)0ααandthevaluesT:=inf{T>0|ΩT=Ω},whicharethetimesittakesforααα–wavesmovingfromΓtofillthewholeofΩ.TherelationcsT.IfT0is’nottoolarge’,thevectorfields∇τανα:=|∇τα|areregularandsatisfyν(x)·ν(x)>0,x∈ΩT.Duetothelatter,eachpspvectorfield(R3−valuedfunction)u=u(x)mayberepresentedintheformu(x)=u(x)+u(x),x∈ΩT(∗)pspwithu(x)pkνp(x)andu(x)s⊥νs(x).3LAMESYSTEM.CONTROLLABILITYConsiderthedynamicalsystemX3u=ρ−1∂c∂u(i=1,2,3)inΩ×(0,T);ittjijkllkj,k,l=1u|t=0=ut|t=0=0inΩ;u=fonΓ×[0,T],(∂:=∂)wherecistheelasticitytensoroftheLamemodel:j∂xjijklcijkl=λδijδkl+µ(δikδjl+δilδjk);letu=uf(x,t)={uf(x,t)}3bethesolution(wave).ii=1DenoteH:=L(Ω;R3)(withmeasureρdx);HT:={y∈H|suppy⊂2,ραΩ¯T}.Aswasshownin[3],themapf7→ufiscontinuousfromL(Γ×α2[0,T];R3)intoC([0,T];H).Byvirtueofthisandduetothefinitenessofthewavevelocities,thereachablesetUT:={uf(·,T)|f∈L(Γ×[0,T];R3)}2isembeddedintoHT.Aswasprovedinthesamepaper,therelationpclosUT⊃HTsisvalidforanyT>0,i.e.,anapproximatecontrollabilityalwaysholdsinthesubdomainΩTfilledwiththeshearwaves,whereastheelementsofthesdefectsubspaceNT:=HTclosUTpH(‘unreachablestates’)canbesupportedonlyin∆ΩT.Ontheotherhand,itisnotdifficulttoshowtheexampleswithNT6={0},T0andΓ⊂∂Ω,0anopennon-emptysubsetoftheboundaryofΩ,weconsiderthelinearheatequation:ut−∆u=0inQu=v1Σ0onΣ(1)u(x,0)=u0(x)inΩ,whereQ=Ω×(0,T),Σ=∂Ω×(0,T)andΣ0=Γ0×(0,T)andwhere1Σ0denotesthecharacteristicfunctionofthesubsetΣ0ofΣ.In(1)v∈L2(Σ)isaboundarycontrolthatactsonthesystemthroughthesubsetΣ0oftheboundaryandu=u(x,t)isthestate.System(1)issaidtobenull-controllableattimeTifforanyu∈L2(Ω)0thereexistsacontrolv∈L2(Σ)suchthatthesolutionof(1)satisfies0u(x,T)=0inΩ.(2)Thisarticleisconcernedwiththenull-controllabilityproblemof(1)whenthedomainΩisunbounded. 164PROBLEM5.32MOTIVATIONANDHISTORYOFTHEPROBLEMWebeginwiththefollowingwell-knownresultTheorem1.WhenΩisaboundeddomainofclassC2system(1)isnull-controllableforanyT>0.WerefertoD.L.Russell[12]forsomeparticularexamplestreatedbymeansofmomentproblemsandFourierseriesandtoA.FursikovandO.Yu.Imanuvilov[3]andG.LebeauandL.Robbiano[7]forthegeneralresultcoveringanyboundedsmoothdomainΩandopen,nonemptysubsetΓ0of∂Ω.Boththeapproachesof[3]and[7]arebasedontheuseofCarlemaninequalities.However,inmanyrelevantproblemsthedomainΩisunbounded.Weaddressthefollowingquestion:ifΩisanunboundeddomain,issystem(1)null-controllableforsomeT>0?.Noneoftheapproachesmentionedaboveapplyinthissituation.Infact,veryparticularcasesbeingexcepted(seethefollowingsection),thereexistnoresultsonthenull-controllabilityoftheheatequation(1)whenΩisunbounded.Theapproachdescribedin[6]and[9]isalsoworthmentioning.Inthisarticleitisprovedthat,foranyT>0,theheatequationhasafundamentalsolutionthatisC∞awayfromtheoriginandwithsupportinthestrip0≤t≤T.Thisfundamentalsolution,ofcourse,growsveryfastas|x|goestoinfinity.Asaconsequenceofthis,aboundarycontrollabilityresultmaybeimmediatelyobtainedinanydomainΩwithcontrolsdistributedallalongitsboundary.Note,however,thatwhenthedomainisunboundedthesolutionsandcontrolsobtainedinthiswaygrowtoofastas|x|→∞and,therefore,thesearenotsolutionsintheclassicalsense.Infact,intheframeofunboundeddomains,onehastobeverycarefulindefiningtheclassofadmissiblecontrolledsolutions.Whenimposing,forinstance,theclassicalintegrabilityconditionsatinfinity,oneisimposingadditionalrestrictionsthatmaydeterminetheanswertothecontrollabilityproblem.Thisisindeedthecase,asweshallexplain.Thereisaweakernotionofcontrollabilityproperty.Itistheso-calledap-proximatecontrollabilityproperty.System(1)issaidtobeapproximatelycontrollableintimeTifforanyu∈L2(Ω)thesetofreachablestates,0R(T;u)={u(T):usolutionof(1)withv∈L2(Σ)},isdenseinL2(Ω).00Withtheaidofclassicalbackwarduniquenessresultsfortheheatequation(see,forinstance,J.L.LionsandE.Malgrange[8]andJ.M.Ghidaglia[4]),itcanbeseenthatnull-controllabilityimpliesapproximatecontrollability.Theapproximatecontrolproblemforthesemilinearheatequationingeneralunboundeddomainswasaddressedin[13]whereanapproximationmethodwasdeveloped.ThedomainΩwasapproximatedbyboundeddomains(es-sentiallybyΩ∩BR,BRbeingtheballofradiusR)andtheapproximatecontrolintheunboundeddomainΩwasobtainedaslimitoftheapproximate NULL-CONTROLLABILITYOFTHEHEATEQUATION165controlontheapproximatingboundeddomainΩ∩BR.Butthisapproachdoesnotapplyinthecontextofthenull-controlproblem.However,takingintoaccountthatapproximatecontrollabilityholds,itisnaturaltoanalyzewhethernull-controllabilityholdsaswell.In[1]itwasprovedthatthenull-controllabilitypropertyholdseveninun-boundeddomainsifthecontrolissupportedinasubdomainthatonlyleavesaboundedsetuncontrolled.Obviously,thisresultisveryclosetothecaseinwhichthedomainΩisboundedanddoesnotanswertothemainissueunderconsiderationofwhetherheatprocessesarenull-controllableinunboundeddomains.3AVAILABLERESULTSToourknowledge,inthecontextofunboundeddomainsΩandtheboundarycontrolproblem,onlytheparticularcaseofthehalf-spacehasbeenconsid-ered:Ω=Rn={x=(x0,x):x0∈Rn−1,x>0}+nn(3)Γ=∂Ω=Rn−1={(x0,0):x0∈Rn−1}0(see[10]forn=1and[11]forn>1).Accordingtotheresultsin[10]and[11],thesituationiscompletelydifferenttothecaseofboundeddomains.InfactasimpleargumentshowsthatthenullcontrollabilityresultwhichthatholdsforthecaseΩboundedisnolonger
true.Indeed,thenull-controllabilityof(1)withinitialdatainL2Rnand+boundarycontrolinL2(Σ)isequivalenttoanobservabilityinequalityfortheadjointsystemϕt+∆ϕ=0onQ(4)ϕ=0onΣ.Moreprecisely,itisequivalenttotheexistenceofapositiveconstantC>0suchthatZ2∂ϕkϕ(0)k2≤Cdx0dt(5)L2(Rn)+Σ∂xnholdsforeverysmoothsolutionof(4).WhenΩisbounded,Carlemaninequalitiesprovidetheestimate(5)and,consequently,null-controllabilityholds(see,forinstance,[3]).Inthecaseofahalf-space,byusingatranslationargument,itiseasytoseethat(5)doesnothold(see[11]).Inthecaseofboundeddomains,byusingFourierseriesexpansion,thecon-trolproblemmaybereducedtoamomentproblem.However,FourierseriescannotbeuseddirectlyinRn.Nevertheless,itwasobservedbyM.Es-+cobedoandO.Kavianin[2]that,onsuitablesimilarityvariablesandatthe 166PROBLEM5.3appropriatescale,solutionsoftheheatequationonconicaldomainsmaybeindeeddevelopedinFourierseriesonaweightedL2−space.Thisideawasusedin[10]and[11]tostudythenull-controllabilitypropertywhenΩisgivenby(3).Firstly,weusesimilarityvariablesandweightedSobolevspacestodevelopthesolutionsinFourierseries.Asequenceofone-dimensionalcontrolledsystemslikethosestudiedin[10]isobtained.Eachofthesesystemsisequivalenttoamomentproblemofthefollowingtype:givenS>0and(an)n≥1(dependingontheFouriercoefficientsoftheinitialdatau0)findf∈L2(0,S)suchthatZSf(s)ensds=a,∀n≥1.(6)n0Thismomentproblemturnsouttobecriticalsinceitconcernsthefamilyofrealexponentialfunctions{eλns}withλ=n,inwhichtheusualn≥1nPsummabilityconditionontheinversesoftheexponents,1<∞,n≥1λndoesnothold.Itwasprovedthat,ifthesequence(an)n≥1hasthepropertythat,foranyδ>0,thereexistsCδ>0suchthat|a|≤Ceδn,∀n≥1,(7)nδproblem(6)hasasolutionifandonlyifan=0foralln≥1.Since(an)n≥1dependontheFouriercoeficientsoftheinitialdata,thefol-lowingnegativecontrollabilityresultfortheone-dimensionalsystemsisob-tained:Theorem2.WhenΩisthehalfline,thereisnonontrivialinitialdatumu0belongingtoanegativeSobolevspacethatisnull-controllableinfinitetimewithL2boundarycontrols.Thisnegativeresultwascomplementedbyshowingthatthereexistini-tialdatawithexponentiallygrowingFouriercoefficientsforwhichnull-controllabilityholdsinfinitetimewithL2−controls.Wementionthatin[10]and[11]wearedealingwithsolutionsdefinedinthesenseoftransposition,andthereforethesolutionsin[6]and[9]thatgrowandoscillateveryfastatinfinityareexcluded.4OPENPROBLEMSAswehavealreadymentioned,thenull-controllabilitypropertyof(1)whenΩisunboundedanddifferentfromahalf-lineorhalf-spaceisstillopen.Theapproachbasedontheuseofthesimilarityvariablesmaystillbeusedingeneralconicaldomains.But,duetothelackoforthogonalityofthetracesofthenormalderivativesoftheeigenfunctions,thecorrespondingmomentproblemismorecomplexandremainstobesolved. NULL-CONTROLLABILITYOFTHEHEATEQUATION167WhenΩisageneralunboundeddomain,thesimilaritytransformationdoesnotseemtobeofanyhelpsincethedomainonegetsaftertransformationdependsontime.Therefore,acompletelydifferentapproachseemstobeneededwhenΩisnotconical.However,onemaystillexpectabadbehaviorofthenull-controlproblem.Indeed,assumeforinstancethatΩcontainsRn.IfoneisabletocontroltozeroinΩaninitialdataubymeansof+0aboundarycontrolactingon∂Ω×(0,T),then,byrestriction,oneisabletocontroltheinitialdatau0|Rnwiththecontrolbeingtherestrictionof+thesolutioninthelargerdomainΩ×(0,T)toRn−1×(0,T).Acarefuldevelopmentofthisargumentandoftheresultitmayleadtoremainstobedone.ACKNOWLEDGEMENTSThefirstauthorwaspartiallysupportedbyGrantPB96-0663ofDGES(Spain)andGrantA3/2002ofCNCSIS(Romania).ThesecondauthorwaspartiallysupportedbyGrantPB96-0663ofDGES(Spain)andtheTMRnetworkoftheEU“HomogenizationandMultipleScales(HMS2000).”BIBLIOGRAPHY[1]V.Cabanillas,S.deMenezesandE.Zuazua,“Nullcontrollabilityinunboundeddomainsforthesemilinearheatequationwithnonlinearitiesinvolvinggradientterms,”J.Optim.TheoryApplications,110(2)(2001),245-264.[2]M.EscobedoandO.Kavian,“Variationalproblemsrelatedtoself-similarso-lutionsoftheheatequation,”NonlinearAnal.TMA,11(1987),1103-1133.[3]A.FursikovandO.Yu.Imanuvilov,“Controllabilityofevolutionequations,”LectureNotesSeries#34,ResearchInstituteofMathematics,GlobalAnalysisResearchCenter,SeoulNationalUniversity,1996.[4]J.M.Ghidaglia,“Somebackwarduniquenessresults,”NonlinearAnal.TMA,10(1986),777-790.[5]O.Yu.ImanuvilovandM.Yamamoto,“CarlemanestimateforaparabolicequationinSobolevspacesofnegativeorderanditsapplications,”In:Con-trolofNonlinearDistributedParameterSystems,G.Chenetal.eds.,Marcel-Dekker,2000,pp.113-137.[6]B.F.Jones,Jr.,“Afundamentalsolutionoftheheatequationwhichissup-portedinastrip,”J.Math.Anal.Appl.,60(1977),314-324.[7]G.LebeauandL.Robbiano,“Contrˆoleexactdel’´equationdelachaleur,”Comm.P.D.E.,20(1995),335-356.[8]J.L.LionsandE.Malgrange,“Surl’unicit´er´etrogradedanslesprobl`emesmixtesparaboliques,”Math.Scan.,8(1960),277-286. 168PROBLEM5.3[9]W.Littman,“Boundarycontroltheoryforhyperbolicandparabolicpartialdif-ferentialequationswithconstantcoefficients,”AnnaliScuolaNorm.Sup.Pisa,SerieIV,3(1978),567-580.[10]S.MicuandE.Zuazua,“Onthelackofnull-controllabilityoftheheatequa-tiononthehalf-line,”Trans.AMS,353(2001),1635-1659.[11]S.MicuandE.Zuazua,“Onthelackofnull-controllabilityoftheheatequa-tiononthehalf-space,”PortugaliaMatematica,58(2001),1-24.[12]D.L.Russell,“Controllabilityandstabilizabilitytheoryforlinearpartialdif-ferentialequations.Recentprogressandopenquestions,”emSIAMRev.,20(1978),639-739.[13]L.deTeresaandE.Zuazua,“Approximatecontrollabilityoftheheatequationinunboundeddomains,”NonlinearAnal.TMA,37(1999),1059-1090. Problem5.4Istheconservativewaveequationregular?GeorgeWeissDept.ofElectricalandElectronicEngineeringImperialCollegeLondonExhibitionRoadLondonSW72BTUKG.Weiss@imperial.ac.uk1DESCRIPTIONOFTHEPROBLEMWeconsideraninfinite-dimensionalsystemdescribedbythewaveequa-tiononann–dimensionaldomain,withmixedboundarycontrolandmixedboundaryobservation,whichhasbeenanalyzed(asanexampleforacertainclassofconservativelinearsystems)in[13].Asomewhatsimplerversionofthissystemhasappeared(alsoasanexample)inthepaper[11,section7]andarelatedsystemhasbeendiscussedin[5].WeassumethatΩ⊂RnisaboundeddomainwithLipschitzboundaryΓ,asdefinedinGrisvard[3].Thismeansthat,locally,afterasuitablerotationoftheorthogonalcoordinatesystem,theboundaryisthegraphofaLipschitzfunctiondefinedonanopensetinRn−1.Suchaboundaryadmitscornersandedges.Γ0andΓ1arenonemptyopensubsetsofΓsuchthatΓ0∩Γ1=∅andΓ0∪Γ1=Γ.Wedenotebyxthespacevariable(x∈Ω).Afunctionb∈L∞(Γ)isgiven,whichintuitivelyexpresseshowstronglytheinput1signalactsondifferentpartsoftheactiveboundaryΓ1.Weassumethatb(x)6=0foralmosteveryx∈Γ1.Theequationsofthesystemarez¨(x,t)=∆z(x,t)onΩ×[0,∞),z(x,t)=0onΓ0×[0,∞),√∂z(x,t)+|b(x)|2z˙(x,t)=2·b(x)u(x,t)onΓ×[0,∞),(1)∂ν1∂2√∂νz(x,t)−|b(x)|z˙(x,t)=2·b(x)y(x,t)onΓ1×[0,∞),z(x,0)=z0(x),z˙(x,0)=w0(x)onΩ,whereuistheinputfunctionandyistheoutputfunction.Thefunctions 170PROBLEM5.4z0andw0aretheinitialstateofthesystem.ThepartΓ0oftheboundaryisjustreflectingwaves,whileinputsandoutputsactthroughthepartΓ1.Foreveryg∈H1(Ω)wedenotebyγgtheDirichlettraceofgonΓ(forg∈C1(Ω)⊂H1(Ω)thiswouldsimplybetherestrictionofgtoΓ).WeregardγgasanelementofL2(Γ).WedefinetheHilbertspace11HΓ0(Ω)={g∈H(Ω)|γg=0onΓ0},kgkH1=k∇gkL2.Proposition1.Theequations(1)determineawell-posedlinearsystemΣwithinputspaceU=L2(Γ),outputspaceY=L2(Γ)andstatespace11X=H1(Ω)×L2(Ω).Γ0Fortheprecisemeaningofawell-posedlinearsystemwereferto[8,9,6].Thesepapersusethesamenotationandterminologythatweusehere,buttheirreferenceswillindicateotherworksinwhichequivalentdefinitionscanbefound.Wegiveashortexplanationofwhatwell-posednessmeansinourcase.Ifwetakex(0)=[zw]T∈X,u∈L2([0,∞);U)andwesolvethe00equations(1)onthetimeinterval[0,∞),thenwegetx(τ)=[z(τ)˙z(τ)]T∈Xforeveryτ≥0.x(τ)iscalledthestateofthesystemattimeτ.Moreover,ifwedenotetherestrictionofyto[0,τ]byPy,thenPy∈L2([0,τ];Y).ττ(Notethatinourparticularcase,U=Y.)WecanintroducefourfamiliesofboundedoperatorsT,Φ,Ψ,andFindexedbyτ≥0suchthatforeverysuchτ,x(τ)=Tτx(0)+ΦτPτu,Pτy=Ψτx(0)+FτPτu.Thus,foreveryτ≥0,theoperatormatrixTτΦτΣτ=ΨτFτdefinesaboundedoperatorfromX×L2([0,τ];U)toX×L2([0,τ];Y).Thisistheessentialfeatureofawell-posedlinearsystem.Infact,in[8,9,6],ΣisdefinedasthefamilyofoperatorsΣτ.Forawell-posedlinearsystem,thefamilyTisastronglycontinuoussemigroupofoperatorsactingonX.Proposition1wasprovedin[13,section7],togetherwiththefollowing:Proposition2.ThesystemΣfromProposition1isconservative.ThefactthatΣisconservativemeansthattheoperatorsΣτareunitary.Inparticular,thefactthatΣτisisometricmeansthatwehaveZτZτkx(τ)k2−kx(0)k2=ku(t)k2dt−ky(t)k2dt,00whichcanbeinterpretedasanenergybalanceequation.Forbackgroundonconservativesystems,wereferto[1,2,4,7,12,13].ThesystemΣhas,likeeveryconservativesystem,atransferfunctionGthatisintheSchurclass.ThismeansthatGisanalyticontheopenright ISTHECONSERVATIVEWAVEEQUATIONREGULAR?171half-planeC0andkG(s)k≤1foralls∈C0.Forthesimpleproofofthisfact,see[13,theorem1.3andproposition4.5].TheboundaryvaluesG(iω)canbedefinedforalmosteveryω∈Rasnontangentiallimits,andwehaveG(iω)∗G(iω)=G(iω)G(iω)∗=Iforalmosteveryω∈R(i.e.,Gisinnerandco-inner).Thisfollowsfrom[10,proposition2.1]oralternativelyfrom[7,corollary7.3].Recallthatawell-posedlinearsystemwithinputspaceU,outputspaceY,andtransferfunctionGiscalledregularifforeveryv∈U,thelimitlimG(s)v=Dvs→+∞,s∈Rexists.Inthiscase,theoperatorD∈L(U,Y)iscalledthefeedthroughoperatorofthesystem(see[8,9,6]forfurtherdetails).Forregularlinearsystems,thetheoriesoflocalrepresentation,feedbackanddynamicstabi-lizationaremuchsimplerthanforwell-posedlinearsystems.Conjecture.ThesystemΣfromProposition1isregularanditsfeedthroughoperatoriszero.ConsidertheparticularsituationwhenΩisone-dimensional:Ω=(0,1),Γ0={0},Γ1={1}andU=Y=C.Nowthefunctionbbecomesanonzeronumber,andwithoutlossofgeneralitywemaytakeb=1.Itiseasytoseethattheinputsignalentersthedomainatx=1,propagatesalongthedomain(withunitspeed)untilitgetsreflectedatx=0andthenitpropagatesbacktoexit(astheoutputsignal)atx=1.Iftheinitialstateiszero,thenfort≥2wehavey(t)=u(t−2),sothatthetransferfunctionisG(s)=e−2s.NotethatGisindeedinneranditisregularwithfeedthroughoperatorzero.Theauthorthinksthathecanprovetheconjectureinthefollowingparticularcase:theactiveboundaryΓ1canbepartitionedintoafiniteunionofopensubsetsthatareeitherplanar(i.e.,anopensubsetofann−1dimensionalhyperplane)orspherical(i.e.,anopensubsetofann−1dimensionalsphere).Theideaistoconstructsolutionsof(1),whichlocally(nearaboundarypoint)looklikeaplanarorsphericalwavemovingintothedomainΩ(theinitialstateiszero)andlocally(intimeandspace),uisastepfunction.Thenlocally(intimeandspace)yiszero,whichprovestheclaim,duetotheequivalentcharacterizationofregularityviathestepresponse,see[8,theorem5.8].BIBLIOGRAPHY[1]D.Z.ArovandM.A.Nudelman,“Passivelinearstationarydynamicalscatteringsystemswithcontinoustime,”IntegralEquationsandOper-atorTheory,24(1996),pp.1–43. 172PROBLEM5.4[2]J.A.Ball,“ConservativedynamicalsystemsandnonlinearLivsic-Brodskiinodes,”OperatorTheory:AdvancesandApplications,73(1994),pp.67–95.[3]P.Grisvard,EllipticProblemsinNonsmoothDomains,Pitman,Boston,1985.[4]B.M.J.MaschkeandA.J.vanderSchaft,“PortcontrolledHamil-tonianrepresentationofdistributedparametersystems,”Proc.oftheIFACWorkshoponLagrangianandHamiltonianMethodsforNonlin-earControl,N.E.LeonardandR.Ortega,eds.,PrincetonUniversityPress,2000,pp.28–38.[5]A.Rodriguez–BernalandE.Zuazua,“Parabolicsingularlimitofawaveequationwithlocalizedboundarydamping,”DiscreteandContinuousDynamicalSystems,1(1995),pp.303–346.[6]O.J.StaffansandG.Weiss,“Transferfunctionsofregularlinearsys-tems.PartII:ThesystemoperatorandtheLax-Phillipssemigroup,”Trans.AmericanMath.Society,354(2002),pp.3229–3262.[7]O.J.StaffansandG.Weiss,“Transferfunctionsofregularlinearsys-tems.PartIII:Inversionsandduality,”submitted.[8]G.Weiss,“Transferfunctionsofregularlinearsystems.PartI:Charac-terizationsofregularity,”Trans.AmericanMath.Society,342(1994),pp.827–854.[9]G.Weiss,“Regularlinearsystemswithfeedback,”MathematicsofCon-trol,SignalsandSystems,7(1994),pp.23–57.[10]G.Weiss,“Optimalcontrolofsystemswithaunitarysemigroupandwithcolocatedcontrolandobservation,”SystemsandControlLetters,48(2003),pp.329–340.[11]G.WeissandR.Rebarber,“Optimizabilityandestimatabilityforinfinite-dimensionallinearsystems,”SIAMJ.ControlandOptimiza-tion,39(2001),pp.1204–1232.[12]G.Weiss,O.J.StaffansandM.Tucsnak,“Well-posedlinearsystems:Asurveywithemphasisonconservativesystems,”AppliedMathematicsandComputerScience,11(2001),pp.101–127.[13]G.WeissandM.Tucsnak,“Howtogetaconservativewell-posedlin-earsystemoutofthinair.PartI:well-posednessandenergybalance,”ESAIMCOCV,vol.9,pp.247-74,2003. Problem5.5Exactcontrollabilityofthesemi-linearwaveequationXuZhangDepartamentodeMatem´aticaAplicadaUniversidadComplutense28040MadridSpainandSchoolofMathematicsSichuanUniversityChengdu610064Chinaxuzhang@fudan.eduEnriqueZuazuaDepartamentodeMatem´aticas,FacultaddeCienciasUniversidadAut´onoma28049MadridSpainenrique.zuazua@uam.es1DESCRIPTIONOFTHEPROBLEMLetT>0andΩ⊂Rn(n∈N)beaboundeddomainwithaC1,1boundary∂Ω.LetωbeapropersubdomainofΩanddenotethecharacteristicfunctionofthesetωbyχ.Fixanonlinearfunctionf∈C1(R).ωWeareconcernedwiththeexactcontrollabilityofthefollowingsemilinearwaveequation:ytt−∆y+f(y)=χω(x)u(t,x)in(0,T)×Ω,y=0on(0,T)×∂Ω,(1)y(0)=y0,yt(0)=y1inΩ.In(1),(y(t,·),yt(t,·))isthestateandu(t,·)isthecontrolthatactsonthesystemthroughthesubsetωofΩ.Inwhatfollows,wechoosethestatespaceandthecontrolspaceasH1(Ω)×0L2(Ω)andL2((0,T)×Ω),respectively.Ofcourse,thechoiceofthesespacesis 174PROBLEM5.5notunique.Butthisoneisverynaturalinthecontextofthewaveequation.ThespaceH1(Ω)×L2(Ω)isoftenreferredtoastheenergyspace.0Theexact(internal)controllabilityproblemfor(1)(inH1(Ω)×L2(Ω))may0beformulatedasfollows:foranygiven(y,y),(z,z)∈H1(Ω)×L2(Ω),01010tofind(ifpossible)acontrolu∈L2((0,T)×Ω)suchthattheweaksolutionyof(1)satisfiesy(T)=z0andyt(T)=z1inΩ.(2)Theexact(boundary)controllabilityproblemof(1)canbeformulatedsimi-larly.Inthatcase,thecontroluentersonthesystemthroughtheboundaryconditions.Thisproducesextratechnicaldifficulties.Themainopenprob-lemonthecontrollabilityofthissemilinearwaveequationweshalldescribeherearisesinbothcases.Weprefertopresentitinthecasewherethecontrolactsontheinternalsubdomainωtoavoidunnecessarytechnicaldifficulties.Firstofall,itiswell-knownthatwhenfgrowstoofast,thesolutionof(1)mayblowup.Inthepresenceofblow-upphenomena,asaconsequenceofthefinitespeedofpropagationofsolutionsof(1),theexactcontrollabilityof(1)doesnotholdunlessω=Ω([13]).ThisexceptionmeansthatthecontrolactsonthesystemeverywhereinΩinwhichcasetheeffectofnonlinearitymaybesuppressedeasily.Therefore,wesupposethat(H1)Thenonlinearityf∈C1(R)issuchthat(1)admitsaglobalweaksolutiony∈C([0,T];H1(Ω))∩C1([0,T];L2(Ω))foranygiven(y,y)∈001H1(Ω)×L2(Ω)andu∈L2((0,T)×Ω).0Therearetwoclassesofconditionsonfguaranteeingthat(H1)holds.Thefirstone,whichwillbecalledmildgrowthcondition,amountstorequestingthatf∈C1(R)grows“mildly”atinfinity(see[2]and[3]),i.e.,Z"∞#−2xYlimf(s)ds|x|log(e+x2)<∞,(3)kk|x|→∞0k=1wheretheiteratedlogarithmfunctionlogjisdefinedbytheformulaslog0s=sandlogj+1s=log(logjs),j=0,1,2,···,thenumberejisdefinedbytheequationslogjej=1.ItisobviousthatanygloballyLipschitzcontinuousfunctionfsatisfies(3).But,ofcourse,(3)allowsftogrowinaslightsuperlinearwayatinfinity.Thesecondone,whichwillbecalledgoodsigngrowthcondition,istheclassoffunctionsf∈C1(R)thatgrowfastatinfinitybutsatisfya“good-sign”condition,i.e.,thereexistconstantsL>0,p∈(1,n/(n−2)]ifn≥3andp∈(1,∞)ifn=1,2,suchthat|f(r)−f(s)|≤L(1+|r|p−1+|s|p−1)|r−s|,∀r,s∈R(4)andZxf(s)ds≥−Lx2as|x|→∞.(5)0 EXACTCONTROLLABILITYOFTHESEMI-LINEARWAVEEQUATION175Atypicalexampleisf(u)=u3forn=1,2,3.(6)Ontheotherhand,itiswell-knownthat,eveninthelinearcasewheref≡0,someconditionsonthecontrollabilitytimeTandthegeometryofthesetωwherethecontrolappliesareneededinordertoguaranteetheexactcontrollabilityproperty.Thus,weassumethat(H2)Tandωaresuchthat(1)withf≡0isexactlycontrollable.TherearealsotwoclassesofconditionsonTandωguaranteeingthat(H2)holds.Thefirstone,whichwewillcalltheclassicalmultipliercon-dition,iswhenωisaneighborhoodofasubsetoftheboundaryoftheformΓ(x)={x∈∂Ω:(x−x)·ν(x)>0}forsomex∈Rn,whereν(x)isthe000unitoutwardnormalvectorto∂Ωatx,andT>2max{|x−x0|:x∈Ωω}.Thisisthetypicalsituationoneencounterswhenapplyingthemultipliertechnique([8]).ThesecondoneiswhenTandΩsatisfytheso-calledGeo-metricControlConditionintroducedin[1].WehavethefollowingOpenProblem:Do(H1)and(H2)implytheexactcontrollabilityof(1)?Theaboveproblemcanalsobeformulatedinthemoregeneralcaseinwhichthenonlinearityisoftheformf(t,x,y,yt,∇y).Ofcourse,theproblemisevenmoredifficultinthatcaseandnewphenomenamayoccurduetothestrongdissipativeeffectsthattermsoftheform|u|p−1umayproduce.ttThus,weshallfocusinthecasef=f(y).Thisopenproblemwillbemademoreprecisebelow.2AVAILABLERESULTSANDOPENPROBLEMSNonlinearitieswithmildgrowthconditionFortheonespacedimensionalcase,bycombiningthesidewiseenergyesti-matesforthe1−dwaveequationsandthefixedpointtechnique,Zuazua([13])obtainedthefollowingresult:Theorem1:Assumen=1andΩ=(0,1).Let(a,b)bea(proper)subin-tervalof(0,1),T>2max(a,1−b)andlim|f(x)||x|−1log−2|x|=0.(7)|x|→∞Then(1)isexactcontrollable.Lateron,basedonamethodduetoEmanuilov([5]),Cannarsa,Komornik,´andLoreti([2])improvedtheorem1byrelaxingthegrowthconditiononf.Themainresultin[2]saysthatthesameconclusionintheorem1holdsifthecondition(7)onfisreplacedby(3).Thegrowthcondition(3)onfis 176PROBLEM5.5sharp(sincesolutionsof(1)mayblowupwheneverfgrowsfasterthan(3)atinfinityandfhasthebadsign).Forthehigherdimensionalcase,LiandZhang([7])provedthefollowingresult:Theorem2:Letωbeaneighborhoodof∂Ω,T>diam(Ωω)andlimf(x)|x|−1log−1/2|x|=0.(8)|x|→∞Then(1)isexactlycontrollable.Aspecialcaseoftheorem2iswhenfisgloballyLipschitzcontinuous,whichgivesthemainresultofZuazuain[12].Themainresultin[12]wasgener-alizedtoanabstractsettingbyLasieckaandTriggiani([6])usingaglobalversionofInverseFunctiontheoremandwasextendedin[9]tothecasewhenTandωsatisfytheclassicalmultipliercondition.ItisnaturaltoconjecturethatthesameconclusioninTheorem2holdsunderthegrowthcondition(3)onfasinonedimension.Butthisisbynowanopenproblem.Ontheotherhand,whetherthesameconclusionintheorem2holdsformoregeneralconditionsonTandω,saytheclassicalmultiplierconditionorGeometricControlCondition,isalsoanopenproblem.Especially,whenTandωsatisfytheGeometricControlCondition,theexactcontrollabilityproblemfor(1)isopenevenforgloballyLipschitzcontinuousnonlinearities.NonlinearitieswithgoodsignandsuperlineargrowthatinfinityInthiscase,therearenoglobalexactcontrollabilityresultsintheliterature.However,usingafixedpointargument,Zuazuaprovedthefollowinglocalexactcontrollabilityresultsfor(1)([10]):Theorem3:Let(H2)hold,f∈C1(R)satisfy(4)andf(0)=0.Thenthereisaδ>0suchthatforany(y,y)and(z,z)inH1(Ω)×L2(Ω)01010with|(y0,y1)|H1(Ω)×L2(Ω)+|(z0,z1)|H1(Ω)×L2(Ω)≤δ,thereisacontrolu∈00L2((0,T)×Ω),suchthat(2)holds.Combiningtheorem3andthestabilizationresultsforthesemilinearwaveequationswith“good-sign”conditiononthenonlinearity([11]and[4]),itiseasytoshowthatTheorem4:LetT0andωsatisfytheclassicalmultiplierconditionandfsatisfy(4)–(5).Thenforany(y,y)and(z,z)inH1(Ω)×L2(Ω),there01010existatimeT≥Tandacontrolu(·)∈L2((0,T)×Ω),suchthat(2)holds.0NotethatthecontrollabilitytimeTintheorem4dependson(y0,y1)and(z0,z1).Accordingto[11],onecanobtainexplicitboundsonT.However,whetherTmaybechosentobeuniform,i.e.,independentofthedata(y0,y1)and(z0,z1),isanopenproblemevenforthenonlinearityin(6)forn=1.Thisiscertainlyoneofthemainopenproblemsinthecontextofcontrolla-bilityofnonlinearPDE. EXACTCONTROLLABILITYOFTHESEMI-LINEARWAVEEQUATION177ACKNOWLEDGEMENTSThisworkwassupportedinpartbythegrantsPB96-0663oftheDGES(Spain),theEUTMRProject“HomogenizationandMultipleScales,”aFoundationfortheAuthorsofExcellentPh.D.ThesesinChina,andtheNSFofChina(19901024).BIBLIOGRAPHY[1]C.Bardos,G.Lebeau,andJ.Rauch,“Sharpsufficientconditionsfortheobservation,control,andstabilizationofwavesfromtheboundary,”SIAMJ.ControlOptim.,30,pp.1024-1065,1992.[2]P.Cannarsa,V.Komornik,andP.Loreti,“One-sidedandinternalcon-trollabilityofsemilinearwaveequationswithinfinitelyiteratedloga-rithms,”Preprint.[3]T.CazenaveandA.Haraux,“Equationsd’´evolutionavecnonlin´earit´e´logarithmique,”Ann.Fac.Sci.Toulouse,2,pp.21-51,1980.[4]B.Dehman,G.Lebeau,andE.Zuazua,“Stabilizationandcontrolforthesubcriticalsemilinearwaveequation,”Preprint,2002.[5]O.Yu.Emanuilov,“Boundarycontrollabilityofsemilinearevolution´equations,”RussianMath.Surveys,44,pp.183-184,1989.[6]I.LasieckaandR.Triggiani,“Exactcontrollabilityofsemilinearabstractsystemswithapplicationtowavesandplatesboundarycontrolprob-lems,”Appl.Math.Optim.,23,pp.109-154,1991.[7]L.LiandX.Zhang,“Exactcontrollabilityforsemilinearwaveequa-tions,”J.Math.Anal.Appl.,250,pp.589-597,2000.[8]J.L.Lions,Contrˆolabilit´eexacte,perturbationsetsyst´emesdistribu´es,Tome1,Rech.Math.Appl.8,Masson,Paris,1988.[9]X.Zhang,“Explicitobservabilityestimateforthewaveequationwithpotentialanditsapplication,”R.Soc.Lond.Proc.Ser.AMath.Phys.Eng.Sci.,456,pp.1101-1115,2000.[10]E.Zuazua,“Exactcontrollabilityforthesemilinearwaveequation,”J.Math.PuresAppl.,69,pp.1-31,1990.[11]E.Zuazua,“Exponentialdecayforsemilinearwaveequationswithlocal-izeddamping,”Comm.PartialDifferentialEquations,15,pp.205-235,1990.[12]E.Zuazua,“Exactboundarycontrollabilityforthesemilinearwaveequation,”In:Nonlinearpartialdifferentialequationsandtheirapplica- 178PROBLEM5.5tions,Coll´egedeFranceSeminar,vol.X(Paris,1987-1988),pp.357-391,PitmanRes.NotesMath.Ser.,220,LongmanSci.Tech.,Harlow,1991.[13]E.Zuazua,“Exactcontrollabilityforsemilinearwaveequationsinonespacedimension,”Ann.Inst.H.Poincar´eAnal.NonLin´eaire,10,pp.109-129,1993. Problem5.6SomecontrolproblemsinelectromagneticsandfluiddynamicsLorellaFatoneDipartimentodiMatematicaPuraedApplicataUniversit`adiModenaeReggioEmiliaViaCampi213/b,41100Modena(MO)Italyfatone.lorella@unimo.itMariaCristinaRecchioniIstitutodiTeoriadelleDecisionieFinanzaInnovativa(DE.F.IN.)Universit`adiAnconaPiazzaMartelli8,60121Ancona(AN)Italyrecchioni@posta.econ.unian.itFrancescoZirilliDipartimentodiMatematica“G.Castelnuovo”Universit`adiRoma“LaSapienza”PiazzaleAldoMoro2,00185RomaItalyf.zirilli@caspur.it1INTRODUCTIONInrecentyears,asaconsequenceofthedramaticincreasesincomputingpowerandofthecontinuingrefinementofthenumericalalgorithmsavail-able,thenumericaltreatmentofcontrolproblemsforsystemsgovernedbypartialdifferentialequation;see,forexample,[1],[3],[4],[5],[8].Theimpor-tanceofthesemathematicalproblemsinmanyapplicationsinscienceandtechnologycannotbeoveremphasized.Themostcommonapproachtoacontrolproblemforasystemgovernedbypartialdifferentialequationsistoseetheproblemasaconstrainednon-linearoptimizationproblemininfinitedimension.Afterdiscretizationthe 180PROBLEM5.6problembecomesafinitedimensionalconstrainednonlinearoptimizationproblemthatcanbeattackedwiththeusualiterativemethodsofnonlin-earoptimization,suchasNewtonorquasi-Newtonmethods.Notethattheproblemoftheconvergence,whenthe“discretizationstepgoestozero,”ofthesolutionscomputedinfinitedimensiontothesolutionoftheinfinitedimensionalproblemisaseparatequestionandmustbesolvedseparately.Whenthisapproachisusedanobjectivefunctionevaluationinthenonlin-earoptimizationprocedureinvolvesthesolutionofthepartialdifferentialequationsthatgovernthesystem.Moreover,theevaluationofthegradientorHessianoftheobjectivefunctioninvolvesthesolutionofsomekindofsensitivityequationsforthepartialdifferentialequationsconsidered.Thenonlinearoptimizationprocedurethatusuallyinvolvesfunction,gradientandHessianevaluationiscomputationallyveryexpensive.Thisfactisaseriouslimitationtotheuseofcontrolproblemsforsystemsgovernedbypartialdifferentialequationsinrealsituations.Howevertheap-proachpreviouslydescribedisverystraightforwardanddoesnotuseanyofthespecialfeaturespresentineverysystemgovernedbypartialdifferentialequations.Sothat,atleastinsomespecialcases,itshouldbepossibletoimproveonthisstraightforwardapproach.Thepurposeofthispaperistopointoutaproblem,see[6],[2],whereanewapproach,thatgreatlyimprovesonthepreviouslydescribedone,hasbeenintroducedandtosuggestsomeotherproblemswhere,hopefully,sim-ilarimprovementscanbeobtained.Inparticular,weproposetwocontrolproblemsofgreatrelevanceinseveralapplicationsinscienceandtechnologyandwesuggestthe(open)questionofcharacterizingtheoptimalsolutionofthesecontrolproblemsasthesolutionofsuitablesystemsofpartialdifferen-tialequations.Ifthisquestionhasanaffirmativeanswer,highperformancealgorithmscanbedevelopedtosolvethecontrolproblemsproposed.Notethatin[6],[2]thischaracterizationhasbeenmadeforsomecontrolproblemsinacoustics,thankstotheuseofthePontryaginmaximumprinciple,andhaspermittedtodevelophighperformancealgorithmstosolvethesecon-trolproblems.Moreover,wesuggestthe(open)questionofusingeffectivelythedynamicprogrammingmethodtoderiveclosedloopcontrollawsforthecontrolproblemsconsidered.Foreffectiveuseofthedynamicprogrammingmethod,wemeanthepossibilityofcomputingaclosedloopcontrollawatapproximatelythesamecomputationalcostofsolvingtheoriginalproblemwhennocontrolstrategyisinvolved.Insection2wesummarizetheresultsobtainedin[6],[2],andinsection3wepresenttwoproblemsthatwebelievecanbeapproachedinawaysimilartotheonedescribedin[6],[2]. SOMECONTROLPROBLEMSIN...1812PREVIOUSRESULTSIn[6],[2]afurtivityproblemintimedependentacousticobstaclescatter-ingisconsidered.Anobstacleofknownacousticimpedanceishitbyaknownincidentacousticfield.Whenhitbytheincidentacousticfield,theobstaclegeneratesascatteredacousticfield.Tomaketheobstaclefurtivemeansto“minimize”thescatteredfield.Thefurtivityeffectisobtainedcirculatingontheboundaryoftheobstaclea“pressurecurrent”thatisaquantitywhosephysicaldimensionis:pressuredividedbytime.Theprob-lemconsistsinfindingtheoptimal“pressurecurrent”that“minimizes”thescatteredfieldandthe“size”ofthepressurecurrentemployed.Themathe-maticalmodelusedtostudythisproblemisacontrolproblemforthewaveequation,wherethecontrolfunction(i.e.,thepressurecurrent)influencesthestatevariable(i.e.,thescatteredfield)throughaboundaryconditionimposedontheboundaryoftheobstacle,andthecostfunctionaldependsexplicitlyfromboththestatevariableandthecontrolfunction.IntroducinganauxiliaryvariableandusingthePontryaginmaximumprinciple(see[7])in[6],[2]itisshownthattheoptimalcontrolofthisproblemcanbeobtainedfromthesolutionofasystemoftwocoupledwaveequations.Thissystemofwaveequationsisequippedwithsuitableinitial,final,andboundarycon-ditions.Thankstothisingeniousconstructionthesolutionoftheoptimalcontrolproblemcanbeobtainedsolvingthesystemofwaveequationswith-outthenecessityofgoingthroughtheiterationsimpliedingeneralbythenonlinearoptimizationprocedure.Thisfactavoidsmanyofthedifficulties,thathavebeenmentionedabove,presentinthegeneralcase.Finally,thesystemofwaveequationsissolvednumericallyusingahighlyparallelizablealgorithmbasedontheoperatorexpansionmethod(formoredetails,see[6],[2]andthereferencestherein).Somenumericalresultsobtainedwiththisalgorithmonsimpletestproblemscanbeseenintheformofcom-puteranimationsinthewebsites:http://www.econ.unian.it/recchioni/w6,http://www.econ.unian.it/recchioni/w8.Inthefollowingsection,wesuggesttwoproblemswherewillbeinterestingtocarryoutasimilaranalysis.3TWOCONTROLPROBLEMSLetRbethesetofrealnumbers,x=(x,x,x)T∈R3(wherethesu-123perscriptTmeanstransposed)beagenericvectorofthethree-dimensionalrealEuclideanspaceR3,andlet(·,·),k·kand[·,·]denotetheEuclideanscalarproduct,theEuclideanvectornormandthevectorproductinR3,respectively.Thefirstproblemsuggestedisa“masking”problemintime-dependentelec-tromagneticscattering.LetΩ⊂R3beaboundedsimplyconnectedopenset(i.e.,theobstacle)withlocallyLipschitzboundary∂Ω.LetΩdenotetheclosureofΩandn(x)=(n(x),n(x),n(x))T∈R3,x∈∂Ωbethe123 182PROBLEM5.6outwardunitnormalvectorinxforx∈∂Ω.Notethatn(x)existsalmosteverywhereinxforx∈∂Ω.WeassumethattheobstacleΩischaracter-izedbyanelectromagneticboundaryimpedanceχ>0.Notethatχ=0(χ=+∞)correspondstoconsideraperfectlyconducting(insulating)obsta-cle.LetR3Ωbefilledwithahomogeneousisotropicmediumcharacterizedbyaconstantelectricpermittivity>0,aconstantmagneticpermeabil-ityν>0,zeroelectricconductivity,zerofreechargedensity,andzerofreecurrentdensity.Let(Ei(x,t),Bi(x,t)),(x,t)∈R3×R(whereEiistheelectricfieldandBiisthemagneticinductionfield)betheincomingelectromagneticfieldprop-agatinginthemediumfillingR3ΩandsatisfyingtheMaxwellequations(1)-(3)inR3×R.Let(Es(x,t),Bs(x,t)),(x,t)∈(R3Ω)×RbetheelectromagneticfieldscatteredbytheobstacleΩwhenhitbytheincomingfield(Ei(x,t),Bi(x,t)),(x,t)∈R3×R.ThescatteredelectricfieldEsandsthescatteredmagneticinductionfieldBsatisfythefollowingequations:ss∂B3curlE+(x,t)=0,(x,t)∈(RΩ)×R,(1)∂tss1∂E3curlB−(x,t)=0,(x,t)∈(RΩ)×R,(2)c2∂tss3divB(x,t)=0,divE(x,t)=0,(x,t)∈(RΩ)×R,(3)ss[n(x),E(x,t)]−cχ[n(x),[n(x),B(x,t)]]=ii(4)−[n(x),E(x,t)]+cχ[n(x),[n(x),B(x,t)]],(x,t)∈∂Ω×R,s1s1s1E(x,t)=O,[B(x,t),xˆ]−E(x,t)=o,r→+∞,t∈R,rcr(5)√where0=(0,0,0)T,c=1/ν,r=kxk,x∈R3,xˆ=x,x6=0,x∈R3,kxkO(·)ando(·)aretheLandausymbols,andcurl·anddiv·denotethecurlandthedivergenceoperatorof·withrespecttothexvariablesrespectively.AclassicalprobleminelectromagneticsconsistsintherecognitionoftheobstacleΩthroughtheknowledgeoftheincomingelectromagneticfieldandofthescatteredfield(Es(x,t),Bs(x,t)),(x,t)∈(R3Ω)×Rsolutionof(1)-(5).Intheabovesituation,Ωplaysa“passive”(“static”)role.WewanttomaketheobstacleΩ“active”(“dynamic”)inthesensethat,thankstoasuitablecontrolfunctionchoseninaproperway,theobstacleitselftriestoreacttotheincomingelectromagneticfieldproducingascatteredfieldthatlookslikethefieldscatteredbyapreassignedobstacleD(the“mask”)withimpedanceχ0.Wesuggesttoconsiderthefollowingcontrolproblem:Problem1:Electromagnetic“Masking”Problem:Givenanincomingelec-iitromagneticfield(E,B),anobstacleΩanditselectromagneticboundaryimpedanceχ,andgivenanobstacleDsuchthatD⊆Ωwithelectromag-neticboundaryimpedanceχ0,chooseavectorcontrolfunctionψdefinedon SOMECONTROLPROBLEMSIN...183theboundaryoftheobstacle∂Ωfort∈Randappearingintheboundaryconditionsatisfiedbythescatteredelectromagneticfieldon∂Ω,inordertominimizeacostfunctionalthatmeasuresthe“difference”betweentheelec-sstromagneticfieldscatteredbyΩ,i.e.,(E,B),andtheelectromagneticfieldssscatteredbyD,i.e.,(ED,BD),whenΩandDrespectivelyarehitbythein-iicomingfield(E,B),andthe“size”ofthevectorcontrolfunctionemployed.Thecontrolfunctionψhasthephysicaldimensionofanelectricfieldandtheactionoftheoptimalcontrolelectricfieldontheboundaryoftheobstaclemakestheobstacle“active”(“dynamic”)andabletoreacttotheincidentelectromagneticfieldtobecome“unrecognizable,”thatis“ΩwilldoitsbesttoappearashismaskD.”Thesecondcontrolproblemwesuggesttoconsiderisacontrolprobleminfluiddynamics.LetusconsideranobstacleΩt,t∈R,thatisarigidbody,as-sumedhomogeneous,movinginR3withvelocityυ˜=υ˜(x,t),(x,t)∈Ω×R.tMoreoverfort∈RtheobstacleΩ⊂R3isaboundedsimplyconnectedtopenset.Fort∈Rletξ=ξ(t)bethepositionofthecenterofmassoftheobstacleΩt.Themotionoftheobstacleiscompletelydescribedbythevelocityw=w(ξ,t),t∈Rofthecenterofmassoftheobstacle(i.e.,dξw=,t∈R),theangularvelocityω=ω(ξ,t),t∈Roftheobstacledtaroundtheinstantaneousrotationaxisgoingthroughthecenterofmassξ=ξ(t),t∈Randthenecessaryinitialconditions.Notethatthevelocitiesofthepointsbelongingtotheobstacleυ˜(x,t),(x,t)∈Ωt×Rcanbeex-pressedintermsofw(ξ,t),ω(ξ,t),t∈R.LetR3Ω,t∈RbefilledwithatNewtonianincompressibleviscousfluidofviscosityη.Weassumethatboththedensityofthefluidandthetemperatureareconstant.Forexample,Ωt,t∈Rcanbeasubmarineoranairfoilimmersedinanincompressibleviscousfluid.Letv=(v,v,v)Tandpbethevelocityfieldandthepres-123surefieldofthefluid,respectively,fbethedensityoftheexternalforcespermassunitactingonthefluid,andv−∞beanassignedsolenoidalvectorfield.Weassumethatinthelimitt→−∞thebodyΩtisatrestinthepositionΩ−∞.Undertheseassumptions,wehavethatinthereferenceframegivenbyx=(x,x,x)TthefollowingsystemofNavier-Stokesequations123holds:∂v(x,t)+(v(x,t),∇)v(x,t)−η∆v(x,t)+∇p(x,t)=f(x,t),∂t(6)(x,t)∈(R3Ω)×R,tdivv(x,t)=0,(x,t)∈(R3Ω)×R,(7)tlimv(x,t)=v(x),x∈R3Ω,v(x,t)=υ˜(x,t),(x,t)∈∂Ω×R.−∞−∞tt→−∞(8)TIn(6)wehave∇=∂,∂,∂and∂x1∂x2∂x3TX3∂vX3∂vX3∂v(v,∇)v=v1,v2,v3.jjj∂xj∂xj∂xjj=1j=1j=1 184PROBLEM5.6Theboundaryconditionin(8)requiresthatthefluidvelocityvandthevelocityoftheobstacleυ˜areequalontheboundaryoftheobstaclefort∈R.Wewanttoconsidertheproblemassociatedtothechoiceofamaneuverw(ξ,t),ω(ξ,t),t∈RconnectingtwogivenstatesthatminimizestheworkdonebytheobstacleΩt,t∈Ragainstthefluidgoingfromtheinitialstatetothefinalstate,andthe“size”ofthemaneuveremployed.Notethatinthiscontextamaneuverconnectingtwogivenstatesismadeoftwofunctionsw(ξ,t),ω(ξ,t),t∈Rsuchthatlimw(ξ,t)=w±t→±∞andlimω(ξ,t)=ω±,wherew±andω±arepreassigned.Thecouplet→±∞(w−,ω−)istheinitialstateandthecouple(w+,ω+)isthefinalstate.Forsimplicity,wehaveassumed(w−,ω−)=(0,0).Weformulatethefollowingproblem:Problem2:“Drag”OptimizationProblem:GivenarigidobstacleΩt,t∈RmovinginaNewtonianfluidcharacterizedbyaviscosityηandtheinitialconditionandforcesactingonthefluid,andgiventheinitialstate(0,0)andthefinalstate(w+,ω+),chooseamaneuverconnectingthesetwostatesinordertominimizeacostfunctionalthatmeasurestheworkthattheobstacleΩt,t∈Rmustexertonthefluidtomakethemaneuver,andthe“size”ofthemaneuveremployed.Fromthepreviousconsiderations,severalproblemsarise.Thefirstoneisconnectedwiththequestionofformulatingproblem1andproblem2ascon-trolproblems.In[2]wesuggestapossibleformulationofafurtivityproblemsimilartoproblem1asacontrolproblem.Inparticular,theopenquestionthatwesuggestishowproblem1andproblem2shouldbeformulatedascontrolproblemswhoseoptimalsolutionscanbedeterminedsolvingsuit-ablesystemsofpartialdifferentialequationsviaaningeniouswayofusingthePontryaginmaximumprincipleasdonein[2],[6].Therelevanceofthisformulationliesinthefactthatavoidscomputationallyexpensiveiterativeprocedurestosolvethecontrolproblemsconsidered.Moreover,asecondopenquestionisthederivationofclosedloopcontrollawsatanaffordablecomputationalcostforthecontrolproblemsassociatedtoProblem1andProblem2.Furthermoremanyvariationsofproblem1and2canbeconsidered.Forexampleinproblem1wehaveassumed,forsimplicity,thatthe“mask”isapassiveobstacle,thatis(Es(x,t),Bs(x,t)),(x,t)∈(R3D)×Ristheso-DDlutionofproblem(1)-(5)whenΩ,χarereplacedwithD,χ0,respectively.Inamoregeneralsituationalsothe“mask”canbeanactiveobstacle.Finally,problem1and2areexamplesofcontrolproblemsforsystemsgovernedbytheMaxwellequationsandtheNavier-Stokesequations,respectively.Manyotherexamplesrelevantinseveralapplicationfieldsinvolvingdifferentsys-temsofpartialdifferentialequationscanbeconsidered. SOMECONTROLPROBLEMSIN...185BIBLIOGRAPHY[1]T.S.Angell,A.Kirsch,andR.E.Kleinman,“Antennacontrolandoptimization,”ProceedingsoftheIEEE,79,pp.1559-1568,1991.[2]L.Fatone,M.C.Recchioni,andF.Zirilli,“Furtivityandmaskingprob-lemsintimedependentacousticobstaclescattering,”forthcominginThirdISAACCongressProceedings.[3]J.W.He,R.Glowinski,R.Metcalfe,A.Nordlander,andJ.Periaux,“Activecontrolanddragoptimizationforflowpastacircularcylinder,”JournalofComputationalPhysics,163,pp.83-117,2000.[4]J.E.Lagnese,“AsingularperturbationprobleminexactcontrollabilityoftheMaxwellsystem,”ESAIMControl,OptimizationandCalculusofVariations,6,pp.275-289,2001.[5]J.LunileyandP.Blossery,“Controlofturbulence,”AnnualReviewofFluidMechanics,30,pp.311-327,1998.[6]F.Mariani,M.C.Recchioni,andF.Zirilli,“TheuseofthePontryaginmaximumprincipleinafurtivityproblemintime-dependentacousticobstaclescattering,”WavesinRandomMedia,11,pp.549-575,2001.[7]L.Pontriaguine,V.Boltianski,R.Gamkr´elidz´e,andF.Micktckenko,Th´eorieMath´ematiquedesProcessusOptimaux,EditionsMir,Moscow,1974.[8]S.S.Sritharan,“Anintroductiontodeterministicandstochasticcontrolofviscousflow,”In:OptimalControlofViscousFlow,S.S.Sritharaneds.,SIAM,Philadelphia,pp.1-42,1998. PART6Stability,Stabilization Problem6.1CopositiveLyapunovfunctionsM.K.C¸amlıbelDepartmentofMathematicsUniversityofGroningenP.O.Box800,9700AVGroningenTheNetherlandsk.camlibel@math.rug.nlJ.M.SchumacherDepartmentofEconometricsandOperationsResearchTilburgUniversityP.O.Box90153,5000LETilburgTheNetherlandsj.m.schumacher@kub.nl1PRELIMINARIESThefollowingnotationalconventionsandterminologywillbeinforce.In-equalitiesforvectorsareunderstoodcomponent-wise.GiventwomatricesMandNwiththesamenumberofcolumns,thenotationcol(M,N)denotesthematrixobtainedbystackingMoverN.LetMbeamatrix.Thesub-matrixMJKofMisthematrixwhoseentrieslieintherowsofMindexedbythesetJandthecolumnsindexedbythesetK.ForsquarematricesM,MJJiscalledaprincipalsubmatrixofM.AsymmetricmatrixMissaidtobenon-negative(nonpositive)definiteifxTMx≥0(xTMx≤0)forallx.Itissaidtobepositive(negative)definiteiftheequalitiesholdonlyforx=0.Sometimes,wewriteM>0(M≥0)toindicatethatMispositivedefinite(non-negativedefinite),respectively.WesaythatasquarematrixMisHurwitzifitseigenvalueshavenegativerealparts.Apairofmatrices(A,C)isobservableifthecorrespondingsystem˙x=Ax,y=Cxisobservable,equivalentlyifcol(C,CA,···,CAn−1)isofranknwhereAisofordern. 190PROBLEM6.12MOTIVATIONLyapunovstabilitytheoryisoneoftheevergreentopicsinsystemsandcontrol.For(finitedimensional)linearsystems,thefollowingtheoremisverywell-known.Theorem1:[3,Theorem1.2]:Thefollowingconditionsareequivalent.1.Thesystem˙x=Axisasymptoticallystable.2.TheLyapunovequationATP+PA=Qhasapositivedefinitesym-metricsolutionPforanynegativedefinitesymmetricmatrixQ.Asarefinement,wecanreplacethelaststatementby20.TheLyapunovequationATP+PA=Qhasapositivedefinitesymmet-ricsolutionPforanynonpositivedefinitesymmetricmatrixQsuchthatthepair(A,Q)isobservable.Aninterestingapplicationistothestabilityoftheso-calledswitchedsystems.Considerthesystemx˙=Aσx(1)wheretheswitchingsignalσ:[0,∞)→{1,2}isapiecewiseconstantfunc-tion.Weassumethatithasafinitenumberofdiscontinuitiesoverfinitetimeintervalsinordertoruleoutinfinitelyfastswitching.Astrongnotionofstabilityforthesystem(1)istherequirementofstabilityforarbitraryswitchingsignals.Thedynamicsof(1)coincideswithoneofthelinearsubsystemsiftheswitch-ingsignalisconstant,i.e.,therearenoswitchingsatall.Thisleadsustoanobviousnecessarycondition:stabilityofeachsubsystem.AnotherextremecasewouldemergeifthereexistsacommonLyapunovfunctionforthesubsystems.Indeed,suchaLyapunovfunctionwouldimmediatelyprovethestabilityof(1).Anearlierpaper[8]pointedouttheimportanceofcommutationrelationsbetweenA1andA2infindingacommonLyapunovfunction.Moreprecisely,ithasbeenshownthatifA1andA2areHur-witzandcommutativethentheyadmitacommonLyapunovfunction.In[1,6],thecommutationrelationsofsubsystemsarestudiedfurtherinaLiealgebraicframeworkandsufficientconditionsfortheexistenceofacommonLyapunovfunctionarepresented.Noticethattheresultsof[1]arestrongerthanthosein[6].However,weprefertorestate[6,Theorem2]forsimplicity.Theorem2:IfAiisaHurwitzmatrixfori=1,2andtheLiealgebra{A1,A2}LAissolvablethenthereexistsapositivedefinitematrixPsuchthatATP+PA<0fori=1,2.ii COPOSITIVELYAPUNOVFUNCTIONS191Sofar,wequotedsomeknownresults.OurmaingoalistoposetwoopenproblemsthatcanbeviewedasextensionsofTheorems2and2foraclassofpiecewiselinearsystems.Moreprecisely,wewillconsidersystemsoftheformx˙=AixforCix≥0i=1,2.(2)HeretheconesCi={x|Cix≥0}donotnecessarilycoverthewholex-space.Weassumethata.thereexistsa(possiblydiscontinuous)functionfsuchthat(2)canbedescribedby˙x=f(x)forallx∈C1∪C2,andb.foreachinitialstatex0∈C1∪C2,thereexistsauniquesolutionRxintthesenseofCarath´eodory,i.e.,x(t)=x0+f(x(τ))dτ.0Anaturalexample[b.]ofsuchpiecewiselineardynamicsisalinearcomple-mentaritysystem(see[9])oftheformx˙=Ax+Bu,y=Cx+Du{(u(t)≥0)and(y(t)≥0)and(u(t)=0ory(t)=0)}forallt≥0whereA∈Rn×n,B∈Rn×1,C∈R1×n,andD∈R.IfD>0thissystemcanbeputintotheformof(2)withA=A,C=C,A=A−BD−1C,112andC2=−C.Equivalently,itcanbedescribedbyx˙=f(x)(3)wheref(x)=AxifCx≥0andf(x)=(A−BD−1C)xifCx≤0.NotethatfisLipschitzcontinuousandhence(3)admitsaunique(continuouslydifferentiable)solutionxforallinitialstatesx0.Onewayofstudyingthestabilityofthesystem(2)issimplytoutilizeTheorem2.However,therearesomeobviousdrawbacks:i.ItrequirespositivedefinitenessofthecommonLyapunovfunctionwhereasthepositivityonaconeisenoughforthesystem(2).ii.Itconsidersanyswitchingsignalwhereastheinitialstatedeterminestheswitchingsignalin(2).Inthenextsection,wefocusonwaysofeliminatingtheconservatismmen-tionedin2.3DESCRIPTIONOFTHEPROBLEMSFirst,weneedtointroducesomenomenclature.AmatrixMissaidtobecopositive(strictlycopositive)withrespecttoaconeCifxTMx≥0CC(xTMx>0)forallnonzerox∈C.WeusethenotationM<0andM0 192PROBLEM6.1respectivelyforcopositivityandstrictcopositivity.WhentheconeCisclearfromthecontextwejustwrite0withassociatedeigenvalue(λ≤0)λ<0.Sincethenumberofprincipalsubmatricesofamatrixofordernisroughly2n,thisresulthasapracticaldisadvantage.Infact,MurtyandKabadi[7]showedthattestingforcopositivityisNP-complete.Aninterestingsubclassofcopositivematricesaretheonesthatareequaltothesumofanonnegativedefinitematrixandanon-negativematrix.Thisclassofmatricesisstudiedin[5]wherearelativelymoretractablealgorithmhasbeenpresentedforcheckingifagivenmatrixbelongstotheclassornot. COPOSITIVELYAPUNOVFUNCTIONS193BIBLIOGRAPHY[1]A.A.AgrachevandD.Liberzon,“Lie-algebraicstabilitycriteriaforswitchedsystems,”SIAMJ.ControlOptim.,40(1):253–269,2001.[2]R.W.Cottle,J.-S.Pang,andR.E.Stone,TheLinearComplementarityProblem,AcademicPress,Inc.,Boston,1992.[3]Z.Gaji´candM.T.J.Qureshi,LyapunovMatrixEquationinSystemStabilityandControl,volume195ofMathematicsinScienceandEngi-neering,AcademicPress,SanDiego,1995.[4]W.Kaplan,“Atestforcopositivematrices,”LinearAlgebraAppl.,313:203–206,2000.[5]W.Kaplan,“Acopositivityprobe,”LinearAlgebraAppl.,337:237–251,2001.[6]D.Liberzon,J.P.Hespanha,andA.S.Morse,“Stabilityofswitchedsystems:ALie-algebraiccondition,”SystemsControlLett.,37:117–122,1999.[7]K.MurtyandS.N.Kabadi,“SomeNP-completeproblemsinquadraticandnonlinearprogramming,”Math.Programming,39:117–129,1987.[8]K.S.NarendraandJ.Balakrishnan,“AcommonLyapunovfunctionforstableLTIsystemswithcommutingA-matrices,”IEEETrans.Au-tomaticControl,39:2469–2471,1994.[9]A.J.vanderSchaftandJ.M.Schumacher,“Complementaritymod-ellingofhybridsystems,”IEEETransactionsonAutomaticControl,43(4):483–490,1998. Problem6.2Thestrongstabilizationproblemforlineartime-varyingsystemsAvrahamFeintuchDepartmentofMathematicsBen-GurionUniversityoftheNegevBeer-ShevaIsraelabie@math.bgu.ac.il1DESCRIPTIONOFTHEPROBLEMIwillformulatethestrongstabilizationproblemintheformalismoftheoperatortheoryofsystems.Inthisframework,alinearsystemisalineartransformationLactingonaHilbertspaceHthatisequippedwithanaturaltimestructure,whichsatisfiesthestandardphysicalrealizabilityconditionknownascausality.Tosimplifytheformulation,wechooseHtobethe2nP2sequencespacel[0,∞)={:xi∈C,kxik<∞}andde-notebyPnthetruncationprojectionontothesubspacegeneratedbythefirstnvectors{e0,···,en}ofthestandardorthonormalbasisonH.CausalityofLisexpressedasPnL=PnLPnforallnon-negativeintegersn.AlinearsystemLisstableifitisaboundedoperatoronH.Afundamentalissuethatwasstudiedinbothclassicalandmoderncontroltheorywasthatofinternalstabilizationofunstablesystemsbyfeedback.ItisgenerallyacknowledgedthatthepaperofYoulaetal.[2]wasalandmarkeventinthisstudyandinfacttheissueofstrongstabilizationwasfirstraisedthere.Itwasquicklyseen[5]thatwhilethispaperrestricteditselftotheclassicalcaseofrationaltransferfunctionsitsideasweregiventoabstractiontomuchmoregeneralframeworks.Webrieflydescribetheonerevelanttoourdiscussion.IForalinearsystemL,itsgraphG(L)istherangeoftheoperatorLdefinedonthedomainD(L)={x∈H:Lx∈H}.G(L)isasubspaceofICH⊕H.TheoperatordefinedonD(L)⊕D(C)iscalledtheL−I STRONGSTABILIZATIONPROBLEM195feedbacksystem{L,C}withplantLandcompensatorC,and{L,C}isstableifithasaboundedcausalinverse.LisstabilizableifthereexistsacausallinearsystemC(notnecessarilystable)suchthat{L,C}isstable.TheanalogueoftheresultofYoulaetal.whichcharacterizesallstabilizablelinearsystemsandparametrizesallstabilizerswasgivenbyDaleandSmith[4]:Theorem1.[[6],p.103]:SupposeLisalinearsystemandthereexistcausalstablesystemsM,N,X,Y,Mˆ,Nˆ,Xˆ,Yˆsuchthat(1)G(L)=−1MM−XˆYXRan=Ker[−NˆMˆ],(2)=.NNYˆ−NˆMˆThen(1)Lisstabilizable(2)CstabilizesLifandonlyifYˆ−NQG(C)=Ran=Ker[−(X+QMˆ)Y−QNˆ],whereQXˆ+MQvariesoverallstablelinearsystems.TheStrongStabilizationProblemis:SupposeLisstabilizable.CaninternalstabilitybeachievedwithCitselfastablesystem?Insuchacase,Lissaidtobestronglystabilizable.Theorem2.[[6],p.108]:AlinearsystemLwithproperty(1),(2)ofTheorem1isstabilizedbyastableCifandonlyifMˆ+NCˆisaninvertibleoperator.Equivalently,astableCstabilizesLifandonlyifM+CNisaninvertibleoperator(byaninvertibleoperatorwemeanthatitsinverseisalsobounded).ItisnothardtoshowthatinfactthesameCworksinbothcases;i.e.,M+CNisinvertibleifandonlyifMˆ+NCˆisinvertible.Sohereistheprecisemathematicalformulationoftheproblem:GivencausalstablesystemsM,N,X,YsuchthatXM+YN=I.DoesthereexistacausalstablesystemCsuchthatM+CNisinvertible?2MOTIVATIONANDHISTORYOFTHEPROBLEMThenotionofstronginternalstabilizationwasintroducedintheclassicalpa-perofYoulaetal.[2]andwassolvedforrationaltransferfunctions.Anotherformulationwasgivenin[1].Anapproachtotheclassicalproblemfromthepointofviewdescribedherewasfirstgivenin[9].Recentlysufficientcon-ditionsfortheexistanceofstronglystabilizingcontrollerswereformulatedfromthepointofviewofH∞controlproblems.Thelatestsucheffortis[7].Itisofinteresttowritethatourformulationofthestrongstabilizationprob-lemconnectsittoanequivalentprobleminBanachalgebras,thequestionof1-stabilityofaBanachalgebra:givenapairofelements{a,b}inaBa- 196PROBLEM6.2nachalgebraBwhichsatisfiestheBezoutidentityxa+yb=1forsomex,y∈B,doesthereexistc∈B:a+cbisaunit?ThiswasshowntobethecaseforB=H∞byTreil[8]andthisprovesthateverystabilizablescalartime-invariantsystemisstronglystabilizableoverthecomplexnumberfield.ThematrixanaloguetoTreil’sresultisnotknown.ItisinterestingthattheBanachalgebraB(H)ofallboundedlinearoperatorsonagivenHilbertspaceHisnot1-stable[3].Ourstrongstabilizationproblemisthequestionwhethernestalgebrasare1-stable.BIBLIOGRAPHY[1]B.D.O.Anderson,“AnoteontheYoula-Bongiorno-Lucondition,”Automatica12(1976),387-388.[2]J.J.Bongiorno,C.N.Lu,D.C.Youla,“Single-loopfeedbackstabiliza-tionoflinearmultivariableplants,”Automatica10(1974),159-173.[3]G.Corach,A.Larotonda,“StablerangeinBanachalgebras,”J.PureandAppliedAlg.32(1984),289-300.[4]W.Dale,M.Smith,“Stabilizabilityandexistanceofsystemrepresen-tationsfordiscrete-time,time-varyingsystems,”SIAMJ.Cont.andOptim.31(1993),1538-1557.[5]C.A.Desoer,R.W.Liu,J.Murray,R.Saeks,“Feedbacksystemdesign:thefactorialrepresentationapproachtoanalysisandsynthesis,”IEEETrans.Auto.ControlAC-25(1980),399-412.[6]A.Feintuch,“RobustControlTheoryonHilbertSpace,”AppliedMath.Sciences130,Springer,1998.[7]H.Ozbay,M.Zeren,“OnthestrongstabilizationandstableH∞-controllerdesignproblemsforMIMOsystems,”Automatica36(2000),1675-1684.[8]S.Treil,“ThestablerankofthealgebraH∞equals1,”J.Funct.Anal.109(1992),130-154.[9]M.Vidyasagar,ControlSystemSynthesis:AFactorizationApproach,M.I.T.Press,1985. Problem6.3RobustnessoftransientbehaviorDiederichHinrichsen,ElmarPlischke,andFabianWirthZentrumf¨urTechnomathematikUniversit¨atBremen28334BremenGermany{dh,elmar,fabian}@math.uni-bremen.de1DESCRIPTIONOFTHEPROBLEMBydefinition,asystemoftheformx˙(t)=Ax(t),t≥0(1)(A∈Kn×n,K=R,C)isexponentiallystableifandonlyiftherearecon-stantsM≥1,β<0suchthatkeAtk≤Meβt,t≥0.(2)Therespectiverolesofthetwoconstantsinthisestimatearequitediffer-ent.Theexponentβ<0determinesthelong-termbehaviorofthesystem,whereasthefactorM≥1boundsitsshort-termortransientbehavior.Inapplicationslargetransientsmaybeunacceptable.Thisleadsustothefollowingstricterstabilityconcept.Definition1:LetM≥1,β<0.AmatrixA∈Kn×niscalled(M,β)-stableif(2)holds.Hereβ<0andM≥1canbechoseninsuchawaythat(M,β)-stabilityguaranteesbothanacceptabledecayrateandanacceptabletransientbe-havior.ForanyA∈Kn×nletγ(A)denotethespectralabscissaofA,i.e.,themaxi-mumoftherealpartsoftheeigenvaluesofA.Itiswell-knownthatγ(A)<0impliesexponentialstability.Moreprecisely,foreveryβ>γ(A)thereex-istsaconstantM≥1suchthat(2)issatisfied.However,itisunknownhowtodeterminetheminimalvalueofMsuchthat(2)holdsforagivenβ∈(γ(A),0). 198PROBLEM6.3Problem1:a)GivenA∈Kn×nandβ∈(γ(A),0),determineanalyticallytheminimalvalueMβ(A)ofM≥1forwhichAis(M,β)-stable.b)ProvideeasilycomputableformulasforupperandlowerboundsforMβ(A)andanalyzetheirconservatism.Associatedtothisproblemisthedesignproblemforlinearcontrolsystemsoftheformx˙=Ax+Bu,(3)where(A,B)∈Kn×n×Kn×m.AssumethatadesiredtransientandstabilitybehaviorfortheclosedloopisprescribedbygivenconstantsM≥1,β<0,thenthepair(A,B)iscalled(M,β)-stabilizable(bystatefeedback),ifthereexistsanF∈Km×nsuchthatA−BFis(M,β)-stable.Problem2:a)GivenconstantsM≥1,β<0,characterizethesetof(M,β)-stabili-zablepairs(A,B)∈Kn×n×Kn×m.b)Provideamethodforthecomputationof(M,β)-stabilizingfeedbacksFfor(M,β)-stabilizablepairs(A,B).Inordertoaccountforuncertaintiesinthemodel,weconsidersystemsdescribedbyx˙=A∆x=(A+D∆E)x,whereA∈Kn×nisthenominalsystemmatrix,D∈Kn×`andE∈Kq×naregivenstructurematrices,and∆∈K`×qisanunknownperturbationmatrixforwhichonlyaboundoftheformk∆k≤δisassumedtobeknown.Problem3:a)GivenA∈Kn×n,D∈Kn×`andE∈Kq×n,determineanalyticallythe(M,β)−stabilityradiusdefinedbynor(A;D,E)=infk∆k∈K`×q,∃τ>0:ke(A+D∆E)τk≥Meβτ.(M,β)(4)b)Provideanalgorithmforthecalculationofthisquantity.c)Determineeasilycomputableupperandlowerboundsforr(M,β)(A;D,E).Thetwopreviousproblemscanbethoughtofasstepstowardsthefollowingfinalproblem.Problem4:Givenasystem(A,B)∈Kn×n×Kn×m,adesiredtransientbehaviordescribedbyM≥1,β<0,andmatricesD∈Kn×`,E∈Kq×ndescribingtheperturbationstructure, ROBUSTNESSOFTRANSIENTBEHAVIOR199a)characterizetheconstantsγ>0forwhichthereexistsastatefeedbackmatrixsuchthatr(M,β)(A−BF;D,E)≥γ.(5)b)ProvideamethodforthecomputationoffeedbackmatricesFsuchthat(5)issatisfied.2MOTIVATIONANDHISTORYOFTHEPROBLEMStabilityandstabilizationarefundamentalconceptsinlinearsystemstheoryandinmostdesignproblemsexponentialstabilityistheminimalrequirementthathastobemet.Fromapracticalpointofview,however,thetransientbehaviorofasystemmaybeofequalimportanceandisoftenoneofthecriteriathatdecidesonthequalityofacontrollerinapplications.Assuch,thenotionof(M,β)−stabilityisrelatedtosuchclassicaldesigncriteriaas“overshoot”ofsystemresponses.Thequestionofhowfartransientsmoveawayfromtheoriginisofinterestinmanysituations;forinstance,ifcertainregionsofthestatespacearetobeavoidedinordertopreventsaturationeffects.Asimilarproblemoccursiflineardesignisperformedasalocaldesignforanonlinearsystem.Inthiscase,largetransientsmayresultinasmalldomainofattraction.ForanintroductiontotherelationoftheconstantMwithestimatesofthedomainofattraction,wereferto[4,Chapter5].ThesolutionofProblem4andalsooftheotherproblemswouldprovideawaytodesignlocallinearfeedbackswithgoodlocalestimatesforthedomainofattractionwithouthavingtoresorttotheknowledgeofLyapunovfunctions.WhilethelattermethodisexcellentifaLyapunovfunctionisknown,itisalsoknownthatitmaybequitehardtofindthemorifquadraticLyapunovfunctionsareusedthentheobtainableestimatesmaybefarfromoptimal,seesection3.Apartfromthesemotivationsfromcontroltherelationbetweendomainsofattractionandtransientbehavioroflinearizationsatfixedpointsisanactivefieldinrecentyearsmotivatedbyproblemsinmathematicalphysics,inparticular,fluiddynamics;see[1,10]andreferencestherein.Relatedproblemsoccurinthestudyofiterativemethodsinnumericalanalysis;seee.g.,[3].Wewouldliketopointoutthattheproblemsdiscussedinthisnotegivepointwiseconditionsintimefortheboundsandarethereforedifferentfromcriteriathatcanbeformulatedviaintegralconstraintsonthepositivetimeaxis.Intheliterature,suchintegralcriteriaaresometimesalsocalledboundsonthetransientbehavior;seee.g.,[9]whereinterestingresultsareobtainedforthiscase.Stabilityradiiwithrespecttoasymptoticstabilityoflinearsystemswereintroducedin[5]andthereisaconsiderablebodyofliteratureinvestigating 200PROBLEM6.3thisproblem.Thequestionsposedinthisnoteareanextensionoftheavailabletheoryinsofarasthetransientbehaviorisneglectedinmostoftheavailableresultsonstabilityradii.3AVAILABLERESULTSAnumberofresultsareavailableforproblem1.EstimatesofthetransientbehaviorinvolvingeitherquadraticLyapunovfunctionsorresolventinequal-itiesareknownbutcanbequiteconservativeorintractable.Moreover,formanyoftheavailableestimates,littleisknownontheirconservatism.TheHille-YosidaTheorem[8]providesanequivalentdescriptionof(M,β)-stabilityintermsofthenormofpowersoftheresolventofA.Namely,Ais(M,β)-stableifandonlyifforalln∈Nandallα∈Rwithα>βitholdsthat−nMk(αI−A)k≤.(α−β)nAcharacterizationofMastheminimaleccentricityofnormsthatareLya-punovfunctionsof(1)ispresentedin[7].Whiletheseconditionsarehardtocheck,thereisaclassical,easilyverifiable,sufficientconditionusingquadraticLyapunovfunctions.Letβ∈(γ(A),0),ifP>0satisfiestheLyapunovinequalityA∗P+PA≤2βP<0,andhasconditionnumberκ(P):=kPkkP−1k≤M2thenAis(M,β)-stable.TheexistenceofP>0satisfyingtheseconditionsmaybeposedasanLMI-problem[2].However,itcanbeshownthatifβ<0isgivenandthespectralboundofAisbelowβthenthismethodisnecessarilyconservative,inthesensethatthebestboundonMobtainableinthiswayisstrictlylargerthantheminimalbound.Furthermore,experimentsshowthatthegapbetweenthesetwoboundscanbequitelarge.Inthiscontext,notethattheproblemcannotbesolvedbyLMItechniquessincethecharacterizationoftheoptimalMforgivenβisnotanalgebraicproblem.ThereisalargenumberoffurtherupperboundsavailableforkeAtk.Thesearediscussedandcomparedindetailin[4,11],seealsothereferencestherein.Anumberoftheseboundsisalsovalidintheinfinite-dimensionalcase.Forproblem2,sufficientconditionsarederivedin[7]usingquadraticLya-punovfunctionsandLMItechniques.TheexistenceofafeedbackFsuchthatP(A−BF)+(A−BF)∗P≤2βPandκ(P)=kPkkP−1k≤M2,(6)or,equivalently,thesolvabilityoftheassociatedLMIproblem,ischaracter-izedingeometricterms.This,however,onlyprovidesasufficientconditionunderwhichProblem2canbesolved.ButtheLMIproblem(6)isfarfrombeingequivalenttoProblem2. ROBUSTNESSOFTRANSIENTBEHAVIOR201Concerningproblem3differentialRiccatiequationswereusedtoderiveboundsforthe(M,β)−stabilityradiusin[6].Supposethereexistposi-tivedefiniteHermitianmatricesP0,Q,RofsuitabledimensionssuchthatthedifferentialRiccatiequationP˙−(A−βI)P−P(A−βI)∗−E∗QE−PDRD∗P=0(7)P(0)=P0(8)hasasolutiononR+whichsatisfiesσ¯(P(t))/σ(P0)≤M2,t≥0.Thenthestructured(M,β)−stabilityradiusisatleastpr(M,β)(A;D,E)≥σ(Q)σ(R),(9)where¯σ(X)andσ(X)denotethelargestandsmallestsingularvalueofX.However,itisunknownhowtochoosetheparametersP0,Q,Rinanoptimalwayanditisunknownwhetherequalitycanbeobtainedin(9)byanoptimalchoiceofP0,Q,R.Tothebestofourknowledge,noresultsareavailabledealingwithproblem4.BIBLIOGRAPHY[1]J.S.BaggettandL.N.Trefethen,“Low-dimensionalModelsofSubcrit-icalTransitiontoTurbulence’”PhysicsofFluids9:1043–1053,1997.[2]S.Boyd,L.ElGhaoui,E.Feron,andV.Balakrishnan,LinearMatrixInequalitiesinSystemsandControlTheory,volume15ofStudiesinAppliedMathematics,SIAM,Philadelphia,1994.[3]T.BraconnierandF.Chaitin-Chatelin,“Roundoffinducesachaoticbe-haviorforeigensolversappliedonhighlynonnormalmatrices,”M.-O.Bristeauetal.(eds.),ComputationalScienceforthe21stCentury.Sym-posium,Tours,France,May5–7,1997.Chichester:JohnWiley&Sons.43-52(1997).[4]M.I.Gil’,StabilityofFiniteandInfiniteDimensionalSystems,KluwerAcademicPublishers,Boston,1998.[5]D.HinrichsenandA.J.Pritchard,“Stabilityradiusforstructuredper-turbationsandthealgebraicRiccatiequation,”Systems&ControlLet-ters,8:105–113,1986.[6]D.Hinrichsen,E.Plischke,andA.J.Pritchard,“LiapunovandRiccatiequationsforpracticalstability,”In:Proc.EuropeanControlConf.ECC-2001,Porto,Portugal,(CD-ROM),pp.2883–2888,2001. 202PROBLEM6.3[7]D.Hinrichsen,E.Plischke,andF.Wirth;“StateFeedbackStabilizationwithGuaranteedTransientBounds,”In:ProceedingsofMTNS-2002,NotreDame,IN,USA,August,2002.[8]A.Pazy,SemigroupsofLinearOperatorsandApplicationstoPartialDifferentialEquations,Springer-Verlag,NewYork,1983.[9]A.Saberi,A.A.Stoorvogel,andP.Sannuti,ControlofLinearSystemswithRegulationandInputConstraints,Springer-Verlag,London,2000.[10]L.N.Trefethen,“Pseudospectraoflinearoperators,”SIAMReview,39(3):383–406,1997.[11]K.Veseli´c,“Boundsforexponentiallystablesemigroups,”Lin.Alg.Appl.,358:195–217,2003. Problem6.4LiealgebrasandstabilityofswitchednonlinearsystemsDanielLiberzonCoordinatedScienceLaboratoryUniversityofIllinoisatUrbana-ChampaignUrbana,IL61801USAliberzon@uiuc.edu1PRELIMINARYDESCRIPTIONOFTHEPROBLEMSupposethatwearegivenafamilyfp,p∈PofcontinuouslydifferentiablefunctionsfromRntoRn,parameterizedbysomeindexsetP.Thisgivesrisetotheswitchedsystemx˙=f(x),x∈Rn(1)σwhereσ:[0,∞)→Pisapiecewiseconstantfunctionoftime,calledaswitch-ingsignal.Impulseeffects(statejumps),infinitelyfastswitching(chatter-ing),andZenobehaviorarenotconsideredhere.Weareinterestedinthefollowingproblem:findconditionsonthefunctionsfp,p∈Pwhichguaran-teethattheswitchedsystem(1)isasymptoticallystable,uniformlyoverthesetofallpossibleswitchingsignals.Ifthispropertyholds,wewillrefertotheswitchedsystemsimplyasbeingstable.Itisclearlynecessaryforeachofthesubsystems˙x=fp(x),p∈Ptobeasymptoticallystable—whichwehenceforthassume—butsimpleexamplesshowthatthisconditionaloneisnotsufficient.Theproblemposedabovenaturallyarisesinthestabilityanalysisofswitchedsystemsinwhichtheswitchingmechanismiseitherunknownortoocom-plicatedtobeexplicitlytakenintoaccount.Thisproblemhasattractedconsiderableattentionandhasbeenstudiedfromvariousangles(see[7]forreferences).Hereweexploreaparticularresearchdirection,namely,theroleofcommutationrelationsamongthesubsystemsbeingswitched.Inthefol-lowingsections,weprovideanoverviewofavailableresultsonthistopicanddelineatetheopenproblemmoreprecisely. 204PROBLEM6.42AVAILABLERESULTS:LINEARSYSTEMSInthissection,weconcentrateonthecasewhenthesubsystemsarelinear.Thisresultsintheswitchedlinearsystemx˙=Ax,x∈Rn.(2)σWeassumethroughoutthat{Ap:p∈P}isacompactsetofstablematrices.Tounderstandhowcommutationrelationsamongthelinearsubsystemsbe-ingswitchedplayaroleinthestabilityquestionfortheswitchedlinearsystem(2),considerfirstthecasewhenPisafinitesetandthematricescommutepairwise:ApAq=AqApforallp,q∈P.Thenitnothardtoshowbyadirectanalysisofthetransitionmatrixthatthesystem(2)isstable.Alternatively,inthiscaseonecanconstructaquadraticcommonLyapunovfunctionforthefamilyoflinearsubsystems˙x=Apx,p∈Passhownin[10],whichiswell-knowntoleadtothesameconclusion.AusefulobjectthatrevealsthenatureofcommutationrelationsistheLiealgebraggeneratedbythematricesAp,p∈P.ThisisthesmallestlinearsubspaceofRn×nthatcontainsthesematricesandisclosedundertheLiebracketoperation[A,B]:=AB−BA(see,e.g.,[11]).Beyondthecommutingcase,thenaturalclassesofLiealgebrastostudyinthepresentcontextarenilpotentandsolvableones.ALiealgebraisnilpotentifallLiebracketsofsufficientlyhighordervanish.SolvableLiealgebrasformalargerclassofLiealgebras,inwhichallLiebracketsofsufficientlyhighorderhavingacertainstructurevanish.IfPisafinitesetandgisanilpotentLiealgebra,thentheswitchedlinearsystem(2)isstable;thiswasprovedin[4]forthediscrete-timecase.Thesystem(2)isstillstableifgissolvableandPisnotnecessarilyfinite(aslongasthecompactnessassumptionmadeatthebeginningofthissectionholds).Theproofofthismoregeneralresult,givenin[6],reliesonthefactsthatmatricesinasolvableLiealgebracanbesimultaneouslyputinthetriangularform(Lie’sTheorem)andthatafamilyoflinearsystemswithstabletriangularmatriceshasaquadraticcommonLyapunovfunction.Itwassubsequentlyshownin[1]thattheswitchedlinearsystem(2)isstableiftheLiealgebragcanbedecomposedintoasumofasolvableidealandasubalgebrawithacompactLiegroup.Moreover,ifgfailstosatisfythiscondition,thenitcanbegeneratedbyfamiliesofstablematricesgivingrisetostableaswellastounstableswitchedlinearsystems,i.e.,theLiealgebraalonedoesnotprovideenoughinformationtodeterminewhetherornottheswitchedlinearsystemisstable(thisistrueundertheadditionaltechnicalrequirementthatI∈g).Byvirtueoftheaboveresults,onehasacompletecharacterizationofallmatrixLiealgebrasgwiththepropertythateverysetofstablegeneratorsforggivesrisetoastableswitchedlinearsystem.TheinterestingandrathersurprisingdiscoveryisthatthispropertydependsonlyonthestructureofgasaLiealgebra,andnotonthechoiceofaparticularmatrixrepresen- STABILITYOFSWITCHEDSYSTEMS205tationofg.Namely,LiealgebraswiththispropertyarepreciselytheLiealgebrasthatadmitadecompositionofthekinddescribedearlier.Thus,inthelinearcase,theextenttowhichcommutationrelationscanbeusedtodistinguishbetweenstableandunstableswitchedsystemsiswellunderstood.Lie-algebraicsufficientconditionsforstabilityaremathematicallyappealingandeasilycheckableintermsoftheoriginaldata(ithastobenoted,how-ever,thattheyarenotrobustwithrespecttosmallperturbationsinthedataandthereforehighlyconservative).3OPENPROBLEM:NONLINEARSYSTEMSWeshallnowturntothegeneralnonlinearsituationdescribedbyequa-tion(1).LinearizingthesubsystemsandapplyingtheresultsdescribedintheprevioussectiontogetherwithLyapunov’sindirectmethod,itisnothardtoobtainLie-algebraicconditionsforlocalstabilityofthesystem(1).Thiswasdonein[6,1].However,theproblemweareposinghereistoinvesti-gatehowthestructureoftheLiealgebrageneratedbytheoriginalnonlinearvectorfieldsfp,p∈Pisrelatedtostabilitypropertiesoftheswitchedsys-tem(1).Takinghigher-ordertermsintoaccount,onemayhopetoobtainmorewidelyapplicableLie-algebraicstabilitycriteriaforswitchednonlinearsystems.Thefirststepinthisdirectionistheresultprovedin[8]thatifthesetPisfiniteandthevectorfieldsfp,p∈Pcommutepairwise,inthesensethat∂fq(x)∂fp(x)n[fp,fq](x):=fp(x)−fq(x)=0∀x∈R,∀p,q∈P∂x∂xthentheswitchedsystem(1)is(globally)stable.Infact,commutativityoftheflowsisallthatisneeded,andthecontinuousdifferentiabilityassumptiononthevectorfieldscanberelaxed.Ifthesubsystemsareexponentiallystable,aconstructionanalogoustothatof[10]canbeappliedinthiscasetoobtainalocalcommonLyapunovfunction;see[12].AlogicalnextstepistostudyswitchednonlinearsystemswithnilpotentorsolvableLiealgebras.Oneapproachwouldbeviasimultaneoustriangu-larization,asdoneinthelinearcase.NonlinearversionsofLie’sTheorem,whichprovideLie-algebraicconditionsunderwhichafamilyofnonlinearsystemscanbesimultaneouslytriangularized,aredevelopedin[3,5,9].However,asdemonstratedin[2],thetriangularstructurealoneisnotsuffi-cientforstabilityinthenonlinearcontext.Additionalconditionsthatcanbeimposedtoguaranteestabilityareidentifiedin[2],buttheyarecoordinate-dependentandsocannotbeformulatedintermsoftheLiealgebra.More-over,theresultsonsimultaneoustriangularizationdescribedinthepapersmentionedaboverequirethattheLiealgebrahavefullrank,whichisnottrueinthecaseofacommonequilibrium.Thusanaltogethernewapproachseemstoberequired. 206PROBLEM6.4Insummary,themainopenquestionisthis:Q:Whichstructuralproperties(ifany)oftheLiealgebrageneratedbyanoncommutingfamilyofasymptoticallystablenonlinearvectorfieldsguaranteestabilityofeverycorrespondingswitchedsystem?Forex-ample,whendoesnilpotencyorsolvabilityoftheLiealgebraimplystability?Tobeginansweringthisquestion,onemaywanttofirstaddresssomespe-cialclassesofnonlinearsystems,suchashomogeneoussystemsorsystemswithfeedbackstructure.Onemayalsowanttorestrictattentiontofinite-dimensionalLiealgebras.AmoregeneralgoalofthispaperistopointoutthefactthatLiealgebrasseemtobedirectlyconnectedtostabilityofswitchedsystemsand,inviewofthewell-establishedtheoryoftheformerandhightheoreticalinterestaswellaspracticalimportanceofthelatter,thereisaneedtodevelopabetterunderstandingofthisconnection.ItmayalsobeusefultopursuepossiblerelationshipswithLie-algebraicresultsinthecontrollabilityliterature(see[1]forabriefpreliminarydiscussiononthismatter).BIBLIOGRAPHY[1]A.A.AgrachevandD.Liberzon,“Lie-algebraicstabilitycriteriaforswitchedsystems,”SIAMJ.ControlOptim.,40:253–269,2001.[2]D.AngeliandD.Liberzon,“Anoteonuniformglobalasymptoticsta-bilityofnonlinearswitchedsystemsintriangularform,”In:Proc.14thInt.Symp.onMathematicalTheoryofNetworksandSystems(MTNS),2000.[3]P.E.Crouch,“DynamicalrealizationsoffiniteVolterraseries,”SIAMJ.ControlOptim.,19:177–202,1981.[4]L.Gurvits,“Stabilityofdiscretelinearinclusion,”LinearAlgebraAppl.,231:47–85,1995.[5]M.Kawski,“NilpotentLiealgebrasofvectorfields‘,”J.ReineAngew.Math.,388:1–17,1988.[6]D.Liberzon,J.P.Hespanha,andA.S.Morse,“Stabilityofswitchedsystems:aLie-algebraiccondition,”SystemsControlLett.,37:117–122,1999.[7]D.Liberzon,SwitchinginSystemsandControl.Birkh¨auser,Boston,2003. STABILITYOFSWITCHEDSYSTEMS207[8]J.L.Mancilla-Aguilar,“Aconditionforthestabilityofswitchednon-linearsystems,”IEEETrans.Automat.Control,45:2077–2079,2000.[9]A.Marigo,“Constructivenecessaryandsufficientconditionsforstricttriangularizabilityofdriftlessnonholonomicsystems‘,”In:Proc.38thIEEEConf.onDecisionandControl,pages2138–2143,1999.[10]K.S.NarendraandJ.Balakrishnan,“AcommonLyapunovfunctionforstableLTIsystemswithcommutingA-matrices,”IEEETrans.Au-tomat.Control,39:2469–2471,1994.[11]H.Samelson,NotesonLieAlgebras.VanNostrandReinhold,NewYork,1969.[12]H.Shim,D.J.Noh,andJ.H.Seo,“CommonLyapunovfunctionforexponentiallystablenonlinearsystems,”In:Proc.4thSIAMConferenceonControlanditsApplications,1998. Problem6.5RobuststabilitytestforintervalfractionalorderlinearsystemsIvoPetr´aˇsDepartmentofInformaticsandProcessControlBERGFaculty,TechnicalUniversityofKoˇsiceB.Nˇemcovej3,04200KoˇsiceSlovakRepublicivo.petras@tuke.skYangQuanChenCenterforSelf-OrganizingandIntelligentSystemDept.ofElectricalandComputerEngineeringUtahStateUniversityLogan,UT-84322-4160USAyqchen@ieee.orgBlasM.VinagreDept.ofElectronicandElectromechanicalEngineeringIndustrialEngineeringSchool,UniversityofExtramaduraAvda.DeElvass/n,06071-BadajozSpainbvinagre@unex.es1DESCRIPTIONOFTHEPROBLEMRecently,arobuststabilitytestprocedureisproposedforlineartime-invari-antfractionalordersystems(LTIFOS)ofcommensurateorderswithpara-metricintervaluncertainties[6].Theproposedrobuststabilitytestmethodisbasedonthecombinationoftheargumentprinciplemethod[2]forLTIFOSandthecelebratedKharitonov’sedgetheorem.Ingeneral,anLTIFOScanbedescribedbythedifferentialequationorthecorrespondingtransfer ROBUSTSTABILITYCHECKMETHODSFORFOS209functionofnoncommensuraterealorders[7]ofthefollowingform:bsβm+...+bsβ1+bsβ0Q(sβk)m10G(s)==,(1)ansαn+...+a1sα1+a0sα0P(sαk)whereαk,βk(k=0,1,2,...)arerealnumbersandwithoutlossofgeneralitytheycanbearrangedasαn>...>α1>α0,βm>...>β1>β0.Thecoefficientsak,bk(k=0,1,2,...)areuncertainconstantswithinaknowninterval.Itiswell-knownthatanintegerorderLTIsystemisstableifalltherootsofthecharacteristicpolynomialP(s)arenegativeorhavenegativerealpartsiftheyarecomplexconjugate(e.g.,[1]).Thismeansthattheyarelocatedontheleftoftheimaginaryaxisofthecomplexs-plane.Whendealingwithnoncommensurateordersystems(or,ingeneral,withfractionalordersystems)itisimportanttobearinmindthatP(sα),α∈Risamultivaluedfunctionofs,thedomainofwhichcanbeviewedasaRiemannsurface(seee.g.,[4]).AquestionofrobuststabilitytestprocedureandproofofitsvalidityforgeneraltypeoftheLTIFOSdescribedby(1)isstillopen.2MOTIVATIONANDHISTORYOFTHEPROBLEMFortheLTIFOSwithnouncertainty,theexistingstabilitytest(orcheck)methodsfordynamicsystemswithinteger-orderssuchasRouthtabletech-nique,cannotbedirectlyapplied.Thisisduetothefactthatthechar-acteristicequationoftheLTIFOSis,ingeneral,notapolynomialbutapseudo-polynomialfunctionofthefractionalpowersofs.Ofcourse,beingthecharacteristicequationafunctionofacomplexvariable,stabilitytestbasedontheargumentprinciplecanbeapplied.Ontheotherhand,ithasbeenshown,byseveralauthorsandbyusingseveralmethods,thatforthecaseofLTIFOSofcommensurateorder,ageometricalmethodbasedontheargumentoftherootsofthecharacteristicequation(apolyno-mialinthisparticularcase)canbeusedforthestabilitycheckintheBIBO(bounded-inputbounded-output)sense(see,e.g.,[3]).Intheparticularcaseofcommensurateordersystems,itholdsthatαk=αk,βk=αk,(0<α<1),∀k∈Z,andthetransferfunctionhasthefollowingformPMαkαk=0bk(s)Q(s)G(s)=K0PNαk=K0P(sα)(2)k=0ak(s)WithN>MthefunctionG(s)becomesaproperrationalfunctioninthecomplexvariablesαandcanbeexpandedinpartialfractionsoftheform"#XNAiG(s)=K0,(3)sα+λii=1 210PROBLEM6.5whereλ(i=1,2,..,N)aretherootsofthepolynomialP(sα)orthesystemipolesthatareassumedtobesimple.Stabilityconditioncanthenbestatedthat[2,3]:Acommensurateordersystemdescribedbyarationaltransferfunction(2)isstableif|arg(λ)|>απ,withλthei-throotofi2iP(sα).FortheLTIFOSwithcommensurateorderwheresystempolesareingeneralcomplexconjugate,thestabilityconditioncanbeexpressedasfollows[2,3]:AcommensurateordersystemdescribedbyarationaltransferfunctionG(σ)=Q(σ),whereσ=sα,α∈R+,(0<α<1),isP(σ)stableif|arg(σ)|>απ,withσthei-throotofP(σ).i2iTherobuststabilitytestprocedurefortheLTIFOSofcommensurateorderswithparametricintervaluncertaintiescanbedividedintothefollowingsteps:•step1:RewritetheLTIFOSG(s)ofthecommensurateorderα,totheequivalencesystemH(σ),wheretransformationis:sα→σ,+α∈R;•step2:WritetheintervalpolynomialP(σ,q)oftheequivalencesys-temH(σ),whereintervalpolynomialisdefinedasXnP(σ,q)=[q−,q+]σi;i=0•step3:ForintervalpolynomialP(σ,q),constructfourKharitonov’spolynomials:p−−(σ),p−+(σ),p+−(σ),p++(σ);•step4:TestthefourKharitonov’spolynomialswhethertheysatisfythestabilitycondition:|arg(σ)|>απ,∀σ∈C,withσthei-throoti2iofP(σ);Notethatforlow-degreepolynomials,lessKharitonov’spolynomialsaretobetested:•Degree5:p−−(σ),p−+(σ),p+−(σ);•Degree4:p+−(σ),p++(σ);•Degree3:p+−(σ).WedemonstratedthistechniquefortherobuststabilitycheckfortheLTIFOSwithparametricintervaluncertaintiesthroughsomeworked-outillus-trativeexamplesin[6].In[6]thetime-domainanalyticalexpressionsareavailableandthereforethetime-domainandthefrequency-domainstabilitytestresults(seealso[5])canbecross-validated. ROBUSTSTABILITYCHECKMETHODSFORFOS2113AVAILABLERESULTSForgeneralLTIFOS,ifthecoefficientsareuncertainbutareknowntoliewithinknownintervals,howtogeneralizetherobuststabilitytestresultbyKharitonov’swell-knownedgetheorem?Thisisdefinitelyanewresearchtopic.Themainfutureresearchobjectivescouldbe:•AproofofvalidityoftherobuststabilitytestprocedurefortheLTIFOSofcommensurateorderswithparametricintervaluncertainties.•AnalgebraicmethodandanexactproofforthestabilityinvestigationfortheLTIFOSofnoncommensurateorderswithknownparameters.•ArobuststabilitytestprocedureofLTIFOSofnoncommensurateor-derswithparametricintervaluncertainties.BIBLIOGRAPHY[1]R.C.DorfandR.H.Bishop,ModernControlSystems,Addison-WesleyPublishingCompany,1990.[2]D.Matignon,“Stabilityresultonfractionaldifferentialequationswithapplicationstocontrolprocessing,”In:IMACS-SMCProceeding,July,Lille,France,pp.963-968,1996.[3]D.Matignon,“Stabilitypropertiesforgeneralizedfractionaldifferentialsystems,”In:ProceedingofFractionalDifferentialSystems:Models,MethodsandApplications,vol.5,pp.145-158,1998.[4]D.A.Pierre,“MinimumMean-Square-ErrorDesignofDistributedPa-rameterControlSystems,”ISATransactions,vol.5,pp.263-271,1966.[5]I.Petr´aˇsandL.Dorˇc´ak,“TheFrequencyMethodforStabilityInves-ˇtigationofFractionalControlSystems,”J.ofSACTA,vol.2,no.1-2,pp.75-85,1999.[6]I.Petr´aˇs,Y.Q.Chen,B.M.Vinagre,“Arobuststabilitytestproce-dureforaclassofuncertainLTIfractionalordersystems,”In:Proc.ofICCC2002,May27-30,Beskydy,pp.247-252,2002.[7]I.Podlubny,FractionalDifferentialEquations,AcademicPress,SanDiego,1999. Problem6.6DelayindependentanddelaydependentAizermanproblemVladimirR˘asvanDepartmentofAutomaticControlUniversityofCraiova13A.I.CuzaStreet1100CraiovaRomaniavrasvan@automation.ucv.ro1INTRODUCTIONThehalf-centuryoldproblemofAizermanconsistsinacomparisonoftheabsolutestabilitysectorwiththeHurwitzsectorofstabilityforthelinearizedsystem.Whilethefirsthasbeenshowntobe,generallyspeaking,smallerthanthesecondone,thiscomparisonstillservesasatestforthesharpnessofsufficientstabilitycriteriaasLiapunovfunctionorPopovinequality.Ontheotherhand,therearenowverypopularforlineartimedelaysystemstwotypesofsufficientstabilitycriteria:delay-independentanddelay-dependent.Thepresentpapersuggestsacomparisonofthesecriteriawiththecorre-spondingonesfornonlinearsystemswithsectorrestrictednonlinearities.Inthisway,aproblemofAizermantypeissuggestedforsystemswithdelay.Someexamplesareanalyzed.2ASIMPLEEXAMPLE.STATEMENTOFTHEPROBLEM.Considerthesimpletimedelayequationx˙+a0x(t)+a1x(t−τ)=0,τ>0(1)witha0,a1,xscalars.Itisawell-knownfact[7,9,10]thatexponentialstabilityof(1)isensuredprovidedthefollowinginequalitieshold:1+a0τ>0,−a0τ0,thisproperty,calleddelay-independentstabilityisensuredprovidedthesimpleinequalitiesa0>0,|a1|ξforξ>0,hencethefulfilmentof(4)impliesthefulfilmentof(2).LetusfollowthewayofBarbashin[6]tointroduceastabilityprobleminthenonlinearcase:givensystem(1)fora0>0,ifwereplacea0xbyϕ(x)whereϕ(x)x>0,theequilibriumattheoriginofthenonlineartimedelaysystemshouldbegloballyasymptoticallystableprovidedϕ(σ)>|a1|(5)σforthedelay-independentstability,orprovidedϕ(σ)1−1>max−a1,ψ(a1τ)(6)στinthedelay-dependentcase.Wemayviewtheaboveprobleminamoregeneralsettingandstateitasfollows:Problem:Giventhedelay-(in)dependentexponentialstabilityconditionsforsometimedelaylinearizedsystem,aretheyvalidinthecasewhenthenonlinearsystemwithasectorrestrictednonlinearity,i.e.,satisfyingϕσ2<ϕ(σ)σ<ϕσ2(7)isconsideredinsteadofthelinearone,orhavetheytobestrengthened?Itisclearthatwehavegatheredhereboththedelay-independentanddelay-dependentcases,thusdefiningastabilityproblemintwodifferentcases.ThisproblemiscalledAizermanproblem,statedhereasdelay-dependent(Aizermanproblem)anddelay-independent(Aizermanproblem).SincethisproblemintheODE(ordinarydifferentialequations)settingisnotonlywell-knownbutalsoquitewell-studied,ashortstateoftheartcouldbeuseful.3THEPROBLEMOFTHEABSOLUTESTABILITY.THEPROB-LEMSOFAIZERMANANDKALMANExactly60yearsagoapaperofB.V.Bulgakov[8]considered,apparentlyforthefirsttime,aproblemofglobalasymptoticstabilityforthezeroequi- 214PROBLEM6.6libriumofafeedbackcontrolsystemcomposedofalineardynamicpartandanonlinearstaticelementx˙=Ax−bϕ(c?x)(8)wherex,b,caren-dimensionalvectors,Aisan×nmatrixandϕ:R→Risacontinuousfunction.Theonlyadditionalassumptionaboutϕwasitslocationinsomesectorasdefinedby(7),wheretheinequalitiesmaybenon-strict.Inthisveryfirstpaper,onlyconditionsfortheabsenceofself-sustainedoscillationswereobtainedbutinanother,morefamouspaperofLurieandPostnikov[17]globalasymptoticstabilityconditionswereobtainedforasystem(8)of3dorderwithϕ(σ)satisfyingϕ(σ)σ>0,i.e.,satisfying(7)withϕ=0,ϕ=+∞.TheconditionsobtainedusingasuitablychosenLiapunovfunctionoftheform“quadraticformofthestatevariablesplusanintegralofthenonlinearity”wereinfactvalidforthewholeclassofnonlinearfunctionsdefinedbyϕ(σ)σ>0.Laterthiswascalledabsolutestabilitybutitisobviouslyarobuststabilityproblemsinceitdealswiththeuncertaintyonthenonlinearfunctiondefinedby(7).Weshallnotinsistmoreonthisproblemandweshallconcentrateonanotherone,connectedwithit,statedbyM.A.Aizerman[1,2].Thislastproblemison(8)anditslinearizedversionx˙=Ax−bhc?x(9)i.e.,system(8)withϕ(σ)=hσ.Itisknownthatthenecessaryandsuffi-cientconditionsofasymptoticstabilityfor(9)willrequirehtoberestricted
tosomeintervalh,hcalledtheHurwitzsector.Ontheotherhand,forsystem(8)theabsolutestabilityproblemisstated:findconditionsofglobalasymptoticstabilityofthezeroequilibriumforallfunctionssatisfying(7).Allfunctionsincludethelinearoneshencetheclassofsystemsdefinedby(8)islargerthantheclassofsystemsdefinedby(9).Consequentlythesec-

torϕ,ϕfrom(7)maybeatmostaslargeastheHurwitzsectorh,h.TheAizermanproblemaskssimply:dothesesectorsalwayscoincide?TheAizermanconjectureassumed:yes.Thefirstcounter-exampletothisconjecturehasbeenproducedbyKrasovskii[16]intheformofa2ndordersystemofspecialform.Themostcelebratedcounterexampleisa3rdordersystemandbelongstoPliss[21].TodayweknowthattheconjectureofAizermandoesnotholdingeneral.Neverthelesstheproblemitselfstimulatedinterestingresearchthatcouldbesummarizedasseekingnecessaryandsufficientconditionsforabsolutestability.Astraightforwardapplicationofthesestudiesischeckingofthesharpnessfor“traditional”absolutestabilitycriteria:theLiapunovfunctionandthefrequencydomaininequalityofPopov.InfactthisisnothingmorebutcomparisonoftheabsolutestabilitysectorwiththeHurwitzsector.OnecanmentionheretheresultsofVoronov[26]andhisco-workersonwhattheycalled“stabilityintheHurwitzsector.”OthernoteworthyresultsbelongtoPyatnitskiiwhofoundnecessaryandsufficientconditionsofabsolutestabilityconnectedtoaspecialvariational DELAYINDEPENDENT215problemandtoN.E.Barabanov(e.g.,[4]).AmongtheresultsofBarabanovwewouldliketomentionthoseconcernedwiththeso-calledKalmanproblemandconjecturetopicsthatdeservesomeparticularattention.Inhispaper[15]R.E.Kalmanreplacedtheclassofnonlinearfunctionsdefinedby(7)bytheclassofdifferentiablefunctionswithsloperestrictionsγ<ϕ0(σ)<γ(10)

TheKalmanproblemasks:docoincidetheintervalsγ,γandh,hthelastonebeingpreviouslydefinedbytheinequalitiesofHurwitz?Theanswertothisquestionisalsonegativebutitsstoryisnotquitestraightforward.AgoodreferenceisthepaperofBarabanov[3]andwewouldliketofollowsomeofthepresentationthere:theonlycounterexampleknownuptothatpaperhadbeenpublishedbyFitts[11]andtheauthorsofawell-knownandcitedmonographinthefield(NarendraandTaylor,[18])werecitingitasabasicargumentforthenegativeanswertoKalmanconjecture.InfacttherewasnoproofinthepaperofFittsbutjustasimulation:aspecificlinearsubsystemhadbeenadopted,aspecificnonlinearityalsoandself-sustainedperiodicoscillationswerecomputedforvariousvaluesofasystem’sparameter.InhisimportantpaperBarabanov[3]wasabletoproverigorouslythefollowing:•theanswertotheproblemofKalmanispositiveforall3dordersys-tems;itfollowsthatthesystemofPlisscounter-exampleisabsolutelystablewithintheHurwitzsectorprovidedtheclassofthenonlinearfunctionsisdefinedby(10)insteadof(7);•thecounterexamplegivenbyFittsisnotcorrectatleastforsomesubsetofitsparametersasisfollowsbysimpleapplicationoftheBrockettWillemsfrequencydomaininequalityforabsolutestabilityofsystemswithsloperestrictednonlinearity.Moreover,thepaperofBarabanovprovidesanalgorithmoffindingsystemswithanon-trivialperiodicsolution;inthisway,aprocedureisgivenforconstructingcounterexamplestothetwoconjecturesdiscussedabove.Ob-viously,thetechniqueofBarabanovseemsanechoofthepioneeringpaperofBulgakov[8],butweshallinsistnomoreonthissubject.4STABILITYANDABSOLUTESTABILITYOFTHESYSTEMSWITHTIMEDELAYA.Weshallconsiderforsimplicityonlythecaseofthesystemsdescribedbyfunctionaldifferentialequationsofdelayedtype(accordingtothewell-knownclassificationoftheseequations;see,forinstance,BellmanandCooke[7])andweshallrestrictourselvestothesingledelaycase.Inthelinearcase,thesystemisdescribedbyx˙=A0x(t)+A1x(t−τ),τ>0(11) 216PROBLEM6.6ExponentialstabilityofthissystemisensuredbythelocationintheLHP(left-handplane)oftherootsofthecharacteristicequation
detλI−A−Ae−λτ=0(12)01wheretheLHS(left-handside)isaquasipolynomial.WehaveheretheRouth-Hurwitzproblemforquasipolynomials.Thisproblemhasbeenstudiedsincethefirstapplicationsof(11);thebasicresultsaretobefoundinthepaperofPontryagin[22]andinthememoirofChebotarevandMeiman[9].AvaluablereferenceisthebookofStepan[25].Fromthistopic,weshallrecallthefollowing.StartingfromtheiralgebraicintuitionChebotarevandMeimanpointedoutthat,accordingtoSturmtheory,theRouth-Hurwitzconditionsforquasi-polynomialshavetobeexpressedasafinitenumberofinequalitiesthatmightbetranscendental.Thedetailedanalysisperformedintheirmemoirforthe1stand2nddegreequasipolynomialsshowedtwotypesofinequalities:oneofthemcontainedonlyalgebraicinequalities,whiletheothercontainedalsotranscendentalinequalities;thefirstonescorrespondtostabilityforarbitraryvaluesofthedelayτ,whilethesecondonesputsomelimitationsonthevaluesofτ>0forwhichexponentialstabilityof(11)holds.Thesystemdescribedby(1)andconditions(2),(3)and(4)aregoodillustrationsofthis.Theaspectisquitetransparentintheexamplesanalysisperformedthroughoutauthor’sbook[23]aswellasthroughoutthebookofStepan[25].Wemayseeherethedifferenceoperatedbetweenwhatwillbecalledlaterdelay-independentanddelay-dependentstability.ThisdifferencewillbecomeimportantafterthepublicationofthepaperofHaleetal.[13],whichwillbeassimilatedbythecontrolcommunityafteritsincorporationinthe3deditionofHale’smonograph,authorizedbyHaleandVerduynLunel[14].Therearebynowdozensofreferencesconcerningdelay-dependentanddelay-independentRouth-Hurwitzproblemfor(11);wesendthereadertothebooksofS.I.Niculescu[19,20]withtheirrichreferencelists.Aspecialcaseof(2)thatisinfacttheunderlyingtopicofmostreferencescitedin[19,20]isstabilityforsmalldelays.Asshownin[10]thestabilityinequalitiesaregivenbyarccos−a0a1a1+a0>0,0≤τ|a0|(otherwise(4)holdsandstabilityisdelay-independent).Infactmostrecentresearchdefinesdelay-dependentstabilityasabove,i.e.,aspreservationofstabilityforsmalldelays(abetternamewouldbe“delayrobuststability”since,accordingtoapaperofJaroslavKurzweil,“smalldelaysdon’tmatter”).B.Sincelinearblockswithdelayareusualincontrol,introductionofsystemswithsectorrestrictednonlinearities(7)isonlynatural.ThemostsuitablereferencesonthisproblemarethemonographsofA.Halanay[12]andofthe DELAYINDEPENDENT217author[23].Ifwerestrictourselvesagaintothecaseofdelayedtypewithasingledelay,thenamodelproblemcouldbethesystemx˙=Ax(t)+Ax(t−τ)−bϕ(c?x(t)+c?x(t−τ))(14)0101wherex,b,c0,c1aren-vectorsandA0,A1aren×nmatrices;thenonlinearfunctionϕ(σ)satisfiesthesectorcondition(7).Followingauthor’sbook[23]weshallconsiderascalarversionof(14):x˙+a0x(t)+ϕ(x(t)+c1x(t−τ))=0(15)whereϕ(σ)σ>0.Assumethata0>0andapplythefrequencydomaininequalityofPopovforϕ=+∞:Re(1+jωβ)H(jω)>0,∀ω≥0(16)Since1+ce−τs1H(s)=s+a0thefrequencydomaininequalityreads
a2+ω2β(1+ccosωτ)+ω(aβ−1)sinωτ010>0a2+ω20−1BychoosingthePopovparameterβ=a0theaboveinequalitybecomes1+c1cosωτ>0,∀ω≥0,(17)whichcannotholdfor∀ωbutonlywith|c1|<1.ThefrequencydomaininequalityofPopovprescribesinthiscaseadelay-independentabsolutestability.5BACKTOTHEEXAMPLEWehavestatedadelay-independentandadelay-dependentAizermanprob-lemforsystemswithtimedelayinarathergeneralsettingthatcouldincluderathergeneralsystemsofdifferentialequationswithdeviatedargumentwhilewechosethestartingsystemasaverysimpleone,ofthedelayedtype.Inthefollowing,weshallillustratethesolvingofaspecificproblemfortheinitialexample.Consider,forinstance,thedelay-independentAizermanproblemdefinedabove,forsystem(1)replacedbyx˙+a1x(t−τ)+ϕ(x(t))=0(18)whereϕ(σ)σ>0.Takingintoaccountthat(4)suggestsϕ(σ)>|a1|σweintroduceanewnonlinearfunctionf(σ)=ϕ(σ)−|a1|σ 218PROBLEM6.6andobtainthetransformedsystem(viaasectorrotation):x˙+|a1|x(t)+a1x(t−τ)+f(x(t))=0(19)Forthissystem,weapplythefrequencydomaininequalityofPopovforϕ=+∞,i.e.,inequality(16);here1H(s)=(20)s+|a1|+a1e−sτandthefrequencydomaininequalityreducestoβω2−(βasinωτ)ω+|a|+acosωτ≥0(21)111whichisfulfilledprovidedthefreePopovparameterβischosenfrom0<β|a1|<2(22)(moredetailsconcerningmanipulationofthefrequencydomaininequalityfortimedelaysystemsmaybefoundinauthor’sbook[23]).Itfollowsthat(19)isabsolutelystableforthenonlinearitiessatisfyingf(σ)σ>0i.e.ϕ(σ)σ>|a|σ2:thejuststateddelay-independentAizermanproblem1for(1)and(18)hasbeenansweredpositively.6CONCLUDINGREMARKSSincetheclassofsystemswithtimedelaysisconsiderablylargerthantheclassofsystemsdescribedbyordinarydifferentialequations,weexpectvar-ioussettingsofAizerman(orKalman)problems.Thecaseoftheequationsofneutraltypethatexpresspropagationphenomenawasnotyetanalyzedfromthispointofvieweveniftheabsolutestabilityhasbeenconsideredforsuchsystems(seeauthor’sbook[23]).Suchavarietyofsystemsandproblemsshouldbestimulatingforthedevelopmentofthetoolsofanalysis.Itisaknownfactthatthefrequencydomaininequalitiesarebettersuitedfordelay-independentresults,aswellasthemostlyusedLiapunov-KrasovskiifunctionalsleadingtofinitedimensionalLMIs(seee.g.,thecitedbooksofNiculescu[19,20]);theLiapunov-Krasovskiiapproachhasneverthelesssome“opening”todelay-dependentresultsanditisworthtryingtoapplyitinsolvingthedelay-dependentAizermanproblem.ThealgebraicapproachsuggestedbythememoirofChebotarevandMeiman[9]couldbealsoappliedaswellasthe(non)-existenceofself-sustainedoscillationsthatsendsbacktoBulgakovandPliss.AsinthecasewithoutdelaythestatementandsolvingoftheAizermanproblemscouldberewardingfromatleasttwopointsofview:extensionoftheclassofthesystemshavingan“almostlinearbehavior”[5,24]andrefinementofanalysistoolsbytestingthe“sharpness”ofthesufficientconditions. DELAYINDEPENDENT219BIBLIOGRAPHY[1]M.A.Aizerman,“Onconvergenceofthecontrolprocessunderlargedeviationsoftheinitialconditions,”(inRussian)Avtom.itelemekh.vol.VII,no.2-3,pp.148-167,1946.[2]M.A.Aizerman,“Onaproblemconcerningstability”inthelarge”ofdynamicalsystems,”(inRussian)Usp.Mat.Naukt.4,no.4,pp.187-188,1949.[3]N.E.Barabanov,“AbouttheproblemofKalman,”(inRussian)Sib.Mat.J.vol.XXIX,no.3,pp.3-11,1988.[4]N.E.Barabanov,“OntheproblemofAizermanfornon-stationarysys-temsof3dorder,”(inRussian)Differ.uravn.vol.29,no.10,pp.1659-1668,1992.[5]I.Barb˘alatandA.Halanay,“Conditionsdecomportementpresquelinairedanslathoriedesoscillations,”Rev.Roum.Sci.Techn.-Electrotechn.etEnerg.vol.29,no.2,pp.321-341,(1974).[6]E.A.Barbashin,Introductiontostabilitytheory,(inRussian)NaukaPubl.House,Moscow,1967.[7]R.E.BellmanandK.L.Cooke,DifferentialDifferenceEquations,Acad.Press,N.Y.,1963.[8]B.V.Bulgakov,“Self-sustainedoscillationsofcontrolsystems,”(inRus-sian)DANSSSRvol.37,no.9,pp.283-287,1942.[9]N.G.ChebotarevandN.N.Meiman,“TheRouth-Hurwitzproblemforpolynomialsandentirefunctions,”(inRussian)TrudyMat.Inst.V.A.Steklov,vol.XXVI,1949.[10]L.E.El’sgol’tsandS.B.Norkin,Introductiontothetheoryandap-plicationsofdifferentialequationswithdeviatingarguments(inRussian),NaukaPubl.House,Moscow,1971;EnglishversionbyAcad.Press,1973.[11]R.E.Fitts,“Twocounter-examplestoAizerman’sconjecture,”IEEETrans.vol.AC-11,no.3,pp.553-556,July1966.[12]A.Halanay,DifferentialEquations.Stability.Oscillations.TimeLags,Acad.Press,N.Y.,1966.[13]J.K.Hale,E.F.InfanteandF.S.P.Tsen,“Stabilityinlineardelayequations,”J.Math.Anal.Appl.vol.105,pp.533-555,1985.[14]J.K.HaleandS.VerduynLunel,IntroductiontoFunctionalDifferentialEquations,SpringerVerlag,1993. 220PROBLEM6.6[15]R.E.Kalman,“Physicalandmathematicalmechanismsofinstabilityinnonlinearautomaticcontrolsystems,”Trans.ASME,vol.79,no.3,pp.553-566,April1957.[16]N.N.Krasovskii,“Theoremsconcerningstabilityofmotionsdeterminedbyasystemoftwoequations,”(inRussian)Prikl.Mat.Mekh.(PMM),vol.XVI,no.5,pp.547-554,1952.[17]A.I.LurieandV.N.Postnikov,“Onthetheoryofstabilityforcontrolsystems,”(inRussian)Prikl.Mat.Mekh.(PMM),vol.VIII,no.3,pp.246-248,1944.[18]K.S.NarendraandJ.H.Taylor,Frequencydomainstabilitycriteria’,Acad.Press,N.Y.,1973.[19]S.I.Niculescu,Syst`emesaretard’,Diderot,Paris,1997.[20]S.I.Niculescu,“Delayeffectsonstability:Arobustcontrolapproach,”LectureNotesinControlandInformationSciences,no.269,SpringerVerlag,2001.[21]V.A.Pliss,SomeProblemsoftheTheoryofStabilityofMotionintheLarge(inRussian),LeningradStateUniv.Publ.House,Leningrad,(1958).[22]L.S.Pontryagin,“Onzerosofsomeelementarytranscendentalfunc-tions,”(inRussian)Izv.ANSSSRSer.Matem.vol.6,no.3,pp.115-134,1942,withanAppendixpublishedinDANSSSR,vol.91,no.6,pp.1279-1280,1953.[23]Vl.R˘asvan,Absolutestabilityofautomaticcontrolsystemswithtimede-lay(inRomanian),EdituraAcademiei,Bucharest,1975;RussianversionbyNaukaPubl.House,Moscow,1983.[24]Vl.R˘asvan,“Almostlinearbehaviorinsystemswithsectorrestrictednonlinearities,”Proc.RomanianAcademySeriesA:Math.,Phys.,Techn.Sci,Inform.Sci.,vol.2,no.3,pp.127-135,2002.[25]G.Stepan,“Retardeddynamicalsystems:stabilityandcharacteristicfunction,”PitmanRes.NotesinMath.vol.210,LongmanScientificandTechnical,1989.[26]A.A.Voronov,Stability,controllability,observability(inRussian),NaukaPubl.House,Moscow,1979. Problem6.7OpenproblemsincontroloflineardiscretemultidimensionalsystemsLiXuDept.ofElectronicsandInformationSystemsAkitaPrefecturalUniversityHonjo,Akita015-0055Japanxuli@akita-pu.ac.jpZhipingLinSchoolofEEENanyangTechnologicalUniversitySingapore639798RepublicofSingaporeEZPlin@ntu.edu.sgJiang-QianYingFacultyofRegionalStudiesGifuUniversity1-1Yanagido,Gifu501-1193Japanying@cc.gifu-u.ac.jpOsamiSaitoFacultyofEngineeringChibaUniversityInageku,Chiba263-8522JapanYoshihisaAnazawaDept.ofElectronicsandInformationSystemsAkitaPrefecturalUniversityHonjo,Akita015-0055Japan 222PROBLEM6.71INTRODUCTIONThischaptersummarizesseveralopenproblemscloselyrelatedtothefol-lowingcontrolproblemsinlineardiscretemultidimensional(nD,n≥2)systems:•outputfeedbackstabilizabilityandstabilization,•strongstabilizabilityandstabilization,or,equivalently,simultaneousstabilizabilityandstabilizationoftwogivennDsystems,•regulationandtrackingcontrol.Thoughsomeoftheopenproblemspresentedherehavebeenscatteredintheliterature(seee.g.,[7,13,24,26]andthereferencestherein),itseemsthattheyhavenotreceivedsufficientattention,andwereevenoccasionallymistakenasknownresults.Thepurposeofthischapteristwofold:first,toclearupsuchconfusionsandtocallformoreeffortsforthesolutiontotheseexistingopenproblems;andsecond,toproposesomerelatednewopenproblems.2DESCRIPTIONOFTHEPROBLEMS4LetR[z],wherez=(z1,...,zn),bethesetofnDpolynomialsinthevari-ablesz1,...,znwithcoefficientsinthefieldofrealnumbersR;R(z)thesetofnDrationalfunctionsoverR;Rs[z],Rs(z)thesetof(structurally)stablenDpolynomialsandrationalfunctions,respectively,i.e.,nDpolyno-n4nmialshavingnozerosinU¯={z∈C:|z1|≤1,...,|zn|≤1}andnDrationalfunctionswhosedenominatorsbelongtoRs[z].Similarly,letC[z]bethesetofnDpolynomialsoverthefieldofcomplexnumbersC,etc.Problem1:Leta1(z),...,aM(z)∈R[z]begiven.LetIdenotetheidealgeneratedbya1(z),...,aM(z),andV(I)thealgebraicvarietyofI,i.e.,V(I)={z∈Cn:a(z)=0,i=1,...,M}.SupposethatV(I)∩U¯n=∅.iFindaconstructivemethodtoobtainh1(z),...,hM(z)∈R[z]suchthata(z)h(z)+···+a(z)h(z)6=0inU¯n(1)11MMor,equivalently,toobtainh˜1(z),...,h˜β(z)∈Rs(z)suchthata1(z)h˜1(z)+···+aβ(z)h˜M(z)=1(2)Thisproblemcanbereducedtoproblem10,inthesensethatoncethefol-lowingproblemissolved,problem1canbesolvedeasilyusingtheGr¨obnerbasisapproach[6,11,23]. OPENPROBLEMSINCONTROLOFNDSYSTEMS223Problem10:UndertheassumptionthatV(I)∩U¯n=∅,findaconstructivemethodtoobtainapolynomials(z)suchthats(z)∈Rs[z]ands(z)vanishesonV(I).Problem2:Letg(z),a2(z),...,aM(z)∈R[z]begiven.Supposethatitisknownthatthereexisth2(z),...,hM(z)∈Rs(z)suchthatXMg(z)+a(z)h(z)6=0inU¯n.(3)iii=2Findaconstructivemethodtoobtainsuchh2(z),...,hM(z).Problem3:LetD(z)∈Rm×m[z],N(z)∈Rm×l[z]begiven.Denoteby4m+l
α1(z),...,αM(z)them×mminorsof[D(z)N(z)]withM=mandα1(z)=detD(z).SupposethatD(z)andN(z)areminorleftcoprime(MLC),i.e.,α1(z),...,αM(z)havenononunitcommonfactorsoverR[z][30].Supposethatsomeh2(z),...,hM(z)∈Rs(z)havebeenfoundsuchthatXMdetD(z)+α(z)h(z)6=0inU¯n,(4)iii=2showwhetherornotthereexistsamatrixC(z)∈Rl×m(z)suchthatsdet(D(z)+N(z)C(z))6=0inU¯n,(5)andfurtherfindaconstructivemethodtoobtainsuchaC(z)whenitsex-istenceisknown.Problem4:LetD(z)∈Rm×m[z],N(z)∈Rl×m[z]begiven.ShowtheconditionfortheexistenceofX(z)∈Rm×m(z),Y(z)∈Rm×l(z)suchthatssD(z)X(z)+Y(z)N(z)=I,(6)andfurtherfindaconstructivemethodtoobtainX(z),Y(z)whentheexistenceisknown.3MOTIVATIONSSincethebeginningof1970s,growinginterestshaveledtoaconsider-ablenumberofcontributionstothetheoryofnDsystems.Thisis,ofcourse,mainlyduetothediversityoftheactualandpotentialapplicationsofnDsystemstheoryembracingnDsignalprocessing,variable-parameterandlumped-distributednetworksynthesis,delay-differentialsystems,linearsys-temsofpartialdifferenceanddifferentialequations,iterativelearningcontrolsystems,linearmultipassprocesses,etc.(see,e.g.,thebooksof[5,10],thespecialissuesof[2,3,15,18]andthereferencestherein).Asitiswell-known,thegeneralizationoftheconventionalone-dimensional(1D)systemstheory 224PROBLEM6.7toitsnDcounterpartisnontrivialbecauseofmanydeepandsubstantialdifferencesbetweenthetwo.Despiteofthetremendousprogressmadeinthepastthreedecades,therearestillmanyopenproblemsintheareaofnDsystems,eithertheoreticallychallengingorpracticallyimportantorboth,remainingtobetackled.Inthischapter,wearemainlyconcernedwithopenproblemsintheareaofnDcontrolsystems,althoughsomeoftheseproblemsarealsocloselyrelatedtonDsignalprocessing,astobediscussedshortly.AnnDMIMO(multi-input-multi-output)systemP(z)∈Rm×l(z)issaidtobeoutputfeedback(structurally)stabilizableifthereisacompensatorC(z)∈Rl×m(z)suchthattheclosed-looptransfermatrixH(z)definedbelowis(structurally)stable,i.e.,eachentryofH(z)isinRs(z):(I+PC)−1−P(I+CP)−1H(z)=−1−1.(7)C(I+PC)(I+CP)IfC(z)itselfcanbefurtherchosentobestable,P(z)issaidtobestronglystabilizable.Itcanbeshownthattwounstablesystemscanbesimultane-ouslystabilizedbyasinglecompensatorifacertainsystemconstructedfromthetwogivenonesisstronglystabilizable[21,20].ConsiderannDsystemgivenbyaleftmatrixfractionaldescription(MFD)P(z)=D(z)−1N(z)withD(z)∈Rm×m[z]andN(z)∈Rm×l[z].Forsimplicity,supposethatD(z)andN(z)areMLC.Then,P(z)isstabilizableifandonlyifV(I)∩U¯n=∅(8)whereV(I)isthealgebraicvarietyoftheidealIgeneratedbythem×mminorsα1(z),...,αM(z)of[D(z)N(z)]asdefinedinproblem3[11,12,19,23].Thisconditionisequivalent[4]tothatthereexisth1(z),...,hM(z)∈R[z]suchthatXMα(z)h(z)6=0inU¯n.(9)iii=1Further,ithasbeenshownthat,onceh1(z),...,hM(z)satisfying(9)havebeenfound,astabilizingcompensatorC(z)=Y(z)X(z)−1∈Rl×m(z)withX(z)∈Rm×m[z],Y(z)∈Rl×m[z]canbeconstructed[12,23].Therefore,thestabilizabilityforagivenP(z)isequivalenttotheconditionof(8)orthesolvabilityof(9),whilethestabilizationproblem,i.e.,theproblemofdesigningastabilizingcompensator,forastabilizableP(z)isreducedtotheproblemofconstructingh1(z),...,hM(z)in(9),whichisjustwhathasbeendescribedinProblem1.Asmentionedpreviously,problem1canbefurtherreducedtoproblem10.Inadditiontotheabove-mentionedstabilizationproblem,problem1alsoplaysanessentialroleinnDsignalprocessing,suchasthedesignofnDfilterbanks(see,e.g.,[1,7,17]).Forthestrongstabilizabilityandstabilizationproblems,itisfurtherrequiredthatC(z)=Y(z)X(z)−1∈Rl×m(z),whichisequivalenttorequiringthats OPENPROBLEMSINCONTROLOFNDSYSTEMS225detX(z)∈Rs[z][26].Itistheneasytosee[28]thatanecessaryconditionforP(z)tobestronglystabilizableisthatthereexiste2(z),...,eM(z)suchthatXMdetD(z)+α(z)e(z)6=0inU¯n(10)iii=2wheretheassumptionα1(z)=detD(z)isused.ThisconditionhasbeenshowntobealsosufficientforSIMO(single-input-multi-output)andMISO(multi-input-single-output)nDsystems[28],andforthesespecialcases,ife2(z),...,eM(z)satisfying(10)canbefound,astablestabilizingcompen-satorC(z)canthenbeconstructivelyobtained.However,thesufficiencyofthisconditionforageneralMIMOnDsystemisstillunknownandtheprob-lemforconstructingastablestabilizingcompensatorforageneralMIMOnDsystemisstillopen,evenife2(z),...,eM(z)havebeenobtained.Prob-lem2correspondstothesolutionproblemof(10),whileproblem3relatestothestrongstabilizabilityandstabilizationofageneralMIMOnDsystem.Itisclearthatthesolutionofproblem2isassumedtobeapreconditionforproblem3.Anotherimportantissueinfeedbacksystemdesignisthetrackinganddis-turbancerejectionproblems.Itcanbeshownthatequation(6)playsacentralroleforvarioustypesofregulationandtrackingproblems(see,e.g,[21,22,25]).So,problem4isrelatetothesolvabilityandsolutionofregu-lationandtrackingproblemsofnDMIMOsystems.4AVAILABLERESULTSProblem1andproblem10:ThetestofthesolvabilityconditionV(I)∩U¯n=∅andthesolutionsofproblem1andproblem10forthecasen=2canbefoundin[11,23]andthereferencestherein.Forthecasen≥3,ifIisofzerodimensional,i.e.,V(I)consistsofonlyafinitenumberofpoints,thesolvabilitytestandsolutionconstructioncanthenbecarriedoutbyutilizingtheGr¨obnerbasisapproach[6,24].Forsomeotherspecialcaseswhenn≥3,see[14].Anothersolutionmethodhasbeensuggestedin[4]byusinganalyticfunctiontheory.However,webelievethatthismethodisnotconstructive.Further,asthedeterminationofwhetherornotV(I)∩U¯n=∅canbeformulatedasatypicalquantifiereliminationproblem,itmaybepos-sibletosolveitbyCylindricalAlgebraicDecomposition(CAD)techniquesdevelopedinthefieldofcomputeralgebra[8,9].Problem2:Problem2ismuchmorecomplicatedanddifficultthanits1Dcounterpart(seee.g.,[21]).Tosolvethisproblem,wehavetofollowtwosteps:first,toseeifthereexisth2(z),...,hM(z)∈Cs(z),andthentoseeifthereexisth2(z),...,hM(z)∈Rs(z),suchthat(3)holds.Itisalso 226PROBLEM6.7interestingtonotethatincontrasttothe1Dcase,Problem2maypossessnosolutiononRs(z),evenifithasasolutiononCs(z)[26,28,29].Necessaryandsufficientconditionsfortheexistenceofh2(z),...,hM(z)∈Cs(z)andRs(z),respectively,havebeenshownin[26,28],whichcanbeverifiedbytheCADbasedmethod[27].Problem3:Problem3hasbeenconsideredandsolvedforthecaseswhenm=1orl=1(i.e.,forSIMOandMISOnDsystems)[28],andforthecasewhenD(z)andN(z)satisfycertainconditionsgivenin[16].Problem4:Necessaryandsufficientsolvabilityconditionsandconstructivesolutionproceduresforthecasen=2canbefoundin[22].BIBLIOGRAPHY[1]S.Basu,“Multi-dimensionalfilterbanksandwavelets:Asystemtheo-reticperspective,”J.FranklinInst.,335B(8),pp.1367-1409,1998.[2]S.BasuandB.C.L´evy,Eds.,“SpecialIssueonMultidimensionalFilterBanksandWavelets,”MultidimensionalSystemsandSignalProcessing,7/8,1996/7.[3]S.BasuandM.N.S.Swamy,Eds.,“SpecialIssueonMultidimensionalSignalsandSystems,”IEEETrans.CircuitsSyst.I,49,2002.[4]C.A.BerensteinandD.C.Struppa,“1-Inversesforpolynomialmatricesofnon-constantrank,”Systems&ControlLetters,6,pp.309–314,1986.[5]N.K.Bose,AppliedMultidimensionalSystemsTheory,NewYork:VanNostrandReinhold,1982.[6]B.Buchberger,“Gr¨obnerbases:Analgorithmicmethodinpolynomialidealtheory,”inMultidimensionalSystemsTheory:Progress,Direc-tionsandOpenProblemsN.K.Bose,ed.,Dordrecht:Reidel,pp.184–232,1985.[7]C.CharoenlarpnopparutandN.K.Bose,“Gr¨obnerBasesforProblemSolvinginMultidimensionalSystems,”MultidimensionalSystemsandSignalProcessing,12(3/4),pp.365-376,2001.[8]G.Collins,“Quantifiereliminationforrealclosedfieldsbycylindricalalgebraicdecomposition,”,LNCS,33,pp.134-183,1975.[9]G.Collins,H.Hong,“Partialcylindricalalgebraicdecompositionandquantifierelimination,”J.SymbolicComputation,12,pp.299-328,1991. OPENPROBLEMSINCONTROLOFNDSYSTEMS227[10]K.GalkowskiandJ.Wood,Eds.,MultidimensionalSignals,CircuitsandSystems,Taylor&Francis,2001.[11]J.P.GuiverandN.K.Bose,“Causalandweaklycausal2-Dfilterswithapplicationsinstabilizations,”In:MultidimensionalSystemsTheory:Progress,DirectionsandOpenProblems,N.K.Bose,ed.,Dordrecht:Reidel,pp.52–100,1985.[12]Z.Lin,FeedbackStabilizabilityofMIMOn-DLinearSystems,Multi-dimensionalSystemsandSignalProcessing,9,pp.149-172,1998.[13]Z.Lin,“OutputfeedbackstabilizabilityandstabilizationoflinearnDsystems,”In:MultidimensionalSignals,CircuitsandSystems,K.GalkowskiandJ.Wood,eds.,Taylor&Francis,pp.59-76,2001.[14]Z.Lin,J.Lam,K.GalkowskiandS.Xu,“AConstructiveApproachtoStabilizabilityandStabilizationofaClassofnDSystems,”Multidi-mensionalSystemsandSignalProcessing,12(3/4),pp.329-344,2001.[15]Z.LinandL.Xu,Eds.,“SpecialIssueonApplicationsofGr¨obnerBasestoMultidimensionalSystemsandSignalProcessing,”MultidimensionalSystemsandSignalProcessing,12,2001.[16]Z.Lin,J.Q.Ying,L.Xu,“AnAlgebraicApproachtoStrongStabiliz-abilityofLinearnDMIMOSystems,”IEEETrans.Automat.Contr.,47(9),pp.1510-1514,2002.[17]H.Park,T.Kalker,M.Vetterli,“Gr¨obnerBasesandMultidimensionalFIRMultirateSystems,”MultidimensionalSystemsandSignalProcess-ing,8(1/2),pp.11-30,1997.[18]E.RogersandP.Rocha,Eds.,“RecentProgressinMultidimensionalControlTheoryandApplications,”MultidimensionalSystemsandSig-nalProcessing,11,2000.[19]S.Shankar,V.R.Sule,“AlgebraicGeometricAspectsofFeedbackSta-bilization,”SIAMJ.Contr.Optim.,30,pp.11-30,1992.[20]S.Shankar,“Anobstructiontothesimultaneousstabilizationoftwon-Dplants,”ActaApplicandaeMathematicae,36,pp.289-301,1994.[21]M.Vidyasagar,ControlSystemSynthesis:AFactorizationApproach,Cambridge,MA:MITPress,1985.[22]L.Xu,O.Saito,andK.Abe.“BilateralPolynomialMatrixEquationsinTwoIndeterminates,”MultidimensionalSystemsandSignalProcessing,1(4),pp.363–37,1990.[23]L.Xu,O.Saito,andK.Abe.“OutputFeedbackStabilizabilityandStabilizationAlgorithmsfor2Dsystems,”MultidimensionalSystemsandSignalProcessing,5(1),pp.41–60,1994. 228PROBLEM6.7[24]L.Xu,J.Q.Ying,O.Saito,“FeedbackStabilizationforaClassofMIMOn-DSystemsbyGr¨obnerBasisApproach,”In:AbstractofFirstInternationalWorkshoponMultidimensionalSystems,pp.88-90,Poland,1998.[25]L.Xu,O.Saito,J.Q.Ying,“2DFeedbackSystemDesign:TheTrackingandDisturbanceRejectionProblems,”ProceedingsofISCAS99,V,pp.13-16,Orlando,USA,1999.[26]J.Q.Ying,“Conditionsforstrongstabilizabilitiesofn-dimensionalsys-tems,”MultidimensionalSystemsandSignalProcessing,9,pp.125–148,1998.[27]J.Q.Ying,L.Xu,Z.Lin,“Acomputationalmethodfordeterminingstrongstabilizabilityofn-Dsystems”,J.SymbolicComputation,27,pp.479-499,1999.[28]J.Q.Ying,“OnthestrongstabilizabilityofMIMOn-dimensionallinearsystems,”SIAMJ.Contr.Optim.,38,pp.313–335,2000.[29]J.Q.Ying,Z.Lin,L.Xu,“Somealgebraicaspectsofthestrongstabi-lizabilityoftime-delaylinearsystems,”IEEETrans.Automat.Contr.,46,pp.454–457,2001.[30]D.C.YoulaandG.Gnavi,“Notesonn-dimensionalsystemtheory,”IEEETrans.CircuitsSyst.,26,pp.105–111,1979. Problem6.8AnopenprobleminadaptativenonlinearcontroltheoryLeonidS.ZhiteckijInt.CentreofInform.TechnologiesandSystemsInstituteofCybernetics40ProspectAkademikaGlushkovaMSP03680Kiev187Ukrainezls@d310.icyb.kiev.ua1STATEMENTOFTHEPROBLEMWedealwiththeproblemofgloballystableadaptivecontrolfordiscrete-time,time-invariant,nonlinear,butlinearlyparameterized(LP)systemsde-scribedbythedifferenceequationy=θTϕ(x)+bu+v,(1)ti−1t−1twherey:Z+→Randu:Z+→Rarethemeasurableoutputandcontrolttinput,respectively,andv:Z+→Ristheunmeasureddisturbance(thetintegertdenotesthediscretetime).θ∈Rdandb∈Raretheunknownparametervectorandscalar(d≥1).f(·):RN→RdrepresentsaknownnonlinearvectorfunctiondependingonthevectorxT=[y,...,y]t−1t−1t−NofNpastoutputs.Itsgrowthisgivenbykϕ(x)k=O(kxkβ)askxk→∞.(2)Assumethatvtisupperboundedbysomefiniteη,i.e.,kvtk∞≤η<∞,(3)wherekvtk∞:=sup0≤t<+∞|vt|denotesthel∞-normofvt.Toregulateytaroundzero,wechoosethewell-knowncertaintyequivalence(CE)feedbackcontrollawu=−b−1θTϕ(x),(4)ttttwherebtandθtaretheestimatesofunknownbandθthataretobeupdatedonlinebyusingeithergradientorleastsquares(LS)basedalgorithms.These 230PROBLEM6.8classicalrecursiveadaptationalgorithmsmaybewritteninageneralformasθ¯t=θ¯t−1+αtΩθ(Pt,yt,ϕ¯(xt−1))(5)Pt=Pt−1−ΩP(Pt−1,ϕ¯(xt−1),αt)(6)withαt=Ωα(yt,ϕ(xt−1)),αt≥0,(7)whereθ¯T=[θT,b],¯ϕT=[ϕT,u]aretheextendedvectors,Pisapositivetttttttdefinite(d+1)×(d+1)matrixandΩ:R(d+1)×(d+1)×R×Rd+1→Rd+1,θΩ:R(d+1)×(d+1)×Rd+1×R→R(d+1)×(d+1)andΩ:R×Rd→R.PαDefinition.System(1)issaidtobegloballystabilizableifthereexistsanadaptivefeedbackcontroloftheform(4)-(7)suchthatlimsupt→∞|yt|<∞foranyinitialx∈RN,θ¯∈Rd+1,α∈R+,someP>0andagiven0000sequenceofthedisturbances{vt}satisfying(3).Now,weformulatetheproblemasfollows:determinethetriple(Ωθ,ΩP,Ωα)suchthattheadaptivefeedbackcontrol(4)-(7)willensuretheglobalstabi-lizabilityofsystem(1)forthegivenclassof{vt}∈l∞providedthatϕ(x)belongstoagivenclassofnonlinearitieshavingagrowthrate(2)withsomeβsatisfying1<β<β?,whereβ?needstobeevaluated.Theproblemstatedthusgeneralizestheproblemsolvedin[1]and[4]totheboundeddisturbancecase.Thisisanopenanddifficultproblemintheadaptivecontroltheory.Tothebestoftheauthor’sknowledge,therearenoavailableresultssolvingitforkvtk∞6=0,whereasthesolutiontoitscontinuous-timecounterpartisknown.2MOTIVATIONIncontrasttotheadaptivecontrolofnonlinearcontinuous-timesystems,wheresubstantialbreakthroughsinthetheoreticalareahavebeenachievedbythemiddleofthe1990s(see,e.g.,[3],[5],etc.),veryfewsimilarworksareavailableintheliteraturethataddresstheglobalstableadaptivecontroldesignfordiscrete-timesystemswithnonlinearities[1],[2],[4],[6]-[8].Oneoftheinherentdifficultiesofdiscrete-timeadaptivecontrolisthattheLya-punovstabilitytechniquestypicallyexploitedinthecontinuoustimecasemaynotbestraightforwardlyextendedtoitsdiscrete-timecounterpart,asdetailedin[4],[6],[8].IthasbeenshowninSectionIIof[4],andin[8]and[9]thattheso-calledKeyTechnicalLemma,whichhasplayedakeyroleinanalyzingtheadaptationalgorithmsoftype(5)-(7)appliedtolineardiscrete-timesystems,canbeusedtoderivethestabilizabilitypropertiesofadaptivenonlinearLPsystemswithanonlinearitywhosegrowthrate(2) ANOPENPROBLEMINADAPTATIVENONLINEARCONTROLTHEORY231islinear(β=1).Unfortunately,thisstabilityanalysistoolisnolongervalidifϕ(x)hasagrowthratefasterthanlinear,i.e.,β>1(see,e.g.,[4],[8]).Insuchasituation,thefollowingquestionsnaturallyarise:Canthelineargrowthrestrictionβ=1berelaxedwithoutgoingtotheinstabilityofclosedloop?WhatarethelimitationsofgradientandLSbasedalgorithms?Ananswertothesequestionscanbepartiallyfoundinrecentworks[2],[7]dealingwithasimilarprobleminthestochasticframework.Althoughtheresultsof[2],[7]shedsomelightonrestrictionsthatmustbeimposedonβtoachieveglobalstability,however,thequestionofhowtheymightbeextendedtothenonstochasticcase,where{vt}∈l∞,hasnotbeenresolvedasyet.3RELATEDRESULTSThefirststepallowingtorelaxthelineargrowthconditionwithrespecttokϕ(x)khasbeenmadebyKanellakopoulos[4]whodealtwiththescalarone-parametricdisturbance-freesystemofform(1)(N=1,d=1,vt≡0)providedthatthegainbisknownandequalto1.InSectionIIIof[8]ithasbeenestablishedthattheLSalgorithm(5),(6)withthenonlineargaindeterminedin(7)asα=1+ϕ2(x)canbeusedtoadaptivelystabilizettsystem(1)foranysmoothnonlinearityϕ(x):R→Rindependentlyofitsgrowthrate.Toderivethisglobalstabilityresult,KanellakopoulosemployedtheLyapunovfunctionV=ln(1+x2)+cP−1θ˜2+P2tttttwithsomec>0,whereθ˜t=θ−θtistheparametererrorvector.Theadaptivecontrolofsystem(1)withnodisturbanceandb=1hasalsobeenstudiedbyGuoandWei[1].Incontrastto[4],theseauthorsusedthestandard(αt≡1)LSbasedalgorithmofform(5),(6).Byexploitinga−1newtheoreticaltoolbasedonsomeboundednesspropertiesof{detPt},theyhaveprovedthatifd=1,thentheclosed-loopadaptivesystem(1),(5)-(7)isgloballystablewheneverβ<8.Ithasbeenalsoestablishedforthemultiparametercase(d>1)thattheglobalstabilityconditionisβd<4(seetheorem3of[4]).ThefundamentallimitationsofthestandardLS-basedadaptivecontrolappliedtosystem(1)withd=1andb=1inthepresenceofstochastic{vt}havebeenestablishedbyGuo[2]whoprovedthatagloballystabilizingadaptiveLS-basedcontrollercanbedesignedifandonlyifβ<4.Recently,XieandGuo[7]showedthatifd1,thenthelineargrowthrestriction(β=1)cannotbeessentiallyrelaxedingeneraltogloballystabilizesystem(1)subjectedtoaGaussianwhitenoise{vt},unlessadditionalconditionsonnumberdandthestructureofϕ(·)areimposed(seeRemark3of[7]).Itseemsthatanewtheoreticaltoolshouldbedevisedtosolvetheproblemformulatedabove. 232PROBLEM6.8BIBLIOGRAPHY[1]L.GuoandC.Wei,“Globalstability/instabilityofLS-baseddiscrete-timeadaptivenonlinearcontrol,”In:Proc.13thIFACWorldCongress,SanFrancisco,CA,USA,Vol.K,pp.277-282,1996.[2]L.Guo,“Oncriticalstabilityofdiscrete-timeadaptiveNnonlinearcon-trol,”IEEETrans.Automat.Contr.,42,pp.1488-1499,1997.[3]I.Kanellakopoulos,P.V.KokotovicandA.S.Morse,“Systematicdesignofadaptivecontrollersforfeedbacklinearizablesystems,”IEEETrans.Automat.Contr.,36,pp.1242-1253,1991.[4]I.Kanellakopoulos,“Discrete-timeadaptivenonlinearsystem,”IEEETrans.Automat.Contr.,39,pp.2362-2365,1994.[5]M.Krstic,I.Kanellakopoulos,andP.V.Kokotovic,NonlinearandAdap-tiveControlDesign,N.Y.:Wiley,1995.[6]Y.SongandJ.W.Grizzle,“Adaptiveoutput-feedbackcontrolofaclassofdiscretetimenonlinearsystems,”In:Proc.1993Amer.Contr.Conf.,,SanFrancisco,USA,pp.1359-1364,1993.[7]L.-L.XieandL.Guo,“Fundamentallimitationsofdiscrete-timeadaptivenonlinearcontrol,”IEEETrans.Automat.Contr.,44,pp.1777-1782,1999.[8]P.-C.YehandP.V.Kokotovic,“Adaptivecontrolofaclassofnonlineardiscrete-timesystems,”Int.J.Control,62,pp.303-324,1995.[9]L.S.Zhiteckij,“Singularity-freestableadaptivecontrolofaclassofnonlineardiscrete-timesystems,”In:Proc.15thIFACWorldCongress,Barcelona,Spain,2002. Problem6.9GeneralizedLyapunovtheoryanditsomega-transformableregionsSheng-GuoWangCollegeofEngineeringUniversityofNorthCarolinaatCharlotteCharlotte,NC28223-0001USAswang@uncc.edu1DESCRIPTIONOFTHEPROBLEMTheopenproblemdiscussedhereisaGeneralizedLyapunovTheoryanditsΩ-transformableregions.First,weprovidethedefinitionoftheΩ-transformableregionsanditsdegrees.Thentheopenproblemispresentedanddiscussed.Definition1:(Gutman&Jury1981)AregionΩ={(x,y)|f(λ,λ∗)=f(x+jy,x−jy)=f(x,y)<0}(1)vxyisΩ-transformableifanytwopointsα,β∈ΩimplyRe[f(α,β∗)]<0,vwherefunctionf(λ,λ∗)=f(x,y)=0istheboundaryfunctionofthexyregionΩvandvisthedegreeofthefunctionf.Otherwise,theregionΩvisnon-Ω-transformable.ItisnoticedthataregionononesideofalineandaregionwithinacircleintheplanebothareΩ-transformableregions.However,someregionsarenon-Ω-transformableregions.OpenProblem:(GeneralizedLyapunovTheory)ConsideramatrixA∈Cn×nandanyΩ-transformableregionΩdescribedbyf(x,y)=f(λ,λ∗)0sothat|y(t)|>ρ(|w(t)|)forallt∈[0,t1],then|y(t)|≤β(|x(0)|,t)∀t∈[0,t1].TheMESpropertyimpliestheSITproperty.Theconversedoesnotholdingeneral,butistrueunderadditionalassumptionsonthesystem.Definition:Letρ∈K.WesaythatalowersemicontinuousfunctionV:Rn→RisalowersemicontinuousSIT-Lyapunovfunctionforsystem(1)≥0withgainρif•thereexistα1,α2∈K∞sothatα1(|h(ξ)|)≤V(ξ)≤α2(|ξ|),∀ξsothat|h(ξ)|>ρ(|g(ξ)|),•thereexistsα3:R≥0→R≥0continuouspositivedefinitesothatforeachξsothat|h(ξ)|>ρ(|g(ξ)|),ζ·v≤−α3(V(ξ))∀ζ∈∂DV(ξ),∀v∈F(ξ).(2)(Here∂Ddenotesaviscositysubgradient.)Theorem:Letasystemoftheform(1)andafunctionρ∈Kbegiven.Thefollowingareequivalent.i.ThesystemsatisfiestheSITpropertywithgainρ.ii.ThesystemadmitsalowersemicontinuousSIT-Lyapunovfunctionwithgainρ. 242PROBLEM6.10iii.ThesystemadmitsalowersemicontinuousexponentialdecaySIT-Lyapunovfunctionwithgainρ.Furtherdetailsareavailablein[3]and[2].BIBLIOGRAPHY[1]R.A.FreemanandP.V.Kokotovi’c,RobustNonlinearControlDesign,State-SpaceandLyapunovTechniques,Birkhauser,Boston,1996.[2]B.Ingalls,ComparisonsofNotionsofStabilityforNonlinearControlSystemswithOutputs,Ph.D.thesis,RutgersUniversity,NewBrunswick,NewJersey,USA,2001.Availableatwww.cds.caltech.edu/∼ingalls.[3]B.Ingalls,E.D.Sontag,andY.Wang,“Measurementtoerrorstability:anotionofpartialdetectabilityfornonlinearsystems,”submitted.[4]A.Isidori,NonlinearControlSystemsII,Springer-Verlag,London,1999.[5]H.K.Khalil,NonlinearSystems,SecondEdition,Prentice-Hall,UpperSaddleRiver,NJ,1996.[6]P.Kokotovi´candM.Arcak,“Constructivenonlinearcontrol:Progressinthe90s,”InvitedPlenaryTalk,IFACCongress,In:Proc.14thIFACWorldCongress,thePlenaryandIndexVolume,pp.49–77,Beijing,1999.