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1、ApplMathOptim(2010)61:191233DOI10.1007/s00245-009-9088-7BoundaryControllabilityfortheQuasilinearWaveEquationPeng-FeiYaoPublishedonline:15September2009©SpringerScience+BusinessMedia,LLC2009AbstractWestudytheboundaryexactcontrollabilityforthequasilinearwaveequa-tion
2、inhighdimensions.Ourmaintoolisthegeometricanalysis.Wederivetheex-istenceoflongtimesolutionsnearanequilibrium,provethelocallyexactcontrolla-bilityaroundtheequilibriumundersomecheckablegeometricalconditions.Wethenestablishthegloballyexactcontrollabilityinsuchawaytha
3、tthestateofthequasi-linearwaveequationmovesfromanequilibriuminonelocationtoanequilibriuminanotherlocationundersomegeometricalconditions.TheDirichletactionandtheNeumannactionarestudied,respectively.OurresultsshowthatexactcontrollabilityisgeometricalcharactersofaRie
4、mannianmetric,givenbythecoefficientsandequi-libriaofthequasilinearwaveequation.Acriterionofexactcontrollabilityisgiven,whichbasedonthesectionalcurvatureoftheRiemannmetric.Someexamplesarepresentedtoverifytheglobalexactcontrollability.KeywordsQuasi-linearwaveequation
5、·Exactcontrollability·SectionalcurvatureAMSSubjectClassification49B·49E·35B35·35L65·35L70·38J451IntroductionandtheMainResultsLet⊂Rnbeanopen,boundedsetwiththesmoothboundary.Supposethatconsistsoftwodisjointparts,0and1with0∩1=∅.LetT>0begiven.WeCommunicatedbyIre
6、naLasiecka.ThisworkissupportedbytheNNSFofChina,grantsno.60225003,no.60334040,no.60821091,no.60774025,andno.10831007andKJCX3-SYW-S01.P.-F.Yao()KeyLaboratoryofControlandSystems,InstituteofSystemsScience,AcademyofMathematicsandSystemsScience,ChineseAcademyofSciences
7、,Beijing100190,PeoplesRepublicofChinae-mail:pfyao@iss.ac.cn192ApplMathOptim(2010)61:191233consideracontrollabilityproblem⎧n⎪⎨u¨=ij=1aij(x,∇u)uxixj+b(x,∇u)on(0,T)×,u=won(0,T)×1,(1.1)⎪⎩u=ϕon(0,T)×0,u(0)=u0,u(˙0)=u1,whereaij(x,y),b(x,y)aresmoothfunctionson×Rnsuc
8、hthatA(x,y)=(an,(1.2)ij(x,y))>0,∀(x,y)∈×Rb(x,0)=0,∀x∈,(1.3)andwisanequilibrium,definedby(1.5)below.Letu0,u1,uˆ0,anduˆ1begivenfunctionsonandT>0begiven.
