(J.P.Puel)Carleman for wave equations

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1、GlobalCarlemaninequalitiesforthewaveequationsandapplicationstocontrollabilityandinverseproblemsJean-PierrePuel11J.-P.Puel:(jppuel@cmapx.polytechnique.fr)LaboratoiredeMathematiquesAp-pliquees,UniversitedeVersaillesSaint-Quentin,45avenuedesEtatsUnis,78035VersaillesCh

2、apter1PresentationoftheproblemsExactcontrollabilityproblemsforwaveequationshavebeenextensivelystudiedinthelastfteenyears(see[13]forexample)andthisproblemcorrespondstomanyapplicationslikehowtostopvibrationsbyaboundaryactionforexampleandmanyothers.Wewillconsiderherethe

3、caseofwaveequationswithboundedpotentialsandpresentadirectapproachtothisproblem.Anotherproblemwhichwillbecon-sideredhereistoretrieveanunknownpotentialinawaveequationfromboundarymeasurements.Thisisnowaninverseproblemandagainitcorrespondstoimpor-tantapplications,evenifam

4、oreinterestingproblemwouldbetoretrieveadiusioncoecientorelasticitycoecients.Thesituationconsideredhereconstitutesonestepinthisdirection.Theresultswepresentherearenotnewbuttheyaregiveninauniedwayandtheattemptistomakethemrathersimpleandcomprehensibledespiteofsomelon

5、gcomputations.ThelinkbetweenthetwoquestionsappearsclearlyfromLions'HilbertUnique-nessMethodforexample.Hereweobtainresultsonthetwoproblemsusingthesamemathematicalmachinery.Wewillpresentthismachinerywhichistheverytechnicalbutpowerfulmath-ematicaltoolgivenbyglobalCarlema

6、nestimatesinthepresentcontextofwaveequationswithpotentials.Wewillgiveacompleteproofoftheseestimatesforstrongsolutionsofwaveequations,followingthemethoddeveloppedin[8].Thenwegiveapplicationstothetwoabovementionnedproblems.Westartwithwellknownpreliminariesonwaveequation

7、swhichwewillneedallalongthisstudy.ThistextcorrespondstolectureswhichhavebeengivenduringtheadvancedschoolControlofSolidsandStructures:MathematicalModellingandEngineeringApplications"attheC.I.S.M.inUdineinJune2004andtoapartofD.E.A.courseattheUniversityPierreetMarieCuri

8、ein2003-2004.11.1BasicpropertiesforthewaveequationInthewholetext,isabounded(regularenough)opensubsetofIRN,denotesitsboundar