[7]M.Krichman,E.D.Sontag,andY.Wang,“Input-output-to-statestabil-ity,”SIAMJournalonControlandOptimization39,2001,pp.1874-1928,2001.[8]M.Krsti´candH.Deng,StabilizationofUncertainNonlinearSystems,Springer-Verlag,London,1998.[9]M.Krsti´c,I.Kanellakopoulos,andP.V.Kokotovi´c,NonlinearandAdap-tiveControlDesign,JohnWiley&Sons,NewYork,1995.[10]R.Sepulchre,M.Jankovic,P.V.Kokotovi´c,ConstructiveNonlinearControl,Springer,1997.[11]E.D.Sontag,“Smoothstabilizationimpliescoprimefactorization,”IEEETransactionsonAutomaticControl34,1989,pp.435–443.[12]E.D.Sontag,“TheISSphilosophyasaunifyingframeworkforstability-likebehavior,”In:NonlinearControlintheYear2000(Volume2)(Lec-tureNotesinControlandInformationSciences,A.Isidori,F.Lamnabhi-Lagarrigue,andW.Respondek,eds.),Springer-Verlag,Berlin,2000,pp.443-468. SMOOTHLYAPUNOVCHARACTERIZATIONOFMES243[13]E.D.SontagandY.Wang,“Oncharacterizationsoftheinput-to-statestabilityproperty,”Systems&ControlLetters24,pp.351–359,1995.[14]E.D.SontagandY.Wang,“Detectabilityofnonlinearsystems,”In:ProceedingsoftheConferenceonInformationSciencesandSystems(CISS96),Princeton,NJ,1996,pp.1031–1036.[15]E.D.SontagandY.Wang,“Output-to-statestabilityanddetectabilityofnonlinearsystems,”Systems&ControlLetters29,pp.279–290,1997.[16]E.D.SontagandY.Wang,“Notionsofinputtooutputstability,”Systems&ControlLetters38,pp.351–359,1999.[17]E.D.SontagandY.Wang,“Lyapunovcharacterizationsofinputtoout-putstability,”SIAMJournalonControlandOptimization39,pp.226–249,2001. PART7Controllability,Observability Problem7.1Timeforlocalcontrollabilityofa1-DtankcontainingafluidmodeledbytheshallowwaterequationsJean-MichelCoronUniversit´eParis-SudD´epartementdeMath´ematiqueBˆatiment42591405OrsayFranceJean-Michel.Coron@math.u-psud.fr1DESCRIPTIONOFTHEPROBLEMWeconsidera1-Dtankcontaininganinviscidincompressibleirrotationalfluid.Thetankissubjecttoone-dimensionalhorizontalmoves.Weassumethatthehorizontalaccelerationofthetankissmallcomparedtothegravityconstantandthattheheightofthefluidissmallcomparedtothelengthofthetank.ThismotivatestheuseoftheSaint-Venantequations[5](alsocalledshallowwaterequations)todescribethemotionofthefluid;see,e.g.,[2,Sec.4.2].Aftersuitablescalingarguments,thelengthofthetankandthegravityconstantcanbetakentobeequalto1;see[1].Thenthedynamicsequationsconsideredare,see[3]and[1],Ht(t,x)+(Hv)x(t,x)=0,(1)v2vt(t,x)+H+(t,x)=−u(t),(2)2xv(t,0)=v(t,1)=0,(3)ds(t)=u(t),(4)dtdD(t)=s(t),(5)dtwhere•H(t,x)istheheightofthefluidattimetandforx∈[0,1],•v(t,x)isthehorizontalwatervelocityofthefluidinareferentialat-tachedtothetankattimetandforx∈[0,1](intheshallowwater 248PROBLEM7.1model,allthepointsonthesameverticalhavethesamehorizontalvelocity),•uisthehorizontalaccelerationofthetankintheabsolutereferential,•sisthehorizontalvelocityofthetank,•Disthehorizontaldisplacementofthetank.Thisisacontrolsystem,denotedΣ,where•thestateisY=(H,v,s,D),•thecontrolisu∈R.Still,byscalingarguments,wemayassumethat,foreverysteadystate,H,whichisthenaconstantfunction,isequalto1;see[1].OneisinterestedinthelocalcontrollabilityofthecontrolsystemΣaroundtheequilibriumpoint(Ye,ue):=((1,0,0,0),0).Ofcourse,thetotalmassofthefluidisconservedsothat,foreverysolutionof(1)to(3),Z1dH(t,x)dx=0.(6)dt0Onegets(6)byintegrating(1)on[0,1]andbyusing(3togetherwithanintegrationbyparts.)Moreover,ifHandvareofclassC1,itfollowsfrom(2)and(3)thatHx(t,0)=Hx(t,1)(=−u(t)).(7)ThereforeweintroducethevectorspaceEoffunctionsY=(H,v,s,D)∈C1([0,1])×C1([0,1])×R×RsuchthatHx(0)=Hx(1),(8)v(0)=v(1)=0,(9)andconsidertheaffinesubspaceY⊂EofY=(H,v,s,D)∈EsatisfyingZ1H(x)dx=1.(10)0Withthesenotations,wecandefineatrajectoryofthecontrolsystemΣ.Definitionofatrajectory:LetT1andT2betworealnumberssatisfyingT16T2.Afunction(Y,u)=((H,v,s,D),u):[T1,T2]→Y×RisatrajectoryofthecontrolsystemΣif(i)thefunctionsHandvareofclassC1on[T,T]×[0,1],12 TIMEFORLOCALCONTROLLABILITYOFA1-DTANK249(ii)thefunctionssandDareofclassC1on[T,T]andthefunctionuis12continuouson[0,T],(iii)theequations(1)to(5)holdforevery(t,x)∈[T1,T2]×[0,1].Forw∈C1([0,1]),let|w|1:=Max{|w(x)|+|wx(x)|;x∈[0,1]}.WenowconsiderthefollowingpropertyoflocalcontrollabilityofΣaround(Ye,ue).DefinitionofP(T):LetT>0.ThecontrolsystemΣsatisfiestheprop-ertyP(T)if,forevery,thereexistsη>0suchthat,foreveryY0=(H0,v0,s0,D0)∈Y,andforeveryY1=(H1,v1,s1,D1)∈Ysuchthat|H0−1|1+|v0|1+|H1−1|1+|v1|1+|s0|+|s1|+|D0|+|D1|<η,thereexistsatrajectory(Y,u):[0,T]→Y×R,t7→((H(t),v(t),s(t),D(t)),u(t))ofthecontrolsystemΣsuchthatY(0)=Y0andY(T)=Y1,(11)and,foreveryt∈[0,T],|H(t)−1|+|v(t)|+|s(t)|+|D(t)|+|u(t)|<.(12)11OuropenproblemistofindforwhichT>0P(T)holds.WeconjecturethatP(T)holdsifandonlyifT>2.2MOTIVATIONANDHISTORYOFTHEPROBLEMTheproblemofcontrollabilityofthesystemΣhasbeenraisedbyF.Dubois,N.Petit,andP.Rouchonin[3].Letusrecallthattheyhavestudiedinthispaperthecontrollabilityofthelinearizedcontrolsystemaround(Ye,ue).Thislinearizedcontrolsystemisht+vx=0,vt+hx=−u(t),v(t,0)=v(t,1)=0,(Σ0)(13)dsd(t)=u(t),tdD(t)=s(t),dtwherethestateis(h,v,s,D)∈Y0,withZ1Y0:=(h,v,s,D)∈E;hdx=0,0andthecontrolisu∈R.Itisprovedin[3]thatΣ0isnotcontrol-lable.Itisalsoprovedin[3]that,evenifΣ0isnotcontrollable,forany 250PROBLEM7.1T>1,onecanmoveduringtheintervaloftime[0,T]fromanysteadystate(h0,v0,s0,D0):=(0,0,0,D0)toanysteadystate(h1,v1,s1,D1):=(0,0,0,D1)forthelinearcontrolsystemΣ0;seealso[4]whenthetankhasanonstraightbottom.Unfortunately,thisdoesnotimplythattherelatedproperty(movefrom(H0,v0,s0,D0):=(0,0,0,D0)to(H1,v1,s1,D1):=(0,0,0,D1)alsoholdsforthenonlinearcontrolsystemΣ,evenif|D1−D0|isarbitrarysmallbutnot0.Infactweconjecturethat,for>0smallenough,evenif|D1−D0|isarbitrarilysmallbutnot0,oneneedsT>2tomovefrom(H0,v0,s0,D0):=(1,0,0,D0)to(H1,v1,s1,D1):=(1,0,0,D1)forthenonlinearcontrolsystemΣifonerequires(12).3AVAILABLERESULTSClearly,P(T)impliesP(T0)forT≤T0.Usingthecharacteristicsofthehyperbolicsystem(1)-(2),oneeasilyseesthatP(T)doesnotholdT<1.Itisprovedin[1]thatP(T)holdsforTlargeenough.Themethodusedin[1]requires,atleast,T>2.BIBLIOGRAPHY[1]J.-M.Coron,“Localcontrollabilityofa1-Dtankcontainingafluidmod-eledbytheshallowwaterequations,”reprintUniversityParis-Sud,2002,acceptedforpublicationinESAIM:COCV.[2]L.Debnath,NonlinearWaterWaves,AcademicPress,SanDiego,1994.[3]F.Dubois,N.Petit,andP.Rouchon,“Motionplanningandnonlinearsimulationsforatankcontainingafluid,”ECC99.[4]N.PetitandP.Rouchon,“Dynamicsandsolutionstosomecontrolprob-lemsforwater-tanksystems,”preprint,CIT-CDS00-004,2000,acceptedforpublicationinIEEETransactionsonAutomaticControl.[5]A.J.C.B.deSaint-Venant,“Th´eoriedumouvementnonpermanentdeseaux,avecapplicationsauxcruesdesrivi`ereset`al’introductiondesmar´eesdansleurlit,”C.R.Acad.Sci.Paris,53pp.147-154,1971. Problem7.2AHautustestforinfinite-dimensionalsystemsBirgitJacobFachbereichMathematikUniversit¨atDortmundD-44221DortmundGermanybirgit.jacob@math.uni-dortmund.deHansZwartDepartmentofAppliedMathematicsUniversityofTwenteP.O.Box217,7500AEEnschedeTheNetherlandsh.j.zwart@math.utwente.nl1DESCRIPTIONOFTHEPROBLEMWeconsidertheabstractsystemx˙(t)=Ax(t),x(0)=x0,t≥0(1)y(t)=Cx(t),t≥0,(2)onaHilbertspaceH.HereAistheinfinitesimalgeneratorofanexpo-nentiallystableC0-semigroup(T(t))t≥0andbythesolutionof(1)wemeanx(t)=T(t)x0,whichistheweaksolution.IfCisaboundedlinearoper-atorfromHtoasecondHilbertspaceY,thenitisstraightforwardtoseethaty(·)in(2)iswell-definedandcontinuous.However,inmanyPDE’s,rewrittenintheform(1)-(2),CisonlyaboundedoperatorfromD(A)toY(D(A)denotesthedomainofA),althoughtheoutputisawell-defined(locally)squareintegrablefunction.Inthefollowing,CwillalwaysbeaboundedoperatorfromD(A)toY.NotethatD(A)isadensesubsetofH.Iftheoutputislocallysquareintegrable,thenCiscalledanadmissibleobservationoperator,seeWeiss[11].ItisnothardtoseethatsincetheC0-semigroupisexponentiallystable,theoutputislocallysquareintegrableifandonlyifitissquareintegrable.Usingtheuniformboundednesstheorem,weseethattheobservationoperatorCisadmissibleifandonlyifthere 252PROBLEM7.2existsaconstantL>0suchthatZ∞kCT(t)xk2dt≤Lkxk2,x∈D(A).(3)YH0AssumingthattheobservationoperatorCisadmissible,system(1)-(2)issaidtobeexactlyobservableifthereisaboundedmappingfromtheoutputtrajectorytotheinitialcondition,i.e.,thereexistsaconstantl>0suchthatZ∞kCT(t)xk2dt≥lkxk2,x∈D(A).(4)YH0Oftentheemphasisisonexactobservabilityonafiniteinterval,whichmeansthattheintegralin(4)isover[0,t0]forsomet0>0.However,forexponen-tiallystablesemigroups,bothnotionsareequivalent,i.e.,if(4)holdsandthesystemisexponentiallystable,thenthereexistsat0>0suchthatthesystemisexactlyobservableon[0,t0].Thereisastrongneedforeasyverifiableequivalentconditionsforexactob-servability.BasedontheobservabilityconjecturebyRussellandWeiss[9]wenowconjecturethefollowing:ConjectureLetAbetheinfinitesimalgeneratorofanexponentiallystableC0-semigrouponaHilbertspaceHandletCbeanadmissibleobservationoperator.Thensystem(1)-(2)isexactlyobservableifandonlyif(C1)(T(t))t≥0issimilartoacontraction,i.e.,thereexistsaboundedoper-atorSfromHtoH,whichisboundedlyinvertiblesuchthat(ST(t)S−1)isacontractionsemigroup;andt≥0(C2)thereexistsam>0suchthatk(sI−A)xk2+|Re(s)|kCxk2≥m|Re(s)|2kxk2(5)HYHforallcomplexswithnegativerealpart,andforallx∈D(A).Ourconjectureisarevisedversionofthe(false)conjecturebyRussellandWeiss;theydidnotrequirethatthesemigroupissimilartoacontraction.2MOTIVATIONANDHISTORYOFTHECONJECTURESystem(1)-(2)withA∈Cn×nandC∈Cp×nisobservableifandonlyifsI−Arank=nforalls∈C.(6)CThisisknownastheHautustest,duetoHautus[2]andPopov[8].IfAisastablematrix,then(6)isequivalenttocondition(C2).AlthoughtherearesomegeneralizationsoftheHautustesttodelaydifferentialequations AHAUTUSTESTFORINFINITE-DIMENSIONALSYSTEMS253(see,e.g.,Klamka[6]andthereferencestherein)thefullgeneralizationoftheHautustesttoinfinite-dimensionallinearsystemsisstillanopenproblem.Itisnothardtoseethatif(1)-(2)isexactlyobservable,thenthesemigroupissimilartoacontraction,seeGrabowskiandCallier[1]andLevan[7].Condition(C2)wasfoundbyRussellandWeiss[9]:theyshowedthatthisconditionisnecessaryforexactobservability,and,undertheextraassump-tionthatAisbounded,theyshowedthatthisconditionalsoissufficient.Withouttheexplicitusageofcondition(C1)itwasshownthatcondition(C2)impliesexactobservabilityif•AhasaRieszbasisofeigenfunctions,Re(λn)=−ρ1,|λn+1−λn|>ρ2,whereλnaretheeigenvaluesofA,andρ1,ρ2>0,[9];orif•minequation(5)isone,[1];orif•Aisskew-adjointandCisbounded,ZhouandYamamoto[12];orif•AhasaRieszbasisofeigenfunctions,andY=Cp,JacobandZwart[5].Recently,weshowedthat(C2)isnotsufficientingeneral,[4].TheC0-semigroupinourcounterexampleisananalyticsemigroup,whichisnotsimilartoacontractionsemigroup.TheoutputspaceintheexampleisjustthecomplexplaneC.3AVAILABLERESULTSANDCLOSINGREMARKSInordertoprovetheconjectureitissufficienttoshowthatforexponentiallystablecontractionsemigroupscondition(C2)impliesexactobservability.Itiswell-knownthatsystem(1)-(2)isexactlyobservableifandonlyifthereexistsaboundedoperatorLthatispositiveandboundedlyinvertibleandsatisfiestheLyapunovequationhAx1,Lx2iH+hLx1,Ax2iH=hCx1,Cx2iY,forallx1,x2∈D(A).(7)FromtheadmissibilityofCandtheexponentialstabilityofthesemigroup,oneeasilyobtainsthatequation(7)hasaunique(non-negative)solution.RussellandWeiss[9]showedthatCondition(C2)impliesthatthissolutionhaszerokernel.ThustheLyapunovequation(2)couldbeastartingpointforaproofoftheconjecture.Wehavestatedourconjectureforinfinite-dimensionaloutputspaces.How-ever,itcouldbethatitonlyholdsforfinite-dimensionaloutputspaces.IftheoutputspaceYisone-dimensionalonecouldtrytoprovetheconjectureusingpowerfultoolsliketheSz.-Nagy-Foiasmodeltheorem(see[10]).Thistoolwasquiteusefulinthecontextofadmissibilityconditionsforcontraction 254PROBLEM7.2semigroups[3].Basedonthisobservation,itwouldbeofgreatinteresttocheckourconjecturefortherightshiftsemigrouponL2(0,∞).Webelievethatexponentialstabilityisnotessentialinourconjecture,andcanbereplacedbystrongstabilityandinfinite-timeadmissibility,see[5].Notethatourconjectureisalsorelatedtotheleft-invertibilityofsemigroups,see[1]and[4]formoredetails.BIBLIOGRAPHY[1]P.GrabowskiandF.M.Callier,“Admissibleobservationoperators,semigroupcriteriaofadmissibility,”IntegralEquationandOperatorTheory,25:182–198,1996.[2]M.L.J.Hautus,“Controllabilityandobservabilityconditionsforlin-earautonomoussystems,”Ned.Akad.Wetenschappen,Proc.Ser.A,72:443–448,1969.[3]B.JacobandJ.R.Partington,“TheWeissconjectureonadmissibilityofobservationoperatorsforcontractionsemigroups,”IntegralEquationsandOperatorTheory,40(2):231-243,2001.[4]B.JacobandH.Zwart,“DisproofoftwoconjecturesofGeorgeWeiss,”Memorandum1546,FacultyofMathemat-icalSciences,UniversityofTwente,2000.Availableathttp://www.math.utwente.nl/publications/[5]B.JacobandH.Zwart,“Exactobservabilityofdiagonalsystemswithafinite-dimensionaloutputoperator,”Systems&ControlLetters,43(2):101-109,2001.[6]J.Klamka,ControllabilityofDynamicalSystems,KluwerAcademicPublisher,1991.[7]N.Levan,“Theleft-shiftsemigroupapproachtostabilityofdistributedsystems,”J.Math.Ana.Appl.,152:354–367,1990.[8]V.M.Popov,HyperstabilityofControlSystems.EdituraAcademiei,Bucharest,1966(inRomanian);Englishtrans.bySpringerVerlag,Berlin,1973.[9]D.L.RussellandG.Weiss,“Ageneralnecessaryconditionforexactobservability,”SIAMJ.ControlOptim.,32(1):1–23,1994.[10]B.Sz.-NagyandC.Foias,HarmonicanalysisofoperatorsonHilbertspaces,Amsterdam-London:North-HollandPublishingCompany.XIII,1970. AHAUTUSTESTFORINFINITE-DIMENSIONALSYSTEMS255[11]G.Weiss,“Admissibleobservationoperatorsforlinearsemigroups,”IsraelJournalofMathematics,65:17–43,1989.[12]Q.ZhouandM.Yamamoto,“Hautusconditionontheexactcontrolla-bilityofconservativesystems,”Int.J.Control,67(3):371–379,1997. Problem7.3ThreeproblemsinthefieldofobservabilityPhilippeJouanLaboratoireR.SalemCNRSUMR6085Universit´edeRouenMath´ematiques,siteColbert76821Mont-Saint-AignanCedexFrancePhilippe.Jouan@univ-rouen.fr1INTRODUCTION.LetXbeaC∞(resp.Cω),connectedmanifold.WeconsideronXthesystemx˙=f(x,u)Σ=(1)y=h(x)wherex∈X,u∈U=[0,1]m,andy∈Rp.TheparametrizedvectorfieldfandtheoutputfunctionhareassumedtobeC∞(resp.Cω).Inordertoavoidcertaincomplications,thestatespaceXisassumedtobecompact,butthisassumptionisnotcrucial(wecanforinstanceassumethatthevectorfieldfvanishesoutofarelativelycompactopensubsetofX).Thethreeproblemsaddressedhereinconcernobservabilityandtheexistenceofobserversforsuchsystems.2PROBLEM1.Wefirstconsideranuncontrolledsystem:x˙=f(x)Σu=(2)y=h(x).Thissystemisassumedtobeobservable(inthefollowingsense:thetra-jectoriesstartingfromtwodifferentinitialstatesaredistinguishedbytheoutput). THREEPROBLEMSINTHEFIELDOFOBSERVABILITY257WheneverthenthderivativeoftheoutputwithrespecttothevectorfieldfisaCr-functionoftheoutputandthen−1previousonesitispossibletoconstructobververs(see[5],[9]).Moreaccurately,theinjectivemappingΦ=(h,Lh,L2h,...,Ln−1h)fffisusedto“immerse”ΣintoRpnwhereaCr-observerisdesigned.Theob-userveddataaretheoutputsofΣtogetherwiththeir(n−1)thfirstderiva-utives.Thestateofthesystem,beingacontinuousmappingof(h,Lh,L2h,...,Ln−1h),isthusestimatedbytheobserver.fffMoregeneralyaCr-observerforΣisasystemΣdefinedinanopensubsetbuVofRnbyΣ=bz˙=F(z,y)(3)xb=θ(z)whereFisaCr-vectorfieldonVandθisacontinuousmappingfromVintoXsuchthat∀x∈X,∀z∈Vlimd(x(t),xb(t))=0t7→+∞foranydistancedonXcompatiblewiththetopologyofX.Thefirstproblemis:DoestheexistenceofaCr-observerforΣimplytheexistenceofanintegerunsuchthatthenthderivativeoftheoutputisaCr-functionoftheoutputandthen−1previousones?InanequivalentwaydoesitexistaCr-functionϕsuchthatLnh=ϕ(h,Lh,L2h,...,Ln−1h)?ffffApositiveanswertothisquestionwouldimplythatalltheobservabilitypropertiesofanuncontrolledsystemarecontainedinthefunctionalrelationLnh=ϕ(h,Lh,L2h,...,Ln−1h).ffffNoticethatwealreadyknowthatthekindofdependencebetweenthenthderivativeoftheoutputandtheprecedingones,thatisthekindoffunctionϕ,determineswhetherthesystemislinearizable,orlinearizablemoduloanoutputinjection(see[2],[7]).3PROBLEM2.Onceweknowthatacontrolledsystemisobservableintheweaksenseof[4](twodifferentinitialstateshavetobedistinguishedbytheoutputforatleastoneinput)aquestionarisesnaturally:whichinputsareuniversal?(Aninputisuniversalifanytwodifferentinitialstatesaredistinguishedbytheoutputforthisinput,see[8].)Anequivalentformulationis:forwhichinputsisthesystemobservable? 258PROBLEM7.3Forgenericreasons,weconsidercontrolledsystemswithmoreoutputsthaninputs:p>m.Problem2is:IsittruethatthesetofC∞-systems(resp.Cω-systems)thatareobservableforeveryC∞inputcontainsanopen(orbetteranopenanddense)subsetofthesetofC∞-systems(resp.thesetofCω-systems)?BothintheC∞andCωcasesdoesobservabilityforeveryC∞-inputimplyobservabilityforeveryL∞-input?Thefollowingfactsareknown:i.ThesetofsystemsobservableforeveryC∞-inputisdenseinthesetofC∞orCω-systems(see[3],[1]).ii.Foragivenbound,thesetofsystemsobservableforeveryC∞-inputwhose2dim(X)firstderivativesareboundedcontainsanopenanddensesubsetofthesetofC∞orCω-systems(see[3],[1]).iii.ACω-systemobservableforeveryC∞-inputisobservableforeveryL∞-input(see[3]).iv.Inthesingle-input,control-affine,C∞-case,theimplicationΣC∞-observable=⇒ΣL∞-observableistrueforanopenanddensesubsetofsystems(see[6]).Ofcourse,apositiveanswertothisproblemwouldmeanthatthepropertyofbeingobservableforeveryL∞-inputispreservedunderslightperturbations.4PROBLEM3.SincethesetofsystemsobservableforeveryL∞-inputisresidual(withmoreouputsthaninputs),itisveryinterestingtodesignobserversforthem,particularlyifthissetcontainsanopensubset.Atthepresenttime,themoregeneralconstructionofobserversfornonlinearsystemsisthehighgainone(see[3]).ButtheobserversdesignedinthiswayhavethedefaulttomakeuseofthederivativesoftheinputandcannotworkifthislastisonlyL∞.Insomeparticularcases(linearizablesystems,linearizablemoduloanoutputinjectionsystems,bilinearsystems,uniformlyobservablesystems...)observersthatworksforeveryinputareknownbuttheycannotbegeneralized.Problem3istherefore:ForsystemsobservableforeveryL∞-input,findageneralconstructionofanobserverwhichworksforeveryL∞-input.Noticethatifthesystemis“immersed”inRN(inasensetomakeprecise)the“immersion”mustnotdependontheinput:inthatcasetheimageof THREEPROBLEMSINTHEFIELDOFOBSERVABILITY259thevectorfieldfinRNwoulddependuponthederivativeoftheinput.InparticularthemappingΦ=(h,Lh,L2h,...,Ln−1h)ffffromXintoRpncannotbeusedbecausethevectorfieldf(x,u)dependsuponuandsoareLh(x,u),L2h(x,u),...,Ln−1h(x,u).fffBIBLIOGRAPHY[1]M.BaldeandPh.Jouan,“Genericityofobservabilityofcontrol-affinesystems,”ControlOptimizationandCalculusofVariations,vol.3,1998,345-359.[2]M.FliessandI.Kupka,“Afinitenesscriterionfornonlinearinput-outputdifferentialsystems,”SIAMJ.onControlandOptimization,21(1983),5,721-728.[3]J.P.GauthierandI.Kupka,DeterministicObservationTheoryandApplications,CambridgeUniversityPress,2001.[4]R.HermannandA.J.Krener,“Nonlinearcontrollabilityandobserv-ability,”IEEETrans.Aut.Contro,AC-22(1977),728-740.[5]Ph.JouanandJ.P.Gauthier,“Finitesingularitiesofnonlinearsystems.Outputstabilization,observabilityandobservers,”JournalofDynam-icalandControlSystems,vol.2,no2,1996,255-288.[6]Ph.Jouan,“C∞andL∞observabilityofsingle-inputC∞-Systems,”JournalofDynamicalandControlSystems,vol.7,no2,2001,151-169.[7]Ph.Jouan,“Immersionofnonlinearsystemsintolinearsystemsmodulooutputinjection,”submittedtoSIAMJ.onControlandOptimization.[8]H.J.Sussmann,“Single-inputobservabilityofcontinuous-timesys-tems,”Math.Syst.Theory12(1979),371-393.[9]X.XiaandM.Zeitz,“Onnonlinearcontinuousobservers,”Int.J.Con-trol,vol.66,no6,1997,943-954. Problem7.4ControloftheKdVequationLionelRosierInstitutElieCartanUniversit´eHenriPoincar´eNancy1B.P.23954506Vandœuvre-l`es-NancyCedexFrancerosier@iecn.u-nancy.fr1DESCRIPTIONOFTHEPROBLEMTheKorteweg-deVries(KdV)equationisthesimplestmodelforunidirec-tionalpropagationofsmallamplitudelongwavesinnonlineardispersivesystems.Itoccursinvariousphysicalcontexts(e.g.,waterwaves,plasmaphysics,nonlinearoptics).Itreadsyt+yxxx+yx+yyx=0,t>0,x∈Ω,(1)∂ythesubscriptsdenotingpartialderivatives(e.g.,yt=∂t).TheKdVequa-tionhasbeenintensivelystudiedsincethe1960sbecauseofitsfascinatingproperties(infinitesetofconservedintegralquantities,integrability,Katosmoothingeffect,etc.).(See[5]andthereferencestherein.)Here,weareconcernedwiththeboundarycontrollabilityoftheKdVequa-tioninthedomainΩ=(0,+∞).Foranypair(a,b)with0≤a0,y∈C∞(0,+∞)andh∈C∞(0,T),itisby000nowwellknown(see[1])thattheinitial-boundary-valueproblemyt+yxxx+yx+yyx=0,00,0∈R(y0,T)foranyinitialstatey0withasmallenoughH3(0,L)-norm.Itmeansthatasolitonmovingtotherightmaybecaughtupandannihilatedbyasetofwavesgeneratedbythewave-maker.ThisresultrestsonaCarlemanestimateforthelinearizedequation(i.e.,(1)withoutthenonlineartermyyx).WhenwelookatthelinearizedequationontheunboundeddomainΩ=(0,+∞),thenthecontrollabilityresultsarenotsogood,duetoalackofcompactness.Indeed,itisprovedin[9]thatthereexistsastatey∈L2(0,+∞)forwhichanytrajectorycon-0nectingytothenullstatedoesnotbelongtoL∞(0,T,L2(0,+∞))(that0R+∞2is,esssup0γ?findA˜(s)∈RHn∞n∞suchthatγ?≤kG(s)−A˜(s)k≤γ.n∞TheoptimalH∞approximationproblemcanbeformallyposedasacon-nstrainedmin-maxproblem.For,notethatanyfunctioninRH∞canbeputinaone-to-onecorrespondencewithapointθofsome(open)setΩ⊂R2n,thereforetheproblemofcomputingγ?canbeposedasnγ?=minmaxkG(jω)−A(jω)k,(2)nθ∈Ωω∈RwhereA(s)=A(s,θ).Theaboveformulationprovidesabruteforceap-proachtothesolutionoftheproblem.Unfortunately,thismethodisnotofanyuseingeneral,becauseofthecomplexityofthesetΩandbecauseofthecurseofdimensionality.However,theformulation(2)suggeststhatpossiblecandidatesolutionsoftheoptimalapproximationproblemarethesaddlepointsofthefunctionkG(jω)−A(jω,θ)k,whichcanbe,inprinciple,computedusingnumericaltools.Itwouldbeinterestingtoprove(ordisprove)thatminmaxkG(jω)−A(jω,θ)k=maxminkG(jω)−A(jω,θ)k.θ∈Ωω∈Rω∈Rθ∈ΩThesolutionmethodbasedonthecomputationofsaddlepointsdoesnotgiveanyinsightintotheproblem,neitherexposesanysystemstheoreticinterpretationoftheoptimalapproximant.Aninterestingpropertyoftheoptimalapproximantisstatedinthefollowingsimplefact,whichcanbeusedtoruleoutthatacandidateapproximantisoptimal.Fact:LetA?(s)∈RHnbesuchthatequation(1)holds.Suppose∞|W(jω?)−A?(jω?)|=γ?,(3)nandA(jω?)6=0(4)forω?=0.Thenthereexistsaconstant˜ω6=ω?suchthat|W(jω˜)−A?(jω˜)|=γ?,ni.e.,ifthevalueγ?isattainedbythefunction|W(jω)−A?(jω)|atω=0itnisalsoattainedatsomeω6=0.Proof:Weprovethestatementbycontradiction.Suppose|W(jω)−A?(jω)|<γ?,(5)n H∞-NORMAPPROXIMATION269forallω6=ω?andconsidertheapproximantA˜(s)=(1+λ)A?(s),withλ∈IR.Byequation(5),condition(4)andbycontinuitywithrespecttoλandωof|W(jω)−A˜(jω)|,thereisaλ?(sufficientlysmall)suchthatmax|W(jω)−(1+λ?)A?(jω)|<γ?,nωor,whatisthesame,itispossibletoobtainanapproximantthatisbetterthanA?(s),henceacontradiction./Itwouldbeinterestingtoshowthattheabovefactholds(oritdoesnothold)whenω?6=0.2AVAILABLERESULTSANDPOSSIBLESOLUTIONPATHSApproximationandmodelreductionhavealwaysbeencentralissuesinsys-temtheory.Forarecentsurveyonmodelreductioninthelarge-scalesetting,wereferthereaderto[1].Thereareseveralresultsinthisarea.Iftheap-proximationisperformedintheHankelnorm,thenanexplicitsolutionoftheoptimalapproximationandmodelreductionproblemshasbeengivenin[3].Notethatthisprocedureprovides,asabyproduct,anupperboundforγ?nandasolutionofthesuboptimalapproximationproblem.Iftheapproxima-tionisperformedintheH2norm,severalresultsandnumericalalgorithmsareavailable[4].ForapproximationintheH∞normaconceptualsolutionisgivenin[5].ThereinitisshownthattheH∞approximationproblemcanbereducedtoaHankelnormapproximationproblemforanextendedsystem(i.e.,asystemobtainedfromastatespacerealizationoftheoriginaltransferfunctionG(s)byaddinginputsandoutputs).TheextendedsystemhastobeconstructedwiththeconstraintthatthecorrespondingGrammiansPandQsatisfyλ(PQ)=(γ?)2withmultiplicityN−n.(6)minnHowever,theaboveprocedure,asalsonotedbytheauthorsof[5],isnotcom-putationallyviable,andpresupposestheknowledgeofγ?.Hencetheneednforfurtherstudyoftheproblem.Intherecentpaper[2],thedecayratesoftheHankelsingularvaluesofstable,single-inputsingle-outputsystems,p(s)arestudied.LetG(s)=bethetransferfunctionunderconsideration.q(s)ThedecayrateoftheHankelsingularvaluesisstudiedbyintroducinganewp(s)setofinput/outputsysteminvariants,namelythequantitiesq∗(s),whereq(s)∗=q(−s),evaluatedatthepolesofG(s).Theseresultsareexpectedtoyieldlightintothestructureoftheaboveproblem(6).Anotherpaperofinterestespeciallyforthesuboptimalapproximationcase,is[6].Inthispapertheset,ofallsystemswhoseH∞normislessthansomepositivenumberγisparameterized.Thusthefollowingproblemcanbeposed:given 270PROBLEM8.1suchasystemwithH∞normlessthanγ,findconditionsunderwhichitcanbedecomposedinthesumoftwosystems,oneofwhichisprespecified.Finally,therearetwospecialclassesofsystemsthatmaybestudiedtoim-proveourinsightintothegeneralproblem.Thefirstclassiscomposedofsingle-inputsingle-outputdiscrete-timestablesystems.Forsuchsystems,aninterestingrelatedproblemistheCarath´eodory-Fej´er(CF)approxima-tionproblemthatisusedforellipticfiltersdesign.In[7]itisshownthatinthescalar,discrete-timecase,optimalapproximantsintheHankelnormapproachasymptoticallyoptimalapproximantsintheH∞norm(theasymp-toticbehaviorbeingwithrespectto→0,where|z|≤<1).TheCFproblemthroughthecontributionofAdamjan-Arov-KreinandlaterGlover,evolvedintowhatisnowadayscalledtheHankel-normapproximationprob-lem.However,noasymptoticresultshavebeenshowntoholdinthegeneralcase.Thesecondspecialclassisthatofsymmetricsystems,thatis,systemswhosestatespacerepresentation(C,A,B)satisfiesA=A0andB=C0.Forinstance,thesesystemshaveapositivedefiniteHankeloperatorandhavefurtherpropertiesthatcanbeexploitedintheconstructionofapproximantsintheH∞sense.BIBLIOGRAPHY[1]A.C.Antoulas,“Lecturesontheapproximationoflargescaledynami-calsystems,”SIAM,Philadelphia,2002.[2]A.C.Antoulas,D.C.Sorensen,andY.Zhou,“OnthedecayrateofHankelsingularvaluesandrelatedissues,”SystemsandControlLetters,2002.[3]K.Glover,“AlloptimalHankel-normapproximationsoflinearmulti-variablesystemsandtheirL∞errorbounds,”InternationalJournalofControl,39:1115-1193,1984.[4]X.-X.Huang,W.-Y.YanandK.L.Teo,“H2nearoptimalmodelre-duction,”IEEETrans.AutomaticControl,46:1279-1285,2001.[5]D.KavranogluandM.Bettayeb,“CharacterizationofthesolutiontotheoptimalH∞modelreductionproblem,”SystemsandControlLet-ters,20:99-108,1993.[6]H.G.SageandM.F.deMathelin,“CanonicalH∞statespaceparametrization,”Automatica,July2000.[7]L.N.Trefethen,“RationalChebyshevapproximationontheunitdisc,”NumerischeMathematik,37:297-320,1981. Problem8.2Non-iterativecomputationofoptimalvalueinH∞controlBenM.ChenDepartmentofElectricalandComputerEngineeringNationalUniversityofSingaporeSingapore117576RepublicofSingaporebmchen@nus.edu.sg1DESCRIPTIONOFTHEPROBLEMWeconsiderann-thordergeneralizedlinearsystemΣcharacterizedbythefollowingstate-spaceequations:x˙=Ax+Bu+EwΣ:y=C1x+D11u+D1w(1)h=C2x+D2u+D22wwherexisthestate,uisthecontrolinput,wisthedisturbanceinput,yisthemeasurementoutput,andhisthecontrolledoutputofΣ.Forsimplicity,weassumethatD11=0andD22=0.WealsoletΣPbethesubsystemcharacterizedbythematrixquadruple(A,B,C2,D2)andΣQbethesubsystemcharacterizedby(A,E,C1,D1).ThestandardH∞optimalcontrolproblemistofindaninternallystabilizingpropermeasurementfeedbackcontrollaw,v˙=Acmpv+BcmpyΣcmp:(2)u=Ccmpv+Dcmpysuchthatwhenitisappliedtothegivenplant(1),theH∞-normoftheresultingclosed-looptransfermatrixfunctionfromwtoh,sayThw(s),isminimized.WenotethattheH∞-normofanasymptoticallystableandpropercontinuous-timetransfermatrixThw(s)isdefinedaskhk2kThwk∞:=supσmax[Thw(jω)]=sup,ω∈[0,∞)kwk2=1kwk2wherewandhare,respectively,theinputandoutputofThw(s). 272PROBLEM8.2TheinfimumortheoptimalvalueassociatedwiththeH∞controlproblemisdefinedasnoγ∗:=infkT(Σ×Σ)k|ΣinternallystabilizesΣ.(3)hwcmp∞cmpObviously,γ∗≥0.Infact,whenγ∗=0,theproblemisreducedtothewell-knownproblemofH∞almostdisturbancedecouplingwithmeasurementfeedbackandinternalstability.WenotethatinordertodesignameaningfulH∞controllawforthegivensystem(1),thedesignershouldknowbeforehandtheinfimumγ∗,whichrep-resentsthebestachievablelevelofdisturbanceattenuation.Unfortunately,theproblemofanoniterativecomputationofthisγ∗forgeneralsystemsstillremainsunsolvedintheopenliterature.2MOTIVATIONANDHISTORYOFTHEPROBLEMOverthelasttwodecades,wehavewitnessedaproliferationofliteratureonH∞optimalcontrolsinceitwasfirstintroducedbyZames[20].Themainfocusoftheworkhasbeenontheformulationoftheproblemforrobustmultivariablecontrolanditssolution.SincetheoriginalformulationoftheH∞probleminZames[20],agreatdealofworkhasbeendoneonfindingthesolutiontothisproblem.Practicallyalltheresearchresultsoftheearlyyearsinvolvedamixtureoftime-domainandfrequency-domaintechniquesincludingthefollowing:1)interpolationapproach(see,e.g.,[13]);chenbm2)frequencydomainapproach(see,e.g.,[5,8,9]);3)polynomialapproach(see,e.g.,[12]);and4)J-spectralfactorizationapproach(see,e.g.,[11]).Recently,considerableattentionhasbeenfocusedonpurelytime-domainmethodsbasedonalgebraicRiccatiequations(ARE)(see,e.g.,[6,7,10,15,16,17,18,19,21]).Alongthislineofresearch,connectionsarealsomadebetweenH∞optimalcontrolanddifferentialgames(see,e.g.,[1,14]).ItisnotedthatmostoftheresultsmentionedabovearefocusingonfindingsolutionstoHcontrolproblems.Manyofthemassumethatγ∗isknown∞orsimplyassumethatγ∗=1.Thecomputationofγ∗intheliteratureareusuallydonebycertainiterationschemes.Forexample,intheregularcaseandutilizingtheresultsofDoyleetal.[7],aniterativeprocedureforapproximatingγ∗wouldproceedasfollows:onestartswithavalueofγanddetermineswhetherγ>γ∗bysolvingtwo“indefinite”algebraicRiccatiequationsandcheckingthepositivesemi-definitenessandstabilizingpropertiesofthesesolutions.Inthecasewhensuchpositivesemi-definitesolutionsexistandsatisfyacouplingcondition,thenwehaveγ>γ∗andonesimplyrepeatstheabovestepsusingasmallervalueofγ.Inprinciple,onecanapproximatetheinfimumγ∗towithinanydegreeofaccuracyinthismanner.However,thissearchprocedureisexhaustiveandcanbeverycostly.Moresignificantly,duetothepossiblehigh-gainoccurrenceasγgetsclosetoγ∗,numericalsolutionsfortheseHAREscanbecomehighlysensitiveand∞ NON-ITERATIVECOMPUTATIONOFOPTIMALVALUEINH∞CONTROL273ill-conditioned.Thisdifficultyalsoarisesinthecouplingcondition.Namely,asγdecreases,evaluationofthecouplingconditionwouldgenerallyinvolvefindingeigenvaluesofstiffmatrices.Thesenumericaldifficultiesarelikelytobemoresevereforproblemsassociatedwiththesingularcase.Thus,ingeneral,theiterativeprocedureforthecomputationofγ∗basedonAREsisnotreliable.3AVAILABLERESULTSTherearequiteafewresearcherswhohaveattemptedtodevelopproceduresforthedeterminationofthevalueofγ∗withoutiterations.Forexample,Petersen[15]hassolvedtheproblemforaclassofone-blockregularcase.Scherer[17,18]hasobtainedapartialanswerforstatefeedbackproblemforalargerclassofsystemsbyprovidingacomputablecandidatevaluetogetherwithalgebraicallyverifiableconditions,andChenandhisco-workers[3,4](seealso[2])havedevelopedanoniterativeproceduresforcomputingthevalueofγ∗foraclassofsystems(singularcase)thatsatisfycertaingeometricconditions.Tobemorespecific,weintroducethefollowingtwogeometricsubspacesoflinearsystems:Givenann-thorderlinearsystemΣ∗characterizedbyamatrixquadruple(A∗,B∗,C∗,D∗),wedefinei.V−(Σ),aweaklyunobservablesubspace,isthemaximalsubspaceof∗Rnwhichis(A+BF)-invariantandcontainedinKer(C+DF)∗∗∗∗∗∗suchthattheeigenvaluesof(A+BF)|V−arecontainedinC−,the∗∗∗open-leftcomplexplane,forsomeconstantmatrixF∗;andii.S−(Σ),astronglycontrollablesubspace,istheminimal(A+KC)-∗∗∗∗invariantsubspaceofRncontainingIm(B+KD)suchthatthe∗∗∗eigenvaluesofthemapwhichisinducedby(A∗+K∗C∗)onthefactorspaceRn/S−arecontainedinC−forsomeconstantmatrixK.∗Theproblemofnoniterativecomputationofγ∗hasbeensolvedbyChenandhisco-workers[3,4](seealso[2])foraclassofsystemsthatsatisfythefollowingconditions:i.Im(E)⊂V−(Σ)+S−(Σ);andPPii.Ker(C)⊃V−(Σ)∩S−(Σ),2QQtogetherwithsomeotherminorassumptions.TheworkofChenetal.in-volvessolvingacoupleofalgebraicRiccatiandLyapunovequations.Thecomputationofγ∗isthendonebyfindingthemaximumeigenvalueofaresultingconstantmatrix.IthasbeendemonstratedbyanexampleinChen[2]thatthenoniterativecomputationofγ∗canbedoneforalargerclassofsystems,whichdonot 274PROBLEM8.2necessarilysatisfytheabovegeometricconditions.Itisbelievedthatthereareroomstoimprovetheexistingresults.BIBLIOGRAPHY[1]T.Ba¸sarandP.Bernhard,H∞OptimalControlandRelatedMinimaxDesignProblems:ADynamicGameApproach,2ndEd.,Birkh¨auser,Boston,1995.[2]B.M.Chen,H∞ControlandItsApplications,Springer,London,1998.[3]B.M.Chen,Y.GuoandZ.L.Lin,“Noniterativecomputationofin-fimumindiscrete-timeH∞-optimizationandsolvabilityconditionsforthediscrete-timedisturbancedecouplingproblem,”InternationalJour-nalofControl,vol.65,pp.433-454,1996.[4]B.M.Chen,A.Saberi,andU.Ly,“Exactcomputationoftheinfi-muminH∞-optimizationviaoutputfeedback,”IEEETransactionsonAutomaticControl,vol.37,pp.70-78,1992.[5]J.C.Doyle,LectureNotesinAdvancesinMultivariableControl,ONR-HoneywellWorkshop,1984.[6]J.C.DoyleandK.Glover,“State-spaceformulaeforallstabilizingcontrollersthatsatisfyanH∞-normboundandrelationstorisksensi-tivity,”Systems&ControlLetters,vol.11,pp.167-172,1988.[7]J.Doyle,K.Glover,P.P.Khargonekar,andB.A.Francis,“StatespacesolutionstostandardH2andH∞controlproblems,”IEEETransactionsonAutomaticControl,vol.34,pp.831-847,1989.[8]B.A.Francis,ACourseinH∞ControlTheory,LectureNotesinCon-trolandInformationSciences,vol.88,Springer,Berlin,1987.[9]K.Glover,“AlloptimalHankel-normapproximationsoflinearmulti-variablesystemsandtheirL∞errorbounds,”InternationalJournalofControl,vol.39,pp.1115-1193,1984.[10]P.Khargonekar,I.R.Petersen,andM.A.Rotea,“H∞-optimalcon-trolwithstatefeedback,”IEEETransactionsonAutomaticControl,vol.AC-33,pp.786-788,1988.[11]H.Kimura,ChainScatteringApproachtoH∞Control,Birkh¨auser,Boston,1997.[12]H.Kwakernaak,“Apolynomialapproachtominimaxfrequencydomainoptimizationofmultivariablefeedbacksystems,”InternationalJournalofControl,vol.41,pp.117-156,1986. NON-ITERATIVECOMPUTATIONOFOPTIMALVALUEINH∞CONTROL275[13]D.J.N.LimebeerandB.D.O.Anderson,“AninterpolationtheoryapproachtoH∞controllerdegreebounds,”LinearAlgebraanditsAp-plications,vol.98,pp.347-386,1988.[14]G.P.PapavassilopoulosandM.G.Safonov,“Robustcontroldesignviagametheoreticmethods,”Proceedingsofthe28thConferenceonDecisionandControl,Tampa,Florida,pp.382-387,1989.[15]I.R.Petersen,“DisturbanceattenuationandH∞optimization:Ade-signmethodbasedonthealgebraicRiccatiequation,”IEEETransac-tionsonAutomaticControl,vol.AC-32,pp.427-429,1987.[16]A.Saberi,B.M.Chen,andZ.L.Lin,“Closed-formsolutionstoaclassofH∞optimizationproblem,”InternationalJournalofControl,vol.60,pp.41-70,1994.[17]C.Scherer,“H∞controlbystatefeedbackandfastalgorithmforthecomputationofoptimalH∞norms,”IEEETransactionsonAutomaticControl,vol.35,pp.1090-1099,1990.[18]C.Scherer,“Thestate-feedbackH∞problematoptimality,”Automat-ica,vol.30,pp.293-305,1994.[19]G.Tadmor,“Worst-casedesigninthetimedomain:ThemaximumprincipleandthestandardH∞problem,”MathematicsofControl,Sig-nalsandSystems,vol.3,pp.301-324,1990.[20]G.Zames,“Feedbackandoptimalsensitivity:Modelreferencetrans-formations,multiplicativeseminorms,andapproximateinverses,”IEEETransactionsonAutomaticControl,vol.26,pp.301-320,1981.[21]K.Zhou,J.Doyle,andK.Glover,RobustandOptimalControl,Pren-ticeHall,NewYork,1996. Problem8.3DeterminingtheleastupperboundontheachievabledelaymarginDanielE.DavisonandDanielE.MillerDepartmentofElectricalandComputerEngineeringUniversityofWaterlooWaterloo,OntarioN2L3G1Canadaddavison@kingcong.uwaterloo.candmiller@hobbes.uwaterloo.ca1MOTIVATIONANDPROBLEMSTATEMENTControlengineershavehadtodealwithtimedelaysincontrolprocessesfordecadesand,consequently,thereisahugeliteratureonthetopic,e.g.,see[1]or[2]forcollectionsofrecentresults.Delaysarisefromavarietyofsources,includingphysicaltransportdelay(e.g.,inarollingmillorinachemicalplant),signaltransmissiondelay(e.g.,inanearth-basedsatellitecontrolsystemorinasystemcontrolledoveranetwork),andcomputationaldelay(e.g.,inasystemwhichusesimageprocessing).Theproblemsposedhereareconcernedinparticularwithsystemswherethetimedelayisnotknownexactly:suchuncertaintyexists,forexample,inarollingmillsystemwherethephysicalspeedoftheprocessmaychangeday-to-day,orinasatellitecontrolsystemwherethesignaltransmissiontimebetweenearthandthesatellitechangesasthesatellitemoves,orinacontrolsystemimplementedontheinternetwherethetimedelayisuncertainbecauseofunknowntrafficloadonthenetwork.Motivatedbytheaboveexamples,wefocushereonthesimplestproblemthatcapturesthedifficultyofcontrolinthefaceofuncertaindelay.Specifi-cally,considertheclassicallineartime-invariant(LTI)unity-feedbackcontrolsystemwithaknowncontrollerandwithaplantthatisknownexceptforanuncertainoutputdelay.Denotetheplantdelaybyτ,theplanttransferfunctionbyP(s)=P0(s)exp(−sτ),andthecontrollerbyC(s).Assumethefeedbacksystemisinternallystablewhenτ=0.Letusdefinethedelaymargin(DM)tobethelargesttimedelaysuchthat,foranydelaylessthan LEASTUPPERBOUNDONDELAYMARGIN277orequaltothisvalue,theclosed-loopsystemremainsinternallystable:DM(P0,C):=sup{τ:forallτ∈[0,τ],thefeedbackcontrolsystemwithcontrollerC(s)andplantP(s)=P0(s)exp(−sτ)isinternallystable}.ComputationofDM(P0,C)isstraightforward.Indeed,theNyquiststabilitycriterioncanbeusedtoconcludethatthedelaymarginissimplythephasemarginoftheundelayedsystemdividedbythegaincrossoverfrequencyoftheundelayedsystem.OthertechniquesforcomputingthedelaymarginforLTIsystemshavealsobeendeveloped,e.g.,see[3],[4],[5],and[6],justtonameafew.Incontrasttotheproblemofcomputingthedelaymarginwhenthecontrollerisknown,thedesignofacontrollertoachieveaprespecifieddelaymarginisnotstraightforward,exceptinthetrivialcasewheretheplantisopen-loopstable,inwhichcasethezerocontrollerachievesDM(P0,C)=∞.Tothebestoftheauthors’knowledge,thereisnoknowntechniquefordesigningacontrollertoachieveaprespecifieddelaymargin.Moreover,thefundamentalquestionofwhetherornotthereexistsafiniteupperboundonthedelaymarginthatisachievablebyaLTIcontrollerhasnotevenbeenaddressed.Hence,therearethreeunsolvedproblems:Problem1:Doesthereexistan(unstable)LTIplant,P0,forwhichthereisafiniteupperboundonthedelaymarginthatisachievablebyaLTIcontroller?Inotherwords,doesthereexistaP0forwhichDMsup(P0):=sup{DM(P0,C):thefeedbackcontrolsystemwithcontrollerC(s)andplantP0(s)isinternallystable}isfinite?Problem2:IftheanswertoProblem1isaffirmative,deviseacomputa-tionallyfeasiblealgorithmthat,givenP0(s),computesDMsup(P0)toagivenprescribeddegreeofaccuracy.Problem3:IftheanswertoProblem1isaffirmative,deviseacomputa-tionallyfeasiblealgorithmthat,givenP0(s)andavalueTintherange0η.Furthermore,Wor(AC,BC,CC,DC)isarealizationofthecontrollerC(z).TherealizationsofC(z)arenotunique.Differentrealizationsareallequivalentiftheyareimplementedininfiniteprecision.Infact,suppose(A0,B0,C0,D0)isarealizationofC(z),thenalltherealizationsofC(z)CCCCformasetI0D0C0I0SC=W:W=−10CC0(12)0TBCAC0TwherethetransformationmatrixT∈Rn×nisanarbitrarynonsingularmatrix.AusefulobservationisthatdifferentWhavedifferentvaluesofυ.Providedthatthevalueofυiscomputationallytractable,anoptimalrealizationofC(z),whichhasamaximumtolerancetoFWLerrors,canbeobtainedviaoptimization.Theopenproblemdefinedinsection1wasfirstseenin[3].Atpresent,thereexistsnoavailableresult.Anapproachtobypassthedifficultyincomputingυistodefinesomeapproximateupperboundofυusingafirst-orderapproximation,whichiscomputationallytractable(see[3]).OneofthethornyitemsintheopenproblemistheHadamardproductW◦∆.Theformofstructuredperturbation,whichwasadoptedinµ-analysismethods[4],maybeusedtodealwiththisHadamardproduct:∆canbetransformedintoageneralizedperturbation∆˜thathascertainstructuresuchasblock-diagonal.ThefixedmatricesA˜,B˜andC˜maybeobtainedsuchthatthestabilityofA˜+B˜∆˜C˜isequivalenttothatofA+B(W◦∆)C.AlthoughthestabilityofA˜+B˜∆˜C˜canbetreated STABLECOEFFICIENTPERTURBATION283satisfactorilybyµ-analysismethods,theopenproblemcannotbesolvedsuccessfullybyµ-analysismethods.Thisisbecauseµ-analysismethodsareconcernedaboutthemaximalsingularvalueσ(∆˜)of∆˜.Infact,thedistancebetweenσ(∆˜)andk∆kmaxcanbequitelarge,andk∆kmaxistheotherthornyitemwhichmakestheopenproblemdifficult.ACKNOWLEDGEMENTSTheauthorsgratefullyacknowledgethesupportoftheUnitedKingdomRoyalSocietyunderaKCWongfellowship(RL/ART/CN/XFI/KCW/11949).JunWuwishestothankthesupportoftheNationalNaturalScienceFoun-dationofChinaunderGrant60174026andtheScientificResearchFounda-tionfortheReturnedOverseasChineseScholarsofZhejiangprovinceunderGrantJ20020546.BIBLIOGRAPHY[1]M.GeversandG.Li,ParameterizationsinControl,EstimationandFilteringProblems:AccuracyAspects,London,SpringerVerlag,1993.[2]R.S.H.IstepanianandJ.F.Whidborne,eds.,DigitalControllerIm-plementationandFragility:AModernPerspective,London,SpringerVerlag,2001.[3]J.Wu,S.Chen,J.F.WhidborneandJ.Chu,“Optimalfloating-pointrealizationsoffinite-precisiondigitalcontrollers,”In:Proc.41stIEEEConferenceonDecisionandControl,LasVegas,USA,Dec.10-13,2002,pp.2570–2575.[4]J.Doyle,“Analysisoffeedbacksystemswithstructureduncertainties,”Proc.IEE,vol.129-D,no6,pp.242–250,1982. PART9Identification,SignalProcessing Problem9.1AconjectureonLyapunovequationsandprincipalanglesinsubspaceidentificationKatrienDeCockandBartDeMoor1Dept.ofElectricalEngineering(ESAT–SCD)K.U.LeuvenKasteelparkArenberg10B-3001LeuvenBelgiumhttp://www.esat.kuleuven.ac.be/sista-cosic-docarchdecock@esat.kuleuven.ac.be,demoor@esat.kuleuven.ac.be1DESCRIPTIONOFTHEPROBLEMThefollowingconjecturerelatestheeigenvaluesofcertainmatricesthatarederivedfromthesolutionofaLyapunovequationthatoccurredintheanal-ysisofstochasticsubspaceidentificationalgorithms[3].First,weformulatetheconjectureasapurematrixalgebraicproblem.InSection2,wewilldescribeitssystemtheoreticconsequencesandinterpretation.Conjecture:LetA∈Rn×nbearealmatrixandv,w∈RnberealvectorsA0sothattherearenotwoeigenvaluesλiandλjofTforwhich0A+vwλiλj=1(i,j=1,...,2n).Ifthen×nmatricesP,QandRsatisfythe1KatrienDeCockisaresearchassistantattheK.U.Leuven.Dr.BartDeMoorisafullprofessorattheK.U.Leuven.Ourresearchissupportedbygrantsfromseveralfundingagenciesandsources:ResearchCouncilKUL:ConcertedResearchActionGOA-Mefisto666,IDO,severalPh.D.,postdoctoral&fellowgrants;FlemishGovernment:FundforScientificResearchFlanders(severalPh.D.andpostdoctoralgrants,projectsG.0256.97,G.0115.01,G.0240.99,G.0197.02,G.0407.02,researchcommunitiesICCoS,ANMMM),AWI(Bil.Int.CollaborationHungary/Poland),IWT(Soft4s,STWW-Genprom,GBOU-McKnow,Eureka-Impact,Eureka-FLiTE,severalPhDgrants);BelgianFederalGovern-ment:DWTC(IUAPIV-02(1996-2001)andIUAPV-22(2002-2006)),ProgramSustain-ableDevelopmentPODO-II(CP/40);Directcontractresearch:Verhaert,Electrabel,Elia,Data4s,IPCOS. 288PROBLEM9.1LyapunovequationPRA0PRAT0RTQ=T
TRTQ0A+vwT0A+vwv
+vTwT,(1)wandP,Qand(I+PQ)arenonsingular,2thenthematricesP−1RQ−1RTnand(I+PQ)−1havethesameeigenvalues.nNotethattheconditionλiλj6=1(∀i,j=1,...,2n)ensuresthatthereexistsPRasolutionToftheLyapunovequation(1)andthatthesolutionisRQunique.WehavecheckedthesimilarityofP−1RQ−1RTand(I+PQ)−1fornumer-nousexamples(“proofbyMatlab”)anditissimpletoprovetheconjectureforn=1.Furthermore,viaalargedetour(see[3])wecanalsoproveitfromthesystemtheoreticinterpretation,whichisgiveninsection5.However,wehavenotbeenabletofindageneralandelegantproof.Wealsoremarkthattherequirementthatvandwarevectorsisnecessaryfortheconjecturetohold.OnecaneasilyfindcounterexamplesforthecaseV,W∈Rn×m,wherem>1.Itisconsequentlyclearthatthisconditiononvandwshouldbeusedintheproof.2BACKGROUNDANDMOTIVATIONAlthoughtheconjectureisformulatedasapurematrixalgebraicproblem,itssystemtheoreticinterpretationisparticularlyinteresting.Inordertoexplaintheconsequences,wefirsthavetointroducesomeconcepts:theprincipalanglesbetweensubspaces(section3)andtheirstatisticalcounterparts,thecanonicalcorrelationsofrandomvariables(section4).Next,insection5wewillshowhowtheconjecture,whenprovedcorrect,wouldenableustoproveinanelegantwaythatthenonzerocanonicalcorrelationsofthepastandthefutureoftheoutputprocessofalinearstochasticmodelareequaltothesinesoftheprincipalanglesbetweentwospecificsubspacesthatarederivedfromthemodel.Thisresult,initsturn,isinstrumentalforfurtherderivationsin[3],whereacepstraldistancemeasureisrelatedtocanonicalcorrelationsandtothemutualinformationoftwoprocesses(seealsosection5).Moreover,bythisnewcharacterizationofthecanonicalcorrelations,wegaininsightinthegeometricpropertiesofsubspacebasedtechniques.2ThematrixInisthen×nidentitymatrix. ACONJECTUREONTHELYAPUNOVEQUATION2893THEPRINCIPALANGLESBETWEENTWOSUBSPACESTheconceptofprincipalanglesbetweenandprincipaldirectionsinsubspacesofalinearvectorspaceisduetoJordaninthenineteenthcentury[8].Wegivethedefinitionandbrieflydescribehowtheprincipalanglescanbecomputed.LetSandSbesubspacesofRnofdimensionpandq,respectively,where12p≤q.Then,thepprincipalanglesbetweenS1andS2,denotedbyθ1,...,θp,andthecorrespondingprincipaldirectionsui∈S1andvi∈S2(i=1,...,p)arerecursivelydefinedascosθ=maxmax|uTv|=uTv111u∈S1v∈S2cosθ=maxmax|uTv|=uTv(k=2,...,p)kkku∈S1v∈S2subjecttokuk=kvk=1,andfork>1:uTu=0andvTv=0,wherei=ii1,...,k−1.IfSandSaretherowspacesofthematricesA∈Rl×nandB∈Rm×n,12respectively,thenthecosinesoftheprincipalanglesθ1,...,θp,canbecom-putedasthelargestpgeneralizedeigenvaluesofthematrixpencil0ABTAAT0T−Tλ.BA00BBFurthermore,ifAandBarefullrowrankmatrices,i.e.,l=pandm=q,thenthesquaredcosinesoftheprincipalanglesbetweentherowspaceofAandtherowspaceofBareequaltotheeigenvaluesof(AAT)−1ABT(BBT)−1BAT.NumericallystablemethodstocomputetheprincipalanglesviatheQRandsingularvaluedecompositioncanbefoundin[5,pp.603–604].4THECANONICALCORRELATIONSOFTWORANDOMVARIABLESCanonicalcorrelationanalysis,duetoHotelling[6],isthestatisticalversionofthenotionofprincipalangles.LetX∈RpandY∈Rq,wherep≤q,bezero-meanrandomvariableswithfullrankjointcovariancematrix3XTT
QxQxyQ=EXY=.YQyxQyThecanonicalcorrelationsofXandYaredefinedasthelargestpeigenvalues0QxyQx0ofthepencil−λ.MoreinformationoncanonicalQyx00Qycorrelationanalysiscanbefoundin[1,6].3E{·}istheexpectedvalueoperator. 290PROBLEM9.15SYSTEMTHEORETICINTERPRETATIONOFCONJECTURELet{y(k)}k∈Zbeareal,discrete-time,scalarandzero-meanstationarystochasticprocessthatisgeneratedbythefollowingsingle-input,single-output(SISO),asymptoticallystablestatespacemodelinforwardinnova-tionform:x(k+1)=Ax(k)+Ku(k),(2)y(k)=Cx(k)+u(k),where{u(k)}istheinnovationprocessof{y(k)},A∈Rn×n,K∈k∈Zk∈ZRn×1istheKalmangainandC∈R1×n.Thestatespacematricesoftheinversemodel(orwhiteningfilter)areA−KC,Kand−C,respectively,asiseasilyseenbywritingu(k)asanoutputwithy(k)asaninput.Bysubstitutingthevectorvin(1)byK,andwby−CT,thematricesP,QandRin(1)canbegiventhefollowinginterpretation.ThematrixPisthecontrollabilityGramianofthemodel(2)andQistheobservabilityGramianoftheinversemodel,whileRisthecrossproductoftheinfinitecontrollabilitymatrixof(2)andtheinfiniteobservabilitymatrixoftheinversemodel.Otherwiseformulated:PRC∞T
T=TC∞Γ∞,RQΓ∞C
C(A−KC)whereC=KAKA2K···andΓ=−2.∞∞C(A−KC)...Duetothestabilityandtheminimumphasepropertyoftheforwardin-novationmodel(2),theseinfiniteproductsresultinfinitematricesandinaddition,theconditionλiλj6=1inconjecture1isfulfilled.Furthermore,underfairlygeneralconditions,P,Q,andIn+PQarenonsingular,whichfollowsfromthepositivedefinitenessofPandQundergeneralconditions.ThematrixP−1RQ−1RTinconjecture1isnowequaltotheproduct(CCT)−1(CΓ)(ΓTΓ)−1(ΓTCT).∞∞∞∞∞∞∞∞Consequently,itsneigenvaluesarethesquaredcosinesoftheprincipalanglesbetweentherowspaceofC∞andthecolumnspaceofΓ∞(seeSection3).Theangleswillbedenotedbyθ1,...,θn(innondecreasingorder).Theeigenvaluesofthematrix(I+PQ)−1,ontheotherhand,arerelatedntothecanonicalcorrelationsofthepastandthefuturestochasticprocessesof{y(k)}k∈Z,whicharedefinedasthecanonicalcorrelationsoftherandomvariablesy(−1)y(0)y(−2)y(1)yp=y(−3)andyf=y(2),...... ACONJECTUREONTHELYAPUNOVEQUATION291anddenotedbyρ1,ρ2,...(innonincreasingorder).Itcanbeshown[3]thatthelargestncanonicalcorrelationsofypandyfareequaltothesquarerootsoftheeigenvaluesofI−(I+PQ)−1.Theothercanonicalcorrelationsarennequalto0.Conjecture1nowgivesusthefollowingcharacterizationofthecanonicalcorrelationsofthepastandthefutureof{y(k)}k∈Z:thelargestncanonicalcorrelationsareequaltothesinesoftheprincipalanglesbetweentherowspaceofC∞andthecolumnspaceofΓ∞andtheothercanonicalcorrelationsareequalto0:ρ1=sinθn,ρ2=sinθn−1,...,ρn=sinθ1,ρn+1=ρn+2=···=0.(3)Thisresultcanbeusedtoprovethatarecentlydefinedcepstralnorm[9]foramodelasin(2)iscloselyrelatedtothemutualinformationofthepastandthefutureofitsoutputprocess.Letthetransferfunctionofthesystemin(2)bedenotedbyH(z).Thenthecomplexcepstrum{c(k)}k∈ZofthemodelisdefinedastheinverseZ-transformofthecomplexlogarithmofH(z):I1k−1c(k)=log(H(z))zdz,2πiCwherethecomplexlogarithmofH(z)isappropriatelydefined(see[10,pp.495–497])andthecontourCistheunitcircle.Thecepstralnormthatweconsider,isdefinedasX∞klogHk2=kc(k)2.k=0Aswehaveprovenin[2],itcanbecharacterizedintermsoftheprincipalanglesθ1,...,θnbetweentherowspaceofC∞andthecolumnspaceofΓ∞asfollows:YnklogHk2=−logcos2θ,ii=1andfrom(3)weobtainYklogHk2=−log(1−ρ2).iP∞2Q2Therelationk=0kc(k)=−log(1−ρi)wasalsoreportedin[7,propo-sition2].Moreover,ifQ{y(k)}k∈ZisaGaussianprocess,thentheexpression−1log(1−ρ2)isequaltothemutualinformationofitspastandfuture2i(see,e.g.,[4]),whichisdenotedbyI(yp;yf).Consequently,X∞klogHk2=kc(k)2=2I(y;y).pfk=06CONCLUSIONSWepresentedamatrixalgebraicconjectureontheeigenvaluesofmatricesthatarederivedfromthesolutionofaLyapunovequation.Weshowedthat 292PROBLEM9.1aproofofconjecture1wouldprovideyetanotherelegantgeometricresultinthesubspacebasedstudyoflinearstochasticsystems.Moreover,itcanbeusedtoexpressacepstraldistancemeasurethatwasdefinedin[9]intermsofcanonicalcorrelationsandalsoasthemutualinformationoftwoprocesses.BIBLIOGRAPHY[1]T.W.Anderson,AnIntroductiontoMultivariateStatisticalAnalysis,JohnWiley&Sons,NewYork,1984.[2]K.DeCockandB.DeMoor,“SubspaceanglesbetweenARMAmod-els,”TechnicalReport400-44a,ESAT-SCD,K.U.Leuven,Leuven,Bel-gium,acceptedforpublicationinSystems&ControlLetters,2002.[3]K.DeCock,PrincipalAnglesinSystemTheory,InformationThe-oryandSignalProcessing,Ph.D.thesis,5FacultyofAppliedSciences,K.U.Leuven,Leuven,Belgium,2002.[4]I.M.Gel’fandandA.M.Yaglom,“Calculationoftheamountofinfor-mationaboutarandomfunctioncontainedinanothersuchfunction,”AmericanMathematicalSocietyTranslations,Series(2),12,pp.199–236,1959.[5]G.H.GolubandC.F.VanLoan,MatrixComputations,TheJohnsHopkinsUniversityPress,Baltimore,1996.[6]H.Hotelling,“Relationsbetweentwosetsofvariates,”Biometrika,28,pp.321–372,1936.[7]N.P.Jewell,P.Bloomfield,andF.C.Bartmann,“Canonicalcorrela-tionsofpastandfuturefortimeseries:Boundsandcomputation,”TheAnnalsofStatistics,11,pp.848–855,1983.[8]C.Jordan,“Essaisurlag´eom´etrie`andimensions,”BulletindelaSoci´et´eMath´ematique,3,pp.103–174,1875.[9]R.J.Martin,“AmetricforARMAprocesses,”,IEEETransactionsonSignalProcessing,48,pp.1164–1170,2000.[10]A.V.OppenheimandR.W.Schafer,DigitalSignalProcessing,Pren-tice/HallInternational,1975.4Thisreportisavailablebyanonymousftpfromftp.esat.kuleuven.ac.beinthedirectorypub/sista/reportsasfile00-44a.ps.gz.5Thisthesiswillalsobemadeavailablebyanonymousftpfromftp.esat.kuleuven.ac.beasfilepub/sista/decock/reports/phd.ps.gz Problem9.2StabilityofanonlinearadaptivesystemforfilteringandparameterestimationMasoudKarimi-GhartemaniDepartmentofElectricalandComputerEngineeringUniversityofTorontoToronto,OntarioCanadaM5S3G4masoud@ele.utoronto.caAlirezaK.ZiaraniDepartmentofElectricalandComputerEngineeringClarksonUniversityPotsdam,NYUSA13699-5720aziarani@clarkson.edu1DESCRIPTIONOFTHEPROBLEMWeareconcernedaboutthemathematicalpropertiesofthedynamicalsys-tempresentedbythefollowingthreedifferentialequations:dA2dt=−2µ1Asinφ+2µ1sinφf(t),dω=−µA2sin(2φ)+2µAcosφf(t),(1)dt22dφ=ω+µdωdt3dtwhereparametersµi,i=1,2,3arepositiverealconstantsandf(t)isafunctionoftimehavingageneralformoff(t)=Aosin(ωot+δo)+f1(t).(2)Ao,ωoandδoarefixedquantitiesanditisassumedthatf1(t)hasnofre-quencycomponentatω.VariablesAandωareinR1andφvariesontheoone-dimensionalcircleS1withradius2π.Thedynamicalsystempresentedby(1)isdesignedto(i)takethesignalf(t)asitsinputsignalandextractthecomponentfo(t)=Aosin(ωot+δo)asits 294PROBLEM9.2outputsignal,and(ii)estimatethebasicparametersoftheextractedsignalfo(t),namelyitsamplitude,phase,andfrequency.Theextractedsignalisy=AsinφandthebasicparametersaretheamplitudeA,frequencyωandphaseangleφ=ωt+δ.Considerthethreevariables(A,ω,φ)inthethree-dimensionalspaceR1×R1×S1.Thesinusoidalfunctionf(t)=Asin(ωt+δ)correspondstotheooooTo-periodiccurveΓo(t)=(Ao,ωo,ωot+δo)(3)inthisspace,withT=2π.oωoThefollowingtheorem,whichtheauthorshaveprovedin[1],presentssomeofthemathematicalpropertiesofthedynamicalsystempresentedby1.Theorem1:Considerthedynamicalsystempresentedbythesetofordi-narydifferentialequations(1)inwhichthefunctionf(t)isdefinedin(2)andf1(t)isaboundedT1-periodicfunctionwithnofrequencycomponentatω.Thethreevariables(A,ω,φ)areinR1×R1×S1.Theparametersoµ,i=1,2,3aresmallpositiverealnumbers.IfT=Toforanyarbi-i1ntraryn∈N,thedynamicalsystemof(1)hasastableTo-periodicorbitinaneighborhoodofΓo(t)asdefinedin(3).Thebehaviorofthesystem,asexaminedwithinthesimulationenviron-ments,hasledtheauthorstothefollowingtwoconjectures,theproofsofwhicharedesired.Conjecture1:WiththesameassumptionsasthosepresentedinTheorem1,ifT=pTforanyarbitrary(p,q)∈N2with(p,q)=1,thedynamical1qosystempresentedby(1)hasastablemTo-periodicorbitwhichliesonatorusinaneighborhoodofΓo(t)asdefinedin(3).Thevalueofm∈Nisdeterminedbythepair(p,q).Conjecture2:WiththesameassumptionsasthosepresentedinTheorem1,ifT1=αToforirrationalα,thedynamicalsystempresentedby(1)hasanattractorsetthatisatorusinaneighborhoodofΓo(t)asdefinedin(3).Inotherwords,theresponseisaneverclosingorbitthatliesonthetorus.Moreover,thisorbitisadensesetonthetorus.Forbothconjectures,theneighborhoodinwhichthetorusislocateddependsonthevaluesofparametersµi,i=1,2,3andthefunctionf1(t).Ifthefunctionf1(t)issmallinorderandtheparametersareproperlyselected,theneighborhoodcanbemadetobeverysmall,meaningthatthefilteringandestimationprocessesmaybeachievedwithahighdegreeofaccuracy.Theorem1dealswiththelocalstabilityanalysisofthedynamicalsystem(1).Inotherwords,theexistenceofanattractor(periodicorbitortorus)andanattractiondomainaroundtheattractorisproved.However,thistheoremdoesnotdealwiththisdomainofattraction.Itisdesirabletospecifythisdomainofattractionintermsofthefunctionf1(t)andparametersµi,i=1,2,3,hencethefollowingopenproblem:OpenProblem:Considerthedynamicalsystempresentedbytheordinary STABILITYOFANONLINEARADAPTIVESYSTEM295differentialequations(1)inwhichthefunctionf(t)isdefinedin(2)andf1(t)isaboundedT1-periodicfunctionwithnofrequencycomponentatωo.Threevariables(A,ω,φ)areinR1×R1×S1.Parametersµ,i=1,2,3iaresmallpositiverealnumbers.ThissystemhasanattractorsetthatmaybeeitheraperiodicorbitoratorusbasedonthevalueofT1.Itisdesiredtospecifytheextentoftheattractiondomainassociatedwiththeattractorintermsofthefunctionf1(t)andtheparametersµi,i=1,2,3.Inotherwords,andinasimplifiedcase,forathree-parameterrepresentationoff1(t)asf1(t)=a1sin(2π/T1t+δ1),itisdesirabletoparameterize,intermsofthenine-parametersetof{µ1,µ2,µ3,Ao,To,δo,a1,T1,δ1},theattractorset,andalsothewholeregionofpoints(A,ω,φ)inR1×R1×S1thatfallsintheattractiondomainoftheattractor.2MOTIVATIONANDHISTORYOFTHEPROBLEMThedynamicalsystempresentedby(1)wasproposedbytheauthorstodeviseasystemfortheextractionofasinusoidalcomponentwithtime-varyingparameterswhenitiscorruptedbyothersinusoidsandnoise[1,2].Thisisofsignificantinterestinpowersystemapplications,forinstance[3].Estimationofthebasicparametersoftheextractedsinusoid,namelytheamplitude,phase,andfrequency,wasanotherobjectofthework.Theseparametersprovideimportantinformationusefulinelectricalengineeringapplications.Someapplicationsofthesysteminbiomedicalengineeringarepresentedin[2,4].Thisdynamicalsystempresentsanalternativestruc-tureforthewell-knownphase-lockedloop(PLL)systemwithsignificantlyadvantageousfeatures.3AVAILABLERESULTSTheorem1,correspondingtothecaseofT=To,hasbeenprovedbythe1nauthorsin[1]wheretheexistence,localuniquenessandstabilityofaTo-periodicorbitareshownusingthePoincar´emaptheoremasstatedin[5,page70].Extensivecomputersimulationsverifiedbylaboratoryexperimentalresultsarepresentedin[1,2].Someofthewide-rangingapplicationsofthedynamicalsystemarepresentedin[2,3,4].Thealgorithmgovernedbytheproposeddynamicalsystempresentsapowerfulsignalprocessingmethodofanalysis/synthesisofnonstationarysignals.Alternatively,itmaybethoughtofasanonlinearadaptivenotchfiltercapableofestimationofparametersoftheoutputsignal. 296PROBLEM9.2BIBLIOGRAPHY[1]M.Karimi-GhartemaniandA.K.Ziarani,“Periodicorbitanalysisoftwodynamicalsystemsforelectricalengineeringapplications,”JournalofEngineeringMathematics,45,pp.135-154,2003.[2]A.K.Ziarani,ExtractionofNonstationarySinusoids,Ph.D.dissertation,UniversityofToronto,2002.[3]M.Karimi-GhartemaniandM.R.Iravani,“ANonlinearAdaptiveFil-terforOn-LineSignalAnalysisinPowerSystems:Applications,”IEEETransactionsonPowerDelivery,17,pp.617-622,2002.[4]A.K.ZiaraniandA.Konrad,“ANonlinearAdaptiveMethodofElim-inationofPowerLineInterferenceinECGSignals,”IEEETransactionsonBiomedicalEngineering,49,pp.540-547,2002.[5]S.Wiggins,IntroductiontoAppliedNonlinearDynamicalSystemsandChaos,NewYork:Springer-Verlag,1983. PART10Algorithms,Computation Problem10.1Root-clusteringformultivariatepolynomialsandrobuststabilityanalysisPierre-AlexandreBlimanINRIARocquencourtBP10578153LeChesnaycedexFrancepierre-alexandre.bliman@inria.fr1DESCRIPTIONOFTHEPROBLEMGiventhe(m+1)complexmatricesA0,...,Amofsizen×nanddenotingD(resp.C+)theclosedunitballinC(resp.theclosedright-halfplane),letusconsiderthefollowingproblem:determinewhetherdefm∀s∈C+,∀z=(z1,...,zm)∈D,det(sIn−A0−z1A1−···−zmAm)6=0.(1)Wehaveprovedin[1]thatproperty(1)isequivalenttotheexistenceofmm−1k∈Nand(m+1)matricesP,Q∈Hkn,Q∈Hk(k+1)n,...,Q∈12mm−1Hk(k+1)n,suchthatP>0kmnandR(P,Q1,...,Qm)<0(k+1)mn.(2)Here,Hnrepresentsthespaceofn×nhermitianmatrices,andRisa(k+1)mndeflinearapplicationtakingvaluesinH,definedasfollows.LetJˆk=def(Ik0k×1),Jˇk=(0k×1Ik),then(usingthepowerofKroneckerproductwiththenaturalmeaning):!HXmRdef=Jˆm⊗⊗A+Jˆ(m−i)⊗⊗Jˇ⊗Jˆ(i−1)⊗⊗APJˆm⊗⊗Ik0kkkikni=1!TXm+Jˆm⊗⊗IPJˆm⊗⊗A+Jˆ(m−i)⊗⊗Jˇ⊗Jˆ(i−1)⊗⊗Aknk0kkkii=1XmTJˆ(m−i+1)⊗Jˆ(m−i+1)⊗+k⊗I(k+1)i−1nQik⊗I(k+1)i−1ni=1 300PROBLEM10.1XmTJˆ(m−i)⊗Jˆ(m−i)⊗−k⊗Jˇk⊗I(k+1)i−1nQik⊗Jˇk⊗I(k+1)i−1n(3)i=1Problem(2,3)isalinearmatrixinequalityinthem+1unknownmatricesP,Q1,...,Qm,aconvexoptimizationproblem.TheLMIs(2,3)obtainedforincreasingvaluesofkconstituteindeedafamilyofweakerandweakersufficientconditionsfor(1).Conversely,property(1)necessarilyimpliessolvabilityoftheLMIsforacertainrankkandbeyond.See[1]fordetails.Numericalexperimentationshaveshownthattheprecisionofthecriteriaobtainedforsmallvaluesofk(2or3)mayberemarkablygoodalready,butrationaluseofthisresultrequirestohaveaprioriinformationonthesizeoftheleastk,ifany,forwhichtheLMIsaresolvable.Bounds,especiallyupperbound,onthisquantityarethushighlydesirable,andtheyshouldbecomputedwithlowcomplexityalgorithms.OpenProblem1:Findaninteger-valuedfunctionk∗(A,A,...,A)de-01mfinedontheproduct(Cn×n)m+1,whoseevaluationnecessitatespolynomialtime,andsuchthatproperty(1)holdsifandonlyifLMI(2,3)issolvablefork=k∗.Onemayimaginethatthepreviousquantityexists,dependinguponnandmonly.∗def∗OpenProblem2:Determinewhetherthequantitykn,m=sup{k(A0,A1,...,A):A,A,...,A∈Cn×n}isfinite.Inthiscase,provideanupperm01mboundofitsvalue.Ifk∗<+∞,then,foranyA,A,...,A∈Cn×n,property(1)holdsifn,m01mandonlyifLMI(2,3)issolvablefork=k∗.n,m2MOTIVATIONSANDCOMMENTSWeexposeheresomeproblemsrelatedtoproperty(1).RobuststabilityProperty(1)isequivalenttoasymptoticstabilityoftheuncertainsystemx˙=(A0+z1A1+···+zmAm)x,(4)mforanyvalueofz∈D.UsualapproachesleadingtosufficientLMIcondi-tionsforrobuststabilityarebasedonsearchforquadraticLyapunovfunc-tionsx(t)HSx(t)withconstantS,seerelatedbibliographyin[2,p.72–73],orparameter-dependentS(z),namelyaffine[8,7,5,6,12]andmorerecentlyquadratic[19,20].MethodsbasedonpiecewisequadraticLyapunovfunc-tions[21,13]andLMIswithaugmentednumberofvariables[9,11]alsoprovidesufficientconditionsforrobuststability. ROOT-CLUSTERINGANDSTABILITYANALYSIS301Theapproachleadingtotheresultexposedin§1systematizestheideaofexpandingS(z)inpowersoftheparameters.Indeed,robuststabilityof(4)guaranteesexistenceofaLyapunovfunctionx(t)HS(z)x(t)withS(z)mpolynomialwithrespecttozandz¯inD,andtheintegerkisrelatedtothedegreeofthispolynomial[1].ComputationofstructuredsingularvalueswithrepeatedscalarblocksProperty(1)isequivalenttoµ∆(A)<1,foracertainmatrixAdeducedfromA0,A1,...,Am,andaset∆ofcomplexuncertaintieswithm+1repeatedscalarblocks.Evaluationofthestructuredsingularvalues(astandardandpowerfultoolofrobustanalysis)hasbeenprovedtobeaNP-hardproblem,see[3,16].Hopehaddawnedthatitsstandard,efficientlycomputable,upperboundcouldbeasatisfyingapproximant[17],butthegapbetweenthetwomeasureshaslatteronbeenprovedinfinite[18,14].Theapproachin§1couldofferattractivenumericalalternativeforthesamepurpose,asresolutionofLMIsisaconvexproblem.Itprovidesafamilyofconvexrelaxations,ofarbitraryprecision,ofaclassofNP-hardproblems.Thecomplexityresultsevokedpreviouslyimplytheexistenceofk∗(A,A,01...,Am)suchthatproperty(1)isequivalenttosolvabilityofLMI(2,3)fork=k∗:first,checkthatµ(A)<1;ifthisistrue,thenassesstok∗the∆valueofthesmallestksuchthatLMI(2,3)issolvable,otherwiseputk∗=1.Thisalgorithmis,ofcourse,adisasterfromthepointofviewofcomplexityandcomputationtime,anditdoesnotanswerProblem1.Concerningthevalueofk∗inProblem2,itsgrowthatinfinityshouldbefasterthananyn,mpowerinn,exceptifP=NP.Delay-independentstabilityofdelaysystemswithnoncommensu-ratedelaysProperty(1)isastrongversionofthedelay-independentstabilityofthefunctionaldifferentialequationofretardedtype˙x=A0x(t)+A1x(t−h1)+···+Amx(t−hm),thatistheasymptoticalstabilityforanyvalueofh1,...,hm≥0,see[10,2,4].ThisproblemhasbeenrecognizedasNP-hard[15].SolvingLMI(2,3)providesexplicitlyaquadraticLyapunov-Krasovskiifunctionalindependentuponthevaluesofthedelays[1].Robuststabilityofdiscrete-timesystemsandstabilityofmultidi-mensional(nD)systemsUnderstandinghowtocopewiththechoiceofktoapplyLMI(2,3),shouldalsoleadtoprogressintheanalysisofthediscrete-timeanalogueof(4),theuncertainsystemxk+1=(A0+z1A1+···+zmAm)xk.Similarly,stabil-ityanalysisformultidimensionalsystems(adiscrete-timeanalogueofthefunctionaldifferentialequationsofneutraltype)wouldbenefitfromsuchcontributions. 302PROBLEM10.1BIBLIOGRAPHY[1]P.-A.Bliman,“Aconvexapproachtorobuststabilityforlinearsystemswithuncertainscalarparameters,”Reportresearchno4316,INRIA,2001.Availableonlineathttp://www.inria.fr/rrrt/rr-4316.html[2]S.Boyd,L.ElGhaoui,E.Feron,V.Balakrishnan,“Linearmatrixin-equalitiesinsystemandcontroltheory,”SIAMStudiesinAppliedMath-ematics,vol.15,1994.[3]R.P.Braatz,P.M.Young,J.C.Doyle,M.Morari,“Computationalcomplexityofµcalculation,”IEEETrans.Automat.Control,39,no5,1000–1002,1994.[4]J.Chen,H.A.Latchman,“Frequencysweepingtestsforstabilityinde-pendentofdelay,”IEEETrans.Automat.Control,40,no9,1640–1645,1995.[5]M.Dettori,C.W.Scherer,“RobuststabilityanalysisforparameterdependentsystemsusingfullblockS-procedure,”Proc.of37thIEEECDC,Tampa,Florida,2798–2799,1998.[6]M.Dettori,C.W.Scherer,“Newrobuststabilityandperformancecon-ditionsbasedonparameterdependentmultipliers,”Proc.of39thIEEECDC,Sydney,Australia,2000.[7]E.Feron,P.Apkarian,P.Gahinet,“Analysisandsynthesisofrobustcontrolsystemsviaparameter-dependentLyapunovfunctions,”IEEETrans.Automat.Control,41,no7,1041–1046,1996.[8]P.Gahinet,P.Apkarian,M.Chilali,“Affineparameter-dependentLya-punovfunctionsandrealparametricuncertainty,”IEEETrans.Au-tomat.Control,41,no3,436–442,1996.[9]J.C.Geromel,M.C.deOliveira,L.Hsu,“LMIcharacterizationofstructuralandrobuststability,”LinearAlgebraAppl.,285,no1-3,69–80,1998.[10]J.K.Hale,E.F.Infante,F.S.P.Tsen,“Stabilityinlineardelayequa-tions,”J.Math.Anal.Appl.,115,533–555,1985.[11]D.Peaucelle,D.Arzelier,O.Bachelier,J.Bernussou,“AnewrobustD-stabilityconditionforrealconvexpolytopicuncertainty,”SystemsandControlLetters,40,no1,21–30,2000.[12]D.C.W.Ramos,P.L.D.Peres,“AnLMIapproachtocomputerobuststabilitydomainsforuncertainlinearsystems,”Proc.ACC,Arlington,Virginia,4073–4078,2001. ROOT-CLUSTERINGANDSTABILITYANALYSIS303[13]A.Rantzer,M.Johansson,“Piecewiselinearquadraticoptimalcontrol,”IEEETrans.Automat.Control,45,no4,629–637,2000.[14]M.Sznaier,P.A.Parrilo,“Onthegapbetweenµanditsupperboundforsystemswithrepeateduncertaintyblocks,”Proc.of38thIEEECDC,Phoenix,Arizona,4511–4516,1999.[15]O.Toker,H.Ozbay,“Complexityissuesinrobuststabilityoflinear¨delay-differentialsystems,”Math.ControlSignalsSystems,9,no4,386–400,1996.[16]O.Toker,H.OzbayOnthecomplexityofpurelycomplex¨µcomputa-tionandrelatedproblemsinmultidimensionalsystems,”IEEETrans.Automat.Control,43,no3,409–414,1998.[17]O.Toker,B.deJager,“Conservatismofthestandardupperboundtest:issup(¯µ/µ)finite?isitboundedby2?,”Openproblemsinmathematicalsystemsandcontroltheory(V.D.Blondel,E.D.Sontag,M.Vidyasagar,J.C.Willemseds),Springer-Verla,gLondon,1999.[18]S.Treil,“Thegapbetweencomplexstructuredsingularvalueµanditsupperboundisinfinite,”1999.Availableonlineathttp://www.math.msu.edu/treil/papers/mu/mu-abs.html[19]A.Trofino,“ParameterdependentLyapunovfunctionsforaclassofuncertainlinearsystems:aLMIapproach,”Proc.of38thIEEECDC,Phoenix,Arizona,2341–2346,1999.[20]A.Trofino,C.E.deSouza,“Bi-quadraticstabilityofuncertainlinearsystems,”Proc.of38thIEEECDC,Phoenix,Arizona,1999.[21]L.Xie,S.Shishkin,M.Fu,“PiecewiseLyapunovfunctionsforrobuststabilityoflineartime-varyingsystems,”Syst.Contr.Lett.,31,no3,165–171,1997. Problem10.2Whenisapairofmatricesstable?VincentD.Blondel,JacquesTheysDepartmentofMathematicalEngineeringUniversityofLouvainAvenueGeorgesLematre,4B-1348Louvain-la-NeuveBelgiumblondel@inma.ucl.ac.be,theys@inma.ucl.ac.beJohnN.TsitsiklisLaboratoryforInformationandDecisionSystemsMassachusettsInstituteofTechnologyCambridge,MA02139USAjnt@mit.edu1STABILITYOFALLPRODUCTSWeconsiderproblemsrelatedtothestabilityofsetsofmatrices.LetΣbeafinitesetofn×nrealmatrices.Givenasystemoftheformxt+1=Atxtt=0,1,...supposethatitisknownthatAt∈Σ,foreacht,butthattheexactvalueofAtisnotaprioriknownbecauseofexogenousconditionsorchangesintheoperatingpointofthesystem.Suchsystemscanalsobethoughtofasatime-varyingsystems.Wesaythatsuchasystemisstableiflimxt=0t→∞forallinitialstatesx0andallsequencesofmatrixproducts.ThisconditionisequivalenttotherequirementlimAit···Ai1Ai0=0t→∞forallinfinitesequencesofindices.Setsofmatricesthatsatisfythiscondi-tionaresaidtobestable. WHENISAPAIROFMATRICESSTABLE?305Problem1.Underwhatconditionsisagivensetofmatricesstable?Conditionforstabilityaretrivialformatricesofdimensionone(allscalarmustbeofmagnitudestrictlylessthanone),andarewell-knownforsetsthatcontainonlyonematrix(theeigenvaluesofthematrixmustbeofmag-nitudestrictlylessthanone).Weareaskingstabilityconditionsformoregeneralcases.Thematricesinthesetmustofcoursehavealltheireigenvaluesofmagnitudestrictlylessthanone.Thisconditiondoesnotsufficeingeneralasitispossibletoobtainanunstabledynamicalsystembyswitchingbetweentwostablelineardynamics.Consider,forinstance,thematrices1110A0=αandA1=α0111Thesematricesarestableiff|α|<1.Consider,then,theproduct221A0A1=α11Itisimmediatetoverifythatthestabilityofthismatrixisequivalenttothecondition|α|<((2/(3+51/2))1/2=0.618andsothestabilityofA,A01doesnotimplythatoftheset{A0,A1}.Exceptforelementarycases,nosatisfactoryconditionsarepresentlyavail-ableforcheckingthestabilityofsetsofmatrices.Infacttheproblemisopeneveninthecaseofmatricesofdimensiontwo.Fromasetofmmatricesofdimensionn,itiseasytoconstructtwomatricesofdimensionnmwhosesta-bilityisequivalenttothatoftheoriginalset.Indeed,letΣ={A1,...,Am}beagivensetanddefineB0=diag(A1,...,Am)andB1=T⊗IwhereTisam×mcyclicpermutationmatrix,⊗istheKroneckermatrixproduct,andIthen×nidentitymatrix.Thenthestabilityofthepairofmatrices{B0,B1}iseasilyseenequivalenttothatofΣ(see[3]foramoredetailledargument).Ourquestionisthus:Whenisapairofmatricesstable?Severalresultsareavailableintheliteratureforthisproblem,see,e.g.,theLiealgebraconditiongivenin[9].Theconditionspresentlyavailableareonlypartlysatisfactoryinthattheyareeitherincomplete(theydonotcoverallcases),theyareapproximate(see,e.g.,[1]and[8]),ortheyarenoteffec-tive.Wesaythataproblemiseffectivelydecidable(or,decidable)ifthereisanalgorithmthat,uponinputofthedataassociatedwithaninstanceoftheproblem,providesayes-noanswerafterafiniteamountofcomputa-tion.Theprecisedefinitionofwhatismeantbyanalgorithmisnotcritical;mostalgorithmmodelsproposedsofarareknowntobeequivalentfromthepointofviewoftheircomputingcapabilities,andtheyalsocoincidewith 306PROBLEM10.2theintuitivenotionofwhatcanbeeffectivelyachieved(see[10]forageneraldescriptionofdecidability,and[4]forasurveyondecidabilityinsystemsandcontrol).Problem1canthusbemademoreexplicitbyaskingforaneffectivedecisionalgorithmforstabilityofarbitraryfinitesets.Problemssimilartothisoneareknowntobeundecidable(see,e.g.[2]and[3]);also,attempts(includingbytheauthorsofthiscontribution)offindingsuchanalgorithmhavesofarfailed,wethereforerisktheconjecture:Conjecture1:Theproblemofdeterminingifagivenpairofmatricesisstableisundecidable.2STABILITYOFALLPERIODICPRODUCTSProblem1isrelatedtothegeneralizedspectralradiusofsetsofmatrices,anotionthatgeneralizestosetsofmatricestheusualnotionofspectralradiusofasinglematrix.Letρ(A)denotethespectralradiusofarealmatrixA,ρ(A):=max{|λ|:λisaneigenvalueofA}.Thegeneralizedspectralradiusρ(Σ)ofafinitesetofmatricesΣisdefinedin[7]byρ(Σ)=limsupρk(Σ),k→∞whereforeachk≥1ρ(Σ)=sup{(ρ(AA···A))1/k:eachA∈Σ}.k12kiWhenΣconsistofjustonesinglematrix,thisquantityisequaltotheusualspectralradius.Moreover,itiseasytoseethat,asforthesinglematrixcase,thestabilityofthesetΣisequivalenttotheconditionρ(Σ)<1,andsoproblem1istheproblemoffindingeffectiveconditionsonΣforρ(Σ)<1.Itisconjecturedin[11]thattheequalityρ(Σ)=ρk(Σ)alwaysoccurforsomefinitek.Thisconjecture,knownasthefinitenessconjecture,canberestatedbysayingthat,ifasetofmatricesΣisunstable,thenthereexistsafiniteunstableproduct,i.e.,ifρ(Σ)≥1,thenthereexistssomek≥1andAi∈Σ(i=1,...,k)suchthatρ(A1A2···Ak)≥1.Theexistenceofafiniteunstableproductisequivalenttotheexistenceofaninfiniteperiodicproductthatdoesnotconvergetozero.Wesaythatasetofmatricesisperiodicallystableifallinfiniteperiodicproductsofmatricestakeninthesetconvergetozero.Stabilityclearlyimpliesperiodicstability;accordingtothefinitenessconjecture,theconverseisalsotrue.Theconjec-turehasbeenprovedtobefalsein[6].Asimplecounterexampleisprovided WHENISAPAIROFMATRICESSTABLE?307in[5],whereitisshownthatthereareuncountablymanyvaluesoftherealparametersaandbforwhichthepairofmatrices1110a,b0111isnotstablebutisperiodicallystable.Sincestabilityandperiodicstabilityarenotequivalent,thefollowingquestionnaturallyarises.Problem2:Underwhatconditionsisagivenfinitesetofmatricesperiod-icallystable?BIBLIOGRAPHY[1]N.E.Barabanov,“AmethodforcalculatingtheLyapunovexponentofadifferentialinclusion,”Avtomat.iTelemekh.,4:53-58,1989;trans-lationinAutomat.RemoteControl50:4,part1,475-479,1989.[2]V.D.BlondelandJ.N.Tsitsiklis,“Whenisapairofmatricesmor-tal?”,InformationProcessingLetters,63:5,283-286,1997.[3]V.D.BlondelandJ.N.Tsitsiklis,“Theboundednessofallproductsofapairofmatricesisundecidable,”SystemsandControlLetters,41(2):135-140,2000.[4]V.D.BlondelandJ.N.Tsitsiklis,“Asurveyofcomputationalcom-plexityresultsinsystemsandcontrol,”Automatica,36(9),1249–1274,2000.[5]V.D.Blondel,J.Theys,andA.A.Vladimirov,“Anelementarycoun-terexampletothefinitenessconjecture,”forthcominginSIAMJournalofMatrixAnalysis.[6]T.BouschandJ.Mairesse,“Asymptoticheightoptimizationfortop-icalIFS,Tetrisheaps,andthefinitenessconjecture,”J.Amer.Math.Soc.,15,77-111,2002.[7]I.DaubechiesandJ.C.Lagarias,“Setsofmatricesallinfiniteproductsofwhichconverge,”LinearAlgebraAppl.,162,227-263,1992.[8]G.Gripenberg,“Computingthejointspectralradius,”LinearAlgebraAppl.,234,43–60,1996.[9]L.Gurvits,“Stabilityofdiscretelinearinclusion,”LinearAlgebraAppl.,231,47–85,1995.[10]J.E.HopcroftandJ.D.Ullman,IntroductiontoAutomataTheory,Languages,andComputation,Addison-Wesley,Reading,MA,1979. 308PROBLEM10.2[11]J.C.LagariasandY.Wang,“Thefinitenessconjectureforthegener-alizedspectralradiusofasetofmatrices,”LinearAlgebraAppl.,214,17-42,1995.[12]C.H.Papadimitriou,ComputationalComplexity,Addison-Wesley,Reading,1994.[13]J.N.Tsitsiklis,“Thestabilityoftheproductsofafinitesetofmatrices,”In:OpenProblemsinCommunicationandComputation,Springer-Verlag,1987.[14]J.N.TsitsiklisandV.D.Blondel,“Spectralquantitiesassociatedwithpairsofmatricesarehard-whennotimpossible-tocomputeandtoapproximate,”MathematicsofControl,Signals,andSystems,10,31-40,1997. Problem10.3FreenessofmultiplicativematrixsemigroupsVincentD.BlondelDepartmentofMathematicalEngineeringUniversityofLouvainAvenueGeorgesLemaitre,4B-1348Louvain-la-NeuveBelgiumblondel@inma.ucl.ac.beJulienCassaigneInstitutdeMath´ematiquesdeLuminyUPR9016-CampusdeLuminy,Case90713288MarseilleCedex9Franceupr9016@iml.univ-mrs.frJuhaniKarhum¨akiDept.ofMath.&TUCSUniversityofTurkuFIN-20014TurkuFinlandkarhumak@cs.utu.fi1DESCRIPTIONOFTHEPROBLEMMatricesplayamajorroleincontroltheory.Inthisnote,weconsiderade-cidabilityquestionforfinitelygeneratedmultiplicativematrixsemigroups.Suchsemigroupsarise,forexample,whenconsideringswitchedlinearsys-tems.WeconsiderembeddingsofthefreesemigroupΣ+={a,...,a}+0k−1intothemultiplicativesemigroupof2×2matricesovernonnegativeintegersN:ϕ:Σ+,→M(N).2×2 310PROBLEM10.3Foratwogeneratorsemigroup,i.e.,Σ+={a,b}+,suchembeddingsaredefined,forexample,bymappings:1120a7−→a7−→0101ϕ1:andϕ2:.(1)1021b7−→b7−→1101Actually,ϕ1providesanembeddingofthetwogeneratorfreegroupintothemultiplicativesemigroupofunimodularmatrices,e.g.,intoSL(2,N).Theembeddingϕ2,inturn,directlyextendstoallfinitelygeneratedfreesemigroups.Indeed,themappingkiϕi:ai7−→fori=0,...,k−101yieldsanembedding{a,...,a}+,→M(N).(2)0k−12×2Toseethis,itisenoughtoverifythatk|w|k(w)ϕi(w)=,01wherek(w)denotesthenumberrepresentedinbasekbythewordw∈{a,...,a}+undertheidentification:acorrespondsthedigiti.Em-0k−1ibeddingsofcountablygeneratedfreesemigroupsareobtainedbyemployingamorphism{a,a,...}+,→{a,b}+,given,forexample,bythemapping01τ:a7→aib.Thenϕ◦τyieldsarequiredembedding.i2Intheaboveexamplesitiseasytocheck,aswedidforϕi,i≥2,thatthemappingsareindeedembeddings.Ingeneral,thesituationisstrikinglydifferent.Infact,weformulate:Problem1:Isitdecidablewhetheragivenmorphismϕ:Σ+→M(N)2×2isanembedding,orequivalently,whetherafinitesetX={A0,...,Ak−1}of2×2matricesoverNisafreegeneratingsetofX+?Problem1deservestwocomments.First,wecouldconsidermatricesoverrationalnumbersratherthannonnegativeintegers.Thisvariantis-asitisnottoodifficulttosee-equivalenttothecasewherematricesareinte-germatrices.Second,theproblemisopenevenifonlytwomatricesareconsidered:Problem2:Isitdecidablewhetherthemultiplicativesemigroupgeneratedbytwo2×2matricesoverNisfree?Ofcourse,thenontrivialpartofproblem2isthecasewhenthesemigroupisofrank2.Inmanyconcreteexamples,thefreenessiseasytoconclude,aswesaw.Amazingly,however,theproblemremainsevenifthematricesareupper-triangular,asisϕ2above. FREENESSOFMULTIPLICATIVEMATRIXSEMIGROUPS3112MOTIVATIONANDHISTORYTheimportanceofproblem1shouldbeobvious,withoutanyfurthermotiva-tion.Indeed,productofmatricesisoneofthemostfundamentaloperationsinmathematics.Inlinearalgebraitwitnessesthecompositionoflinearmap-pings,andinautomatatheoryitdefinesthebehavioroffiniteautomaton,cf.[7],tomentionjusttwoexamples.However,theimportanceofProblem1goesfarbeyondthesegeneralreasons.Theexistenceofembeddingslike(2)havebeenknownforalongtime.Al-readyinthe1920sJ.Neilsen[12]wasusingthesewhenstudyingfreegroups.Suchembeddingsareextremelyusefulforboththetheoriesinvolved.Inonehand,thesecanbeusedtotransferresultsonwordsintothoseofmatrices.Theundecidabilityisanexampleofapropertythatisnaturalandcommoninthetheoryofwords,andtranslatabletomatricesviatheseembeddings.This,indeed,isessentialinthespiritofthisnote.Ontheotherhand,thereexistmanydeepresultsonmatricesthathaveturnedoutusefulforsolvingproblemsonwords.AsplendidexampleisHilbertBasesTheorem,whichimplies-againviaaboveembeddings-afun-damentalcompactnesspropertyofwordequations,so-calledEhrenfeuchtCompactnessProperty,cf.[5].Accordingtotheknowledgeoftheauthor,theproblemsmentionedwerefirstdiscussedin[10],whereproblem1wasexplicitlystated,anditsvariantfor3×3matricesoverNwasshowntobeundecidable.In[4]theundecidabilitywasextendedtoupper-triangular3×3matricesoverN,andmoreoverproblem2wasformulated.Similarproblemsonmatriceshavebeenconsideredmuchearlier.AmongtheoldestresultsisaremarkablepaperbyM.Paterson[13],whereheshowsthatitisundecidablewhetherthemultiplicativesemigroupgeneratedbyafinitesetof3×3integermatricescontainsthezeromatrix.Inotherwords,themortalityproblem,cf.[16],for3×3integermatricesisundecidable.Accordingtothecurrentknowledge,itremainsundecidableeveninthecaseswhenafinitesetconsistsofonlyseven3×3integermatricesorofonlytwo21×21integermatrices,cf.[11]and[8,3,2].For2×2matrices,themortalityproblemisopen.Theaboveundecidabilityresultscanbeinterpretedasfollows.First,theexistenceofthezeroelementinatwogenerator(matrix)semigroupisun-decidable,cf.[3].Second,itisalsoundecidablewhethersomecompositionofaneffectivelygivenfinitesetoflineartransformationofEuclidianspaceR3equalstothezeromapping.Theabovemotivatesarelatedquestion:isitdecidablewhetherafinitelygeneratedsemigroupcontainstheunitelement?Intermsofmatrices,westate:Problem3:IsitdecidablewhetherthemultiplicativesemigroupSgener- 312PROBLEM10.3atedbyafinitesetofn×nintegermatricescontainstheunitmatrix?Forn=2thisisshowntobedecidableinthecaseoftwomatricesin[11],andinthecaseofanarbitrarynumberofmatricesin[6],butingeneraltheproblemisopen.Arelatedproblem,alsoopenatthemoment,askswhetherthesemigroupScontainsadiagonalmatrix.Inthiscontext,thefollowingexampleisofinterest.Example1:Fortwomorphismsh,g:{a,...,a}+→{2,3}+define0k−1thematrices10|h(ai)|10|h(ai)|−10|g(ai)|h(a)−g(a)iiM(i)=010|g(ai)|g(ai)001fori=0,...,k−1.Astraightforwardcomputationshowsthat,foranyw=ai1...ait:10|h(w)|10|h(w)|−10|g(w)|h(w)−g(w)M(i1)...M(it)=010|g(w)|g(w).001Consequently,duetotheundecidabilityofPostCorrespondenceProblem,cf.[14],itisalsoundecidablewhetherthemultiplicativesemigroupgeneratedbyafinitesetof3×3integermatricescontainsamatrixoftheformα000βδ.00γWedonotknowhowtogetridofδ.3AVAILABLERESULTSDuetotheembeddingΣ+,→M(N),onewaytoviewProblem1isto2×2consideritasanextensionoftheproblemaskingtodecidewhetherafinitesetofwordsofΣ+generatesafreesubsemigroupofΣ+.Thisproblemisbasicinthetheoryofcodes,cf.[1].Itisdecidable,evenefficiently,asitisnottoodifficulttosee,cf.e.g.[15].Verylittleseemstobeknownaboutproblem1.Aswealreadysaidthecorrespondingproblemfor3×3matricesisundecidable,theproofbeingareductiontoPostCorrespondenceProblem,asinexample1.Abitmoreintriguingreductiontechniqueswereusedin[4]inordertoshowthattheundecidabilityholdsevenforupper-triangular3×3matricesoverN.AfundamentalobservationintheseproofsisthattheproductmonoidΣ+×∆+isnotembeddableonlyintothesemigroupofmatricesofdimensionfourbutalsointothatofdimensionthree.Inotherwords,thereexistsanembeddingϕ:Σ+×∆+,→M(N).3×3 FREENESSOFMULTIPLICATIVEMATRIXSEMIGROUPS313Ontheotherhand,asalsoshownin[4],theredoesnotexistanysuchembeddingintothesemigroupof2×2matrices,i.e.,intoM2×2(N).Problem2wasformulatedaftervainattemptstosolveitin[4].Actually,eventhecasewhenbothofthematricesareupper-triangular,i.e.,oftheformαβ0γremainedunanswered.Onlyseveralsufficient(effective)conditionsforthefreenesswereestablished.Evenforsomeveryparticularcases,wedonotknowifthesemigroupisfree.Inparticular,wedonotknowwhetherthematrices2035A=andB=0305generateafreesemigroup.Weonlyknowthatthesematricesdonotsatisfyanyequationwherebothsidesareoflengthatmost20.Asaconclusion,wehope,wehavebeenabletopointoutaproblemthatisnotonlyverysimplyformulated,butalsofundamentalandchallenging.BIBLIOGRAPHY[1]J.BerstelandD.Perrin,TheoryofCodes,AcademicPress,1986.[2]V.BlondelandJ.Tsitsiklis,“Whenisapairofmatricesmortal,”In-form.Process.Lett.63,283–286,1997.[3]J.CassaigneandJ.Karhum¨aki,“Examplesofundecidableproblemsfor2-generatormatrixsemigroups,”Theoret.Comput.Sci.204,29–34,1998.[4]J.Cassaigne,T.HarjuandJ.Karhum¨aki,“Ontheundecidabilityoffreenessofmatrixsemigroups,”Int.J.AlgebraComp.9,295–305,1999.[5]C.ChoffrutandJ.Karhum¨aki,“Combinatoricsofwords,”In:G.Rozen-bergandA.Salomaa(eds.),HandbookofFormalLanguages,vol.I,Springer,1997,329–438.[6]C.ChoffrutandJ.Karhum¨aki,“Ondecisionproblemsonintegerma-trices,”manuscript.[7]S.Eilenberg,Automata,LanguagesandMachines,vol.A,AcademicPress,1974.[8]V.HalavaandT.Harju,“Mortalityinmatrixsemigroups,”Amer.Math.Monthly108,649–653,2001. 314PROBLEM10.3[9]T.HarjuandJ.Karhum¨aki,Morphisms,In:G.RozenbergandA.Salomaa(eds.),HandbookofFormalLanguages,vol.I,Springer,1997,439–510.[10]D.Klarner,J.-C.BirgetandW.Satterfield,“Ontheundecidabilityofthefreenessofintegermatrixsemigroups,”Int.J.AlgebraComp.1,223–226,1991.[11]F.Mazoit,“Autourdequelquesprobl`emesded´ecidabilit´esurdessemi-groupesdematrices,”manuscript,1998.[12]J.Nielsen,“DieIsomorphismengruppederfreienGruppen,”Math.Ann.91,169–209,1924.[13]M.Paterson,“Unsolvabilityin3×3matrices,”StudiesAppl.Math.49,105–107,1970.[14]E.Post,“Avariantofrecursivelyunsolvableproblem,”Bull.Amer.Math.Soc.52,264–268,1949.[15]W.Rytter,“Thespacecomplexityoftheuniquedecipheribilityprob-lem,”Inform.Process.Lett.23,1–3,1986.[16]P.Schultz,“Mortalityof2×2matrices,”Amer.Math.Monthly84,463–464,1977. Problem10.4Vector-valuedquadraticformsincontroltheoryFrancescoBulloCoordinatedScienceLaboratoryUniversityofIllinoisUrbana-Champaign,IL61801USAbullo@uiuc.eduJorgeCort´esCoordinatedScienceLaboratoryUniversityofIllinoisUrbana-Champaign,IL61801USAjcortes@uiuc.eduAndrewD.LewisMathematics&StatisticsQueen’sUniversityKingston,ONK7L3N6Canadaandrew@mast.queensu.caSoniaMart´ınezMatem´aticaAplicadaIVUniversidadPolit´ecnicadeCatalu˜naBarcelona,08800Spainsoniam@mat.upc.es1PROBLEMSTATEMENTANDHISTORICALREMARKSForfinitedimensionalR-vectorspacesUandV,weconsiderasymmetricbilinearmapB:U×U→V.ThisthendefinesaquadraticmapQB:U→VbyQ(u)=B(u,u).Correspondingtoeachλ∈V∗isaR-valuedquadraticBformλQBonUdefinedbyλQB(u)=λ·QB(u).Bisdefiniteifthere 316PROBLEM10.4existsλ∈V∗sothatλQispositive-definite.BisindefiniteifforeachBλ∈V∗ann(image(Q)),λQisneitherpositivenornegative-semidefinite,BBwhereanndenotestheannihilator.GivenasymmetricbilinearmapB:U×U→V,theproblemsweconsiderareasfollows.i.FindnecessaryandsufficientconditionscharacterizingwhenQBissurjective.ii.IfQBissurjectiveandv∈V,designanalgorithmtofindapoint−1u∈Q(v).Biii.FindnecessaryandsufficientconditionstodeterminewhenBisin-definite.Fromthecomputationalpointofview,thefirsttwoquestionsarethemoreinterestingones.BothcanbeshowntobeNP-complete,whereasthethirdonecanberecastasasemidefiniteprogrammingproblem.1Actually,ourmaininterestisinageometriccharacterizationoftheseproblems.Section4belowconstitutesaninitialattempttounveiltheessentialgeometrybehindthesequestions.Byunderstandingthegeometryoftheproblemproperly,onemaybeleadtosimplecharacterizationsliketheonepresentedinPropo-sition3,whichturnouttobecheckableinpolynomialtimeforcertainclassesofquadraticmappings.Beforewecommentonhowourproblemimpingesoncontroltheory,letusprovidesomehistoricalcontextforitasapurelymathematicalone.TheclassificationofR-valuedquadraticformsiswellunderstood.However,forquadraticmapstakingvaluesinvectorspacesofdimensiontwoorhigher,theclassificationproblembecomesmoredifficult.ThetheorycanbethoughtofasbeginningwiththeworkofKronecker,whoobtainedafiniteclassifi-cationforpairsofsymmetricmatrices.Forthreeormoresymmetricma-trices,thattheclassificationproblemhasanuncountablenumberofequiv-alenceclassesforagivendimensionofthedomainfollowsfromtheworkofKac[12].Forquadraticforms,inaseriesofpapersDines(see[8]andreferencescitedtherein)investigatedconditionswhenafinitecollectionofR-valuedquadraticmapsweresimultaneouslypositive-definite.Thestudyofvector-valuedquadraticmapsisongoing.Arecentpaperis[13],towhichwereferforotherreferences.2CONTROLTHEORETICMOTIVATIONInterestingly,andperhapsnotobviously,vector-valuedquadraticformscomeupinavarietyofplacesincontroltheory.Welistafewofthesehere.1Wethankananonymousrefereefortheseobservations. VECTOR-VALUEDQUADRATICFORMSINCONTROLTHEORY317Optimalcontrol:Agraˇchev[2]explicitlyrealizessecond-orderconditionsforoptimalityintermsofvector-valuedquadraticmaps.Thegeometricapproachleadsnaturallytotheconsiderationofvector-valuedquadraticmaps,andherethenecessaryconditionsinvolvedefinitenessofthesemaps.AgraˇchevandGamkrelidze[1,3]lookatthemapλ7→λQfromV∗intoBthesetofvector-valuedquadraticmaps.SinceλQBisaR-valuedquadraticform,onecantalkaboutitsindexandrank(thenumberof−1’sandnonzeroterms,respectively,alongthediagonalwhentheformisdiagonal-ized).In[1,3]thetopologyofthesurfacesofconstantindexofthemapλ7→λQBisinvestigated.Localcontrollability:Theuseofvector-valuedquadraticformsarisesfromtheattempttoarriveatfeedback-invariantconditionsforcontrolla-bility.Basto-Gon¸calves[6]givesasecond-ordersufficientconditionforlo-calcontrollability,oneofwhosehypothesesisthatacertainvector-valuedquadraticmapbeindefinite(althoughtheconditionisnotstatedinthisway).Thisconditionissomewhatrefinedin[11],andanecessaryconditionforlocalcontrollabilityisalsogiven.Includedinthehypothesesofthelatteristheconditionthatacertainvector-valuedquadraticmapbedefinite.Controldesignviapowerseriesmethodsandsingularinversion:Numerouscontroldesignproblemscanbetackledusingpowerseriesandinversionmethods.Theearlyreferences[5,9]showhowtosolvetheoptimalregulatorproblemandtherecentworkin[7]proposeslocalsteeringalgo-rithms.Thesestrongresultsapplytolinearlycontrollablesystems,andnogeneralmethodsareyetavailableunderonlysecond-ordersufficientcontrol-labilityconditions.Whileforlinearlycontrollablesystemstheclassicinversefunctiontheoremsuffices,thekeyrequirementforsecond-ordercontrollablesystemsistheabilitytochecksurjectivityandcomputeaninversefunctionforcertainvector-valuedquadraticforms.Dynamicfeedbacklinearisation:In[14]Sluisgivesanecessaryconditionforthedynamicfeedbacklinearizationofasystemx˙=f(x,u),x∈Rn,u∈Rm.Theconditionisthatforeachx∈Rn,thesetD={f(x,u)∈TRn|u∈xxRm}admitsaruling,thatis,afoliationofDbylines.Somemanipulationsxwithdifferentialformsturnsthisnecessaryconditionintooneinvolvinga−1symmetricbilinearmapB.Thecondition,itturnsout,isthatQ(0)6={0}.BThisisshownbyAgraˇchev[1]togenericallyimplythatQBissurjective.3KNOWNRESULTSLetusstateafewresultsalongthelinesofourproblemstatementthatareknowntotheauthors.Thefirstisreadilyshowntobetrue(see[11]fortheproof).IfXisatopologicalspacewithsubsetsA⊂S⊂X,wedenotebyintS(A)theinteriorofArelativetotheinducedtopologyonS.IfS⊂V, 318PROBLEM10.4aff(S)andconv(S)denote,respectively,theaffinehullandtheconvexhullofS.Proposition1:LetB:U×U→VbeasymmetricbilinearmapwithUandVfinite-dimensional.Thefollowingstatementshold:(i)Bisindefiniteifandonlyif0∈intaff(image(QB))(conv(image(QB)));(ii)BisdefiniteifandonlyifthereexistsahyperplaneP⊂Vsothatimage(QB)∩P={0}andsothatimage(QB)liesononesideofP;(iii)ifQBissurjectivethenBisindefinite.Theconverseof(iii)isfalse.ThequadraticmapfromR3toR3definedbyQB(x,y,z)=(xy,xz,yz)maybeshowntobeindefinitebutnotsurjective.AgraˇchevandSarychev[4]provethefollowingresult.Wedenotebyind(Q)theindexofaquadraticmapQ:U→RonavectorspaceU.Proposition2:LetB:U×U→VbeasymmetricbilinearmapwithVfinite-dimensional.Ifind(λQ)≥dim(V)foranyλ∈V∗{0}thenQisBBsurjective.Thissufficientconditionforsurjectivityisnotnecessary.ThequadraticmapfromR2toR2givenbyQ(x,y)=(x2−y2,xy)issurjective,butdoesnotBsatisfythehypothesesofProposition2.4PROBLEMSIMPLIFICATIONOneofthedifficultieswithstudyingvector-valuedquadraticmapsisthattheyaresomewhatdifficulttogetone’shandson.However,itturnsouttobepossibletosimplifytheirstudybyareductiontoaratherconcreteproblem.Herewedescribethisprocess,onlysketchingthedetailsofhowtogofromagivensymmetricbilinearmapB:U×U→Vtothereformulatedendproblem.WefirstsimplifytheproblembyimposinganinnerproductonUandchoosinganorthonormalbasissothatwemaytakeU=Rn.WeletSymn(R)denotethesetofsymmetricn×nmatriceswithentriesinR.OnSymn(R)weusethecanonicalinnerproducthA,Bi=tr(AB).Weconsiderthemapπ:Rn→Sym(R)definedbyπ(x)=xxt,wheretndenotestranspose.Thustheimageofπisthesetofpositivesemidefinitesymmetricmatricesofrankatmostone.IfweidentifySym(R)'Rn⊗Rn,nthenπ(x)=x⊗x.LetKnbetheimageofπandnotethatitisaconeofdimensionninSymn(R)havingasingularityonlyatitsvertexattheorigin.Furthermore,Knmaybeshowntobeasubsetofthehyperconein VECTOR-VALUEDQUADRATICFORMSINCONTROLTHEORY319Sym(R)definedbythosematricesAinSym(R)forminganglearccos(√1)nnnwiththeidentitymatrix.ThustherayfromtheorigininSymn(R)throughtheidentitymatrixisanaxisfortheconeKN.Inalgebraicgeometry,theimageofKnundertheprojectivizationofSymn(R)isknownastheVeronesesurface[10],andassuchiswell-studied,althoughperhapsnotalonglinesthatbeardirectlyontheproblemsofinterestinthisarticle.WenowletB:Rn×Rn→VbeasymmetricbilinearmapwithVfinite-dimensional.Usingtheuniversalmappingpropertyofthetensorproduct,BinducesalinearmapB˜:Sym(R)'Rn⊗Rn→Vwiththeprop-nertythatB˜◦π=B.ThedualofthismapgivesaninjectivelinearmapB˜∗:V∗→Symn(R)(hereweassumethattheimageofBspansV).ByanappropriatechoiceofinnerproductonV,onecanrendertheembeddingB˜∗anisometricembeddingofVinSymn(R).LetusdenotebyLBtheimageofVunderthisisometricembedding.Onemaythenshowthatwiththeseidentifications,theimageofQBinVistheorthogonalprojectionofKnontothesubspaceLB.Thuswereducetheproblemtooneoforthogo-nalprojectionofacanonicalobject,Kn,ontoasubspaceinSymn(R)!Tosimplifythingsfurther,wedecomposeLBintoacomponentalongtheiden-titymatrixinSymn(R)andacomponentorthogonaltotheidentitymatrix.However,thematricesorthogonaltotheidentityarereadilyseentosimplybethetracelessn×nsymmetricmatrices.UsingourpictureofKnasasubsetofahyperconehavingasanaxistheraythroughtheidentitymatrix,weseethatquestionsofsurjectivity,indefiniteness,anddefinitenessofBimpactonlyontheprojectionofKnontothatcomponentofLBorthogonaltotheidentitymatrix.Thefollowingsummarizestheabovediscussion.Theproblemofstudyingtheimageofavector-valuedquadraticformcanbereducedtostudyingtheorthogonalprojectionofKn⊂Symn(R),theunprojectivizedVeronesesurface,ontoasubspaceofthespaceoftracelesssymmetricmatrices.Thisis,wethink,abeautifulinterpretationofthestudyofvector-valuedquadraticmappings,andwillsurelybeausefulformulationoftheproblem.Forexample,withitoneeasilyprovesthefollowingresult.Proposition3:Ifdim(U)=dim(V)=2withB:U×U→Vasymmetricbilinearmap,thenQBissurjectiveifandonlyifBisindefinite.BIBLIOGRAPHY[1]A.A.Agraˇche,“ThetopologyofquadraticmappingsandHessiansofsmoothmappings,”J.SovietMath.,49(3):990–1013,1990. 320PROBLEM10.4[2]A.A.Agraˇchev,“Quadraticmappingsingeometriccontroltheory,”J.SovietMath.,51(6):2667–2734,1990.[3]A.A.AgraˇchevandR.V.Gamkrelidze,“Quadraticmappingsandvectorfunctions:Eulercharacteristicsoflevelsets,”J.SovietMath.,55(4):1892–1928,1991.[4]A.A.AgraˇchevandA.V.Sarychev,“Abnormalsub-Riemanniangeodesics:Morseindexandrigidity,”Ann.Inst.H.Poincar´e.Anal.NonLin´eaire,13(6):635–690,1996.[5]E.G.Al’brekht,“Ontheoptimalstabilizationofnonlinearsystems,”`J.Appl.Math.andMech.,25:1254–1266,1961.[6]J.Basto-Gon¸calves,“Second-orderconditionsforlocalcontrollability,”SystemsControlLett.,35(5):287–290,1998.[7]W.T.CervenandF.Bullo,“Constructivecontrollabilityalgorithmsformotionplanningandoptimization,”IEEETransAutomatControl,Forethcoming.[8]L.L.Dines,“Onlinearcombinationsofquadraticforms,”Bull.Amer.Math.Soc.(N.S.),49:388–393,1943.[9]A.Halme,“Onthenonlinearregulatorproblem,”J.Optim.TheoryAppl.,16(3-4):255–275,1975.[10]J.Harris,AlgebraicGeometry:AFirstCourse,Number133inGrad-uateTextsinMathematics,Springer-Verlag,NewYorkHeidelbergBerlin,1992.[11]R.M.HirschornandA.D.Lewis,“Geometriclocalcontrollability:Second-orderconditions,”preprint,February2002.[12]V.G.Kac,“Rootsystems,representationsofquiversandinvarianttheory,”In:InvariantTheory,number996inLectureNotesinMath-ematics,pages74–108,Springer-Verlag,NewYork-Heidelberg-Berlin,1983.[13]D.B.LeepandL.M.Schueller,“Classificationofpairsofsymmetricandalternatingbilinearforms,”Exposition.Math.,17(5):385–414,1999.[14]W.M.Sluis,“Anecessaryconditionfordynamicfeedbacklineariza-tion,”SystemsControlLett.,21(4):277–283,1993. Problem10.5NilpotentbasesofdistributionsHenryG.HermesDepartmentofMathematicsUniversityofColoradoatBoulder,Boulder,CO80309USAhermes@euclid.colorado.eduMatthiasKawski1DepartmentofMathematicsandStatisticsArizonaStateUniversityTempe,AZ85287-1804USAkawski@asu.edu1DESCRIPTIONOFTHEPROBLEMWhenmodelingcontrolleddynamicalsystemsonecommonlychoosesindi-vidualcontrolvariablesu1,...umthatappearnaturalfromaphysical,orpracticalpointofview.InthecaseofnonlinearmodelsevolvingonRn(ormoregenerally,ananalyticmanifoldMn)thatareaffineinthecontrol,suchachoicecorrespondstoselectingvectorfieldsf0,f1,...fm:M7→TM,andthesystemtakestheformXmx˙=f0(x)+ukfk(x).(1)k=1Fromageometricpointofviewsuchachoiceappearsarbitrary,andthenaturalobjectsarenotthevectorfieldsthemselvesbuttheirlinearspan.Formally,forasetF={v1,...vm}ofvectorfieldsdefinethedistributionspannedbyFas∆F:p7→{c1v1(p)+...+cmvm(p):c1,...cm∈R}⊆TpM.Forsystemswithdriftf0,thegeometricobjectisthemap∆˜F(x)={f0(x)+c1f1(x)+...+cmfm(x):c1,...cm∈R}whoseimageateverypointxisanaffinesubspaceofTxM.Thegeometriccharacterofthedistributioniscapturedbyitsinvarianceunderthegroupoffeedbacktransformations.1SupportedinpartbyNSF-grantDMS00-72369. 322PROBLEM10.5Intraditionalnotation(hereformulatedforsystemswithdrift)theseare(analytic)maps(definedonsuitablesubsets)α:Mn×Rm7→Rmsuchthatforeachfixedx∈Mnthemapv7→α(x,v)isaffineandinvertible.Customarily,oneidentifiesα(x,·)withamatrixandwritesuk(x)=α0k(x)+v1α1k(x)+...vmαmk(x)fork=1,...m.(2)Thistransformationofthecontrolsinducesacorrespondingtransforma-Pm!tionofthevectorfieldsdefinedby˙Px=f0(x)+k=1ukfk(x)=g0(x)+mk=1vkgk(x)g0(x)=f0(x)+α01(x)f1(x)+...α0m(x)fm(x)(3)gk(x)=αk1(x)f1(x)+...αkm(x)fm(x),k=1,...mAssuminglinearindependenceofthevectorfieldssuchfeedbacktransfor-mationsamounttochangesofbasisoftheassociateddistributions.Onenaturallystudiestheorbitsofanygivensystemunderthisgroupaction,i.e.,thecollectionofequivalentsystems.Ofparticularinterestarenormalforms,i.e,naturaldistinguishedrepresentativesforeachorbit.Geometrically(i.e.,withoutchoosinglocalcoordinatesforthestatex),thesearecharacterizedbypropertiesoftheLiealgebraL(g0,g1,...gm)generatedbythevectorfieldsgk(acknowledgingthespecialroleofg0ifpresent).RecallthataLiealgebraLiscallednilpotent(solvable)ifitscentralde-scendingseriesL(k)(derivedseriesL)isfinite,i.e.,thereexistsr<∞suchthatL(r)={0}(L={0}).HereL=L(1)=L<1>andinductivelyL(k+1)=[L(k),L(1)]andL=[L,L].Themainquestionsofpracticalimportanceare:Problem1:Findnecessaryandsufficientconditionsforadistribution∆FspannedbyasetofanalyticvectorfieldsF={f1,...fm}toadmitaba-sisofanalyticvectorfieldsG={g1,...gm}thatgenerateaLiealgebraL(g1,...gm)thathasadesirablestructure,i.e.,thatisa.nilpotent,b.solv-able,orc.finitedimensional.Problem2:DescribeanalgorithmthatconstructssuchabasisGfromagivenbasisF.2MOTIVATIONANDHISTORYOFTHEPROBLEMThereisanabundanceofmathematicalproblems,whicharehardasgiven,yetarealmosttrivialwhenwrittenintherightcoordinates.Classicalexam-plesoffindingtherightcoordinates(or,rather,therightbases)aretransfor-mationsthatdiagonalizeoperatorsinlinearalgebraandfunctionalanalysis.Similarly,everysystemof(ordinary)differentialequationisequivalent(viaachoiceoflocalcoordinates)tothesystem˙x1=1,x˙2=0,...x˙n=0(intheneighborhoodofeverypointthatisnotanequilibrium).Incontrol,formanypurposesthemostconvenientformisthecontrollercanonicalform NILPOTENTBASESOFDISTRIBUTIONS323(e.g.,inthecaseofm=1)˙x1=uand˙xk=xk−1for11.)Whatmakessuchlinearorpolynomialcascadeformsoattractiveforbothanalysisanddesignisthattrajectoriesx(t,u)maybecomputedfromcontrolsu(t)byquadraturesonly,obviatingtheneedtosolvenonlinearODEs.Typicalexamplesincludepoleplacementandpathplanning[11,16,19].Inparticular,iftheLiealgebraisnilpotent(orsimilarlynice),thegeneralsolutionformulaforx(·,u)asanex-ponentialLieseries[20](whichgeneralizesmatrixexponentialstononlinearsystems)collapsesandbecomesinnatelymanageable.Itiswell-knownthatasystemcanbebroughtintosuchpolynomialcascadeformviaacoordinatechangeifandonlyiftheLiealgebraL(f1,...fm)isnilpotent[9].SimilarresultsforsolvableLiealgebrasareavailable[1].Thisleavesopenonlythegeometricquestionaboutwhendoesadistributionadmitanilpotent(orsolvable)basis.3RELATEDRESULTSIn[5]itisshownthatforevery2≤k≤(n−1)thereisak-distribution∆onRnthatdoesnotadmitasolvablebasisinaneighborhoodofzero.Thisshowstheproblemsofnilpotentandsolvablebasesarenottrivial.Geometricproperties,suchassmall-timelocalcontrollability(STLC)are,bytheirverynature,unaffectedbyfeedbacktransformations.Thuscondi-tionsforSTLCprovidevaluableinformationwhetheranytwosystemscanbefeedbackequivalent.Typicalsuchinformation,generalizingthecontrolla-bilityindicesoflinearsystemstheory,iscontainedinthegrowthvector,thatisthedimensionsofthederiveddistributionsthataredefinedinductivelyby∆(1)=∆and∆(k+1)=∆(k)+{[v,w]:v∈∆(k),w∈∆(1)}.Ofhighestpracticalinterestisthecasewhenthesystemis(locally)equiva-lenttoalinearsystem˙x=Ax+Bu(forsomechoiceoflocalcoordinates).Necessaryandsufficientconditionsforsuchexactfeedbacklinearizationto-getherwithalgorithmsforconstructingthetransformationandcoordinateswereobtainedinthe1980s[6,7].Thegeometriccriteriaarenicelystatedintermsoftheinvolutivity(closednessunderLiebracketing)ofthedistribu-jtionsspannedbythesets{(adf0,f1):0≤j≤k}for0≤k≤m.AnecessaryconditionforexactnilpotentizationisbasedontheobservationthateverynilpotentLiealgebracontainsatleastoneelementthatcommuteswitheveryotherelement[4].Awell-studiedspecialcaseisthatofnilpotentsystemswhichthatcanbebroughtintochained-form,compare[16].Thisiscloselyrelatedtodifferen- 324PROBLEM10.5tiallyflatsystems,compare[2,8],whichhavebeenthefocusofmuchstudyinthe1990s.Thekeypropertyistheexistenceofanoutputfunctionsuchthatallsystemvariablescanbeexpressedintermsoffunctionsofafinitenumberofderivativesofthisoutput.Thisworkismorenaturallyperformedusingadualdescriptionintermsofexteriordifferentialsystemsandco-distributions∆⊥={ω:M7→T∗M:hω,fi=0forallf∈∆}.Thisdescriptionispar-ticularlyconvenientwhenworkingwithsmallco-dimensionn−m,compare[12]forarecentsurvey.(Specialcareneedstobetakenatsingularpointswherethedimensionsof∆(k)arenonconstant.)ThislanguageallowsonetodirectlyemploythemachineryofCartan’smethodofequivalence[3].How-ever,anicedescriptionofasystemintermsofdifferentialformsdoesnotnecessarilytranslateinastraightforwardmannerintoanicedescriptionintermsofvectorfields(thatgenerateafinitedimensional,ornilpotentLiealgebra).SomeofthemostnotablerecentprogresshasbeenmadeinthegeneralframeworkofGoursatdistributions,see,e.g.,[13,14,15,17,18,21]fordetaileddescriptions,themostrecentresultsandfurtherrelevantreferences.BIBLIOGRAPHY[1]P.Crouch,“Solvableapproximationstocontrolsystems,”SIAMJ.Con-trol&Optim.,22,pp.40-45,1984.[2]M.Fliess,J.Levine,P.Martin,andP.Rouchon,“Someopenquestionsrelatedtoflatnonlinearsystems,”OpenproblemsinMathematicalSys-temsandControlTheory,V.Blondel,E.Sontag,M.Vidyasagar,andJ.Willems,eds.,Springer,1999.[3]R.Gardener,“Themethodofequivalenceanditsapplications,”CBMSNSFRegionalConferenceSeriesinAppliedMathematics,SIAM,58,1989.[4]H.Hermes,A.Lundell,andD.Sullivan,“Nilpotentbasesfordistribu-tionsandcontrolsystems,”J.ofDiff.Equations,55,pp.385–400,1984.[5]H.Hermes,“Distributionsandtheliealgebrastheirbasescangenerate,”Proc.AMS,106,pp.555–565,1989.[6]R.Hunt,R.Su,andG.Meyer,“Designformulti-inputnonlinearsys-tems,”DifferentialGeometricControlTheory,R.Brockett,R.Millmann,H.Sussmann,eds.,Birkh¨auser,pp.268–298,1982.[7]B.JakubvzykandW.Respondek,“Onlinearizationofcontrolsystems,”Bull.Acad.Polon.Sci.Ser.Sci.Math.,28,pp.517–522,1980.[8]F.Jean,“Thecarwithntrailers:Characterizationfothesingularcon-figuartions,”ESAIMControlOptim.Calc.Var.1,pp.241–266,1996. NILPOTENTBASESOFDISTRIBUTIONS325[9]M.Kawski“NilpotentLiealgebrasofvectorfields,”Journalf¨urdieRieneundAngewandteMathematik,188,pp.1-17,1988.[10]M.KawskiandH.J.Sussmann“NoncommutativepowerseriesandformalLie-algebraictechniquesinnonlinearcontroltheory,”Operators,Systems,andLinearAlgebra,U.Helmke,D.Pr¨atzel-WoltersandE.Zerz,eds.,Teubner,111–128,1997.[11]G.LaffariereandH.Sussmann,“Adifferentialgeometricapproachtomotionplanning,”IEEEInternationalConferenceonRoboticsandAu-tomation,pages1148–1153,Sacramento,CA,1991.[12]R.Montgomery,“ATourofSubriemannianGeometries,TheirGeodesicsandApplications,”AMSMathematicalSurveysandMono-graphs,91,2002.[13]P.Mormul,“Goursatflags,classificationofco-dimensiononesingulari-ties,”J.DynamicalandControlSystems,6,2000,pp.311–330,2000.[14]P.Mormul,“Multi-dimensionalCaratnprolongationandspecialk-flags,”technicalreport,UniversityofWarsaw,Poland,2002.[15]P.Mormul,“Goursatdistributionsnotstronglynilpotentindimensionsnotexceedingseven,”LectureNotesinControlandInform.Sci.,281,pp.249–261,Springer,Berlin,2003.[16]R.Murray,“Nilpotentbasesforaclassofnon-integrabledistributionswithapplicationstotrajectorygenerationfornonholonomicsystems,”MathematicsofControls,Signals,andSystems,7,pp.58–75,1994.[17]W.Pasillas-Lpine,andW.Respondek,“OnthegeometryofGoursatstructures,”ESAIMControlOptim.Calc.Var.6,pp.119–181,2001.[18]W.Respondek,andW.Pasillas-Lpine,“ExtendedGoursatnormalform:ageometriccharacterization,”In:LectureNotesinControlandInform.Sci.,259pp.323–338,Springer,London,2001.[19]J.StrumperandP.Krishnaprasad,“ApproximatetrackingforsystemsonthreedimensionalLiematrixgroupsviafeedbacknilpotentization,”IFACSymposiumRobotControl(1997).[20]H.Sussmann,“AproductexpansionoftheChenseries,”TheoryandApplicationsofNonlinearControlSystems,C.ByrnesandA.Lindquisteds.,Elsevier,pp.323–335,1986.[21]M.Zhitomirskii,“Singularitiesandnormalformsofsmoothdistribu-tions,”BanachCenterPubl.,32,pp.379-409,1995. Problem10.6Whatisthecharacteristicpolynomialofasignalflowgraph?AndrewD.LewisDepartmentofMathematics&StatisticsQueen’sUniversityKingston,ONK7L3N6Canadaandrew@mast.queensu.ca1PROBLEMSTATEMENTSupposeoneisgivensignalflowgraphGwithnnodeswhosebrancheshavegainsthatarerealrationalfunctions(theopenlooptransferfunctions).ThegainofthebranchconnectingnodeitonodejisdenotedRji,andwewriteNjiRji=Dasacoprimefraction.Theclosed-looptransferfunctionfromjinodeitonodejfortheclosed-loopsystemisdenotedTji.Theproblemcanthenbestatedasfollows:IsthereanalgorithmicprocedurethattakesasignalflowgraphGandreturnsa“characteristicpolynomial”PGwiththefollowingproperties:i.PGisformedbyproductsandsumsofthepolynomialsNjiandDji,i,j=1,...,n;ii.allclosed-looptransferfunctionsTji,i,j=1,...,n,areanalyticintheclosedrighthalf-plane(CRHP)ifandonlyifPGisHurwitz?ThegistofconditioniisthattheconstructionofPGdependsonlyonthetopologyofthegraph,andnotonmanipulationsofthebranchgains.Thatis,thedefinitionofPGshouldnotdependonthechoiceofbranchgainsRji,i,j=1,...,n.Forexample,onewouldbeprohibitedfromfactoringpolynomialsorfromcomputingtheGCDofpolynomials.Thisexcludesunhelpfulsolutionsoftheproblemoftheform,“LetPGbetheproductofthecharacteristicpolynomialsoftheclosed-looptransferfunctionsTji,i,j=1,...,n.” SIGNALFLOWGRAPHS3272DISCUSSIONSignalflowgraphsformodellingsysteminterconnectionsareduetoMa-son[3,4].Ofcourse,whenmakingsuchinterconnections,thestabilityoftheinterconnectionisnontriviallyrelatedtotheopen-looptransferfunc-tionsthatweightthebranchesofthesignalflowgraph.Thereareatleasttwothingstoconsiderinthecourseofmakinganinterconnection:(1)istheinterconnectionBIBOstableinthesensethatallclosed-looptransferfunctionsbetweennodeshavenopolesintheCRHP?;and(2)isthein-terconnectionwell-posedinthesensethatallclosed-looptransferfunctionsbetweennodesareproper?Theproblemstatedaboveconcernsonlythefirstofthesematters.Well-posednesswhenallbranchgainsRji,i,j=1,...,n,areproperisknowntobeequivalenttotheconditionthatthedeterminantofthegraphbeabiproperrationalfunction.Wecommentthatotherformsofstabilityforsignalflowgraphsarepossible.Forexample,Wang,Lee,andHo[5]considerinternalstabilty,whereinnotthetransferfunctionsbetweensignalsareconsidered,butratherthatallsignalsinthesignalflowgraphremainboundedwhenboundedinputsareprovided.InternalstabilityasconsideredbyWang,Lee,andHoandBIBOstabilityasconsideredherearedifferent.Thesourceofthisdifferenceaccountsforthesourceoftheopenproblemofourpaper,sinceWang,Lee,andHoshowthatinternalstabilitycanbedeterminedbyanalgorithmicprocedurelikethatweaskforforBIBOstability.Thisisdiscussedalittlefurtherinsection3.Asanillustrationofwhatweareafter,considerthesingle-loopconfigurationoffigure10.6.1.Asiswell-known,ifwewriteR=Ni,i=1,2,ascoprimeiDir-d-R1(s)-R2(s)-y−6Figure10.6.1Single-loopinterconnectionfractions,thenallclosed-looptransferfunctionshavenopolesintheCRHPifandonlyifthepolynomialPG=D1D2+N1N2isHurwitz.ThusPGservesasthecharacteristicpolynomialinthiscase.TheessentialfeatureofPGisthatonecomputesitbylookingatthetopologyofthegraph,andtheexactcharacterofR1andR2areofnoconsequence.Forexample,pole/zerocancellationsbetweenR1andR2areaccountedforinPG.3APPROACHESTHATDONOTSOLVETHEPROBLEMLetusoutlinetwoapproachesthatyieldsolutionshavingoneofpropertiesiandii,butnottheother. 328PROBLEM10.6Theproblemsofinternalstabilityandwell-posednessforsignalflowgraphscanbehandledeffectivelyusingthepolynomialmatrixapproach,e.g.,[1].Suchanapproachwillinvolvethedeterminationofacoprimematrixfrac-tionalrepresentationofamatrixofrationalfunctions.Thiswillcertainlysolvetheproblemofdetermininginternalstabilityforanygivenexample.Thatis,itispossibleusingmatrixpolynomialmethodstoprovideanalgo-rithmicconstructionofapolynomialsatisfyingpropertyiiabove.However,thealgorithmicprocedurewillinvolvecomputingGCDsofvariousofthepolynomialsNjiandDji,i,j=1,...,n.Thustheconditionsdevelopedinthismannerhavetodonotonlywiththetopologyofthesignalflowgraph,butalsothespecificchoicesforthebranchgains,thusviolatingconditioniabove.Theproblemweposedemandsasimpler,moredirectanswertothequestionofdeterminingwhenaninterconnectionisBIBOstable.Wang,Lee,andHe[5]provideapolynomialforasignalflowgraphusingthedeterminantofthegraphwhichwedenoteby∆G(see[3,4]).Specifically,theydefineapolynomialYP=Dji∆G.(1)(i,j)∈{1,...,n}2Thusonemultipliesthedeterminantbyalldenominators,arrivingatapoly-nomialintheprocess.Thispolynomialhasthepropertyiabove.However,whileitistruethatifthispolynomialisHurwitzthenthesystemisBIBOstable,theconverseisgenerallyfalse.ThuspropertyiiisnotsatisfiedbyP.Whatisshownin[5]isthatallsignalsinthegraphareboundedforboundedinputsifandonlyifPisHurwitz.Thisisdifferentfromwhatweareaskinghere,i.e.,thatallclosed-looptransferfunctionshavenoCRHPpoles.ItistruethatthepolynomialPin(1)givesthedesiredcharacteristicpolyno-mialfortheinterconnectionofFigure10.6.1.Itisalsotruethatifthesignalflowgraphhasnoloops(inthiscase∆G=1)thenthepolynomialPof(1)satisfiesconditionii.WecommentthattheconditionofWang,Lee,andHoisthesameconditiononewouldobtainbyconverting(inaspecificway)thesignalflowgraphtoapolynomialmatrixsystem,andthenascertainingwhentheresultingpolynomialmatrixsystemisinternallystable.Theproblemstatedisverybasic,oneforwhichaninquisitiveundergraduatewoulddemandasolution.Itwaswithsomesurprisethattheauthorwasunabletofinditssolutionintheliterature,andhopefullyoneofthereadersofthisarticlewillbeabletoprovideasolution,orpointoutanexistingone.BIBLIOGRAPHY[1]F.M.CallierandC.A.Desoer,MultivariableFeedbackSystems,Springer-Verlag,NewYork-Heidelberg-Berlin,1982.[2]A.D.Lewis,“OnthestabilityofinterconnectionsofSISO,time-invariant,finite-dimensional,linearsystems,”preprint,July2001. SIGNALFLOWGRAPHS329[3]S.J.Mason,“Feedbacktheory:Somepropertiesofsignalflowgraphs,”Proc.IRE,41:1144–1156,1953.[4]S.J.Mason,“Feedbacktheory:Furtherpropertiesofsignalflowgraphs,”Proc.IRE,44:920–926,1956.[5]Q.-G.Wang,T.-H.Lee,andJ.-B.He,“Internalstabilityofintercon-nectedsystems,”IEEETrans.Automat.Control,44(3):593–596,1999. Problem10.7OpenproblemsinrandomizedµanalysisOnurTokerCollegeofComputerSciencesandEngineeringKFUPM,P.O.Box14Dhahran31261SaudiArabiaonur@kfupm.edu.sa1INTRODUCTIONInthischapter,wereviewthecurrentstatusoftheproblemreportedin[5],anddiscusssomeopenproblemsrelatedtorandomizedµanalysis.Basically,whatremainsstillunknownafterTreil’sresult[6]arethegrowthrateoftheµ/µratio,andhowlikelyitistoobserveahighconservatism.Inthecontextofrandomizedµanalysis,wediscusstwoopenproblems(i)ExistenceofpolynomialtimeLasVegastyperandomizedalgorithmsforrobuststabilityagainststructuredLTIuncertainties,and(ii)Theminimumsamplesizetoguaranteethatµ/µˆconservatismwillbeboundedbyg,withaconfidencelevelof1−.2DESCRIPTIONANDHISTORYOFTHEPROBLEMThestructuredsingularvalue[1]isaquitegeneralframeworkforanaly-sis/designagainstcomponentlevelLTIuncertainties.Althoughforsmallnumberofuncertainblocks,theproblemisofreasonabledifficulty,allini-tialstudiesimpliedthatthesameisnotlikelytobetrueforthegeneralcase.Undertheseobservations,convexupperboundtestsbecamepopularalternativesforthestructuredsingularvalue.Later,ithasbeenprovedthattheseupperboundtestsareindeednonconservativerobustnessmeasuresfordifferentclassesofcomponentleveluncertainties,andthestructuredsin-gularvalueanalysisproblemisNP-hard.Seethepaper[5]andreferencesthereinforfurtherdetailsonthehistoryoftheproblem. OPENPROBLEMSINRANDOMIZEDµANALYSIS331WhatremainsstillopenafterTreil’sresult?Animportantopenproblemwastheconservativenessofthestandardupperboundtestforthecomplexµ[5].Recently,Treilshowedthatthegapbetweenµanditsupperboundµcanbearbitrarilylarge[6].Despitethisnegativeresult,computa-tionalexperimentsshowthatmostofthetimethegapisveryclosetooneformatricesofreasonablesize.Thefollowingarestillopenproblems:•Whatisthegrowthrateofthegap?Inotherwords,whatisthegrowthrateofµ(M)supµ(M)6=0µ(M)asafunctionofn=dim(M).ItissuspectedthatthegrowthrateisO(log(n))[6].•Howlikelyitistoobservelowconservatism?Inotherwords,whatistherelativevolumeoftheset{M:(1+)µ(M)≥µ(M)≥µ(M)}inthesetofalln×nmatriceswithallentrieshavingabsolutevalueatmost1.RandomizedalgorithmsandsomeopenproblemsRandomizedal-gorithmsareknowntobequitepowerfultoolsfordealingwithdifficultproblems.ArecentpaperofVidyasagarandBlondel[8]hasbothanicesummaryofearlierresultsinthisarea,andprovidesastrongjustificationfortheimportanceofrandomizedalgorithmsfortacklingdifficultcontrolrelatedproblems.Randomizedstructuredsingularvalueanalysisisstudiedindetailintherecentpaper[3],whichalsohasmanyreferencestorelatedworkinthisarea.LasVegastypealgorithmforµanalysisTherearetwopossiblewaysofutilizingtheresultsofrandomizedalgorithms,inparticulartherandomizedµanalysis.Letusassumethatseveralrandomuncertaintymatriceswithnormboundedby1,∆k’s,k=1,···,S,aregen-eratedaccordingtosomeprobabilitydistribution,andˆµ(M)issettoµˆ(M)=maxρ(M∆k).1≤k≤SThefirstinterpretationisthefollowing:withahighprobability,theinequal-ityρ(M∆)≤µˆ(M)issatisfiedforall∆’sexceptforasetofsmallrelativevolume[7].Thesecondinterpretationistoconsiderthewholeprocessofgeneratingrandom∆samplesandcheckingtheconditionρ(M∆)<1,asaMonteCarlotypealgorithmforthecomplementofrobuststability[2].Indeed,aftergeneratingseveral∆samples,ifˆµ(M)≥1,thenthe(M,∆)systemisnotrobustlystable,otherwiseonecaneithersaythetestisin-conclusiveorconcludethatthe(M,∆)systemisrobustlystable(which 332PROBLEM10.7sometimescanbethewrongconclusion).ThisunpleasantphenomenaisastandardcharacteristicofMonteCarloalgorithms.WhatisnotknowniswhetherthereisalsoapolynomialtimeMonteCarlotyperandomizedalgo-rithmfortherobuststabilityconditionitself.CombiningthesetwoMonteCarloalgorithmswillresultanalgorithmthatnevergivesafalseanswer,andtheprobabilityofgettinginconclusiveanswersgoestozeroaswegeneratemoreandmorerandomsamples.Problem1(LasVegastypealgorithmforµanalysis):IsthereapolynomialtimeLasVegastyperandomizedalgorithmfortestingrobuststabilityagainststructuredLTIuncertainties?Whythisproblemisimportant?Analgorithmlikethiscanbeusedtocheckwhetherthe(M,∆)systemisrobustlystableornotbygeneratingrandom∆matrices:Thereisnopossibilityofgettingfalseanswers,andprobabilityofgettinginconclusiveanswersgoestozeroasthesamplesizegoestoinfinity.However,therateofconvergenceoftheprobabilityofgettinginconclusiveanswerstozero,isalsoanimportantfactorforthealgorithmtobepractical.Relationshipbetweentheconservatismofµ/µˆ,samplesize,andconfidencelevelsConservativenessoftherandomizedlowerboundˆµisalsoanopenproblem.Moreprecisely,wehaveverylittleknowledgeabouttherelationshipbetweentheconservatismratioµ/µˆ,thesamplesizeS,andtheconservatismboundg.Forsimplicity,letndenotesthedimensionofthematrixMfortherestofthissection.Thefollowingisamajoropenproblem:Problem2:Findthebestlowerfound,S(n,g,),suchthatgeneratingS≥S(n,g,)randomsamplesisenoughtoclaimthat,forallM,µ(M)≥maxρ(M∆)≥g−1µ(M),withconfidencelevel≥1−.k1≤k≤S(n,g,)Inotherwords,theprobabilityinequalityProb∆,···,∆:µ(M)≥maxρ(M∆)≥g−1µ(M)≥1S(n,g,)k1≤k≤S(n,g,)≥1−,issatisfiedforallMmatrices.Whythisisanimportantproblem?Inarobuststabilityanalysis,onecansetaconfidencelevelverycloseone,say1−10−6,generatemanyrandom∆samples,andcomputetherandomized OPENPROBLEMSINRANDOMIZEDµANALYSIS333lowerboundˆµ.Ideally,acontrolengineerwouldlikeknowhowconservativeistheobtainedˆµcomparedtotheactualµ,inordertohaveabetterfeelingofthesystemathand.Thereisverylittleknownaboutthisproblem,andsomepartialresultsarereportedin[4].Thefollowingaresimplecorollaries:Result1(Polynomialnumberofsamples):ForanypositiveuniversalconstantsC,α∈Z,generatingS=Cnαrandomsamplesisnotenoughtonclaimthat,forallM,µ(M)≥maxρ(M∆)≥0.99µ(M),withconfidencelevel≥1−10−6.k1≤k≤SnResult2(Exponentialnumberofsamples):Thereisauniversalcon-2.01stantCsuchthat,generatingS=Cenrandomsamplesisenoughtonclaimthat,forallM,µ(M)≥maxρ(M∆)≥0.99µ(M),withconfidencelevel≥1−10−6.k1≤k≤SnAlternatively,onecanfixaconfidencelevel,say1−10−6,andstudytherelationshipbetweenthesamplesizeS,andthebestconservatismbound,g(S),thatcanbeguaranteedwiththisconfidencelevel.Againnotmuchisknownabouthowfast/slowthebestconservatismboundg(S)convergesto1asthesamplesizeSgoestoinfinity.BIBLIOGRAPHY[1]A.PackardandJ.C.Doyle,“Thecomplexstructuredsingularvalue,”Automatica,29,pp.71–109,1993.[2]Papadimitriou,C.H.,ComputationalComplexity,Addison-Wesley,1994.[3]G.C.Calafiore,F.Dabbene,andR.Tempo,“Randomizedalgorithmsforprobabilisticrobustnesswithrealandcomplexstructureduncertainty,”IEEETransactionsonAutomaticControl,vol.45,pp.2218–2235,2000.[4]O.Toker,“ConservatismofRandomizedStructuredSingularValue,”acceptedforpublicationinIEEETransactionsonAutomaticControl.[5]O.Toker,andB.Jager,“ConservatismoftheStandardUpperBoundTest:Issup(µ/µ)finite?Isitboundedby2?,”In:OpenProblemsinMathematicalSystemsandControlTheory,V.Blondel,etal.eds.,chapter43,pp.225–228.[6]S.Treil,“Thegapbetweencomplexstructuredsingularvalueµanditsupperboundisinfinite,”ForthcomingIn:IEEETransactionsonAutomaticControl. 334PROBLEM10.7[7]M.Vidyasagar,ATheoryofLearningandGeneralization,Springer-Verlag,London,1997.[8]M.VidyasagarandV.Blondel,“ProbabilisticsolutionstosomeNP-hardmatrixproblems,”Automatica,vol.37,pp.1397–1405,2001.