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linear Wave Equations and Klein-Gordon Equations

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linear Wave Equations and Klein-Gordon Equations

    linear Wave Equations and Klein-Gordon

    Equations

    (n,Ann.Math.

    31B(1),2010,35-58

    DOI:10.1007/s114010080426x

    ChineseAnnalsof

    Mathernatics,SeriesB

    ?TheEditorialOfficeofCAMand

    SpringerVerlagBerlinHeidelberg2010

    GlobalExactBoundaryControllabilityfor

    CubicSemilinearWaveEquations

    andKleinGordonEquations

    YiZHOUWeiXuZhenLEI

    AbstractTheauthorsprovetheglobalexactboundarycontroIabiIitvf0rthecubicsemi

    linearwaveequationinthreespacedimensions,subjecttoDirichlet,Neumann,orany otherkindofboundarycontrolswhichresultinthewellposednessofthecorresponding

    initialboundaryvalueproblem.Theexponentialdecayofenergyisfirstestablishedforthe cubicsemilinearwaveequationwithsomeboundaryconditionbythemultipliermethod, whichreducestheglobalexactboundarycontroUabmtvproblemtoalocalone.Theproof iscarriedoutinlinewithf2,15f.Thenaconstructivemethodthathasbeendeveloped inI13lisusedtostudythelocalproblem.Especiallywhentheregionisstarcomplemented, itisobtainedthatthecontrolfunctiononlyneedtobeappliedonarelativelyopensubset oftheboundary.ForthecubicKleinGordonequation.similarresultsoftheglobalexact boundarycontrollabilityareprovedbysuchanidea.

    KeywordsGlobalexactboundarycontrollability,Cubicsemilinearwaveequations,

    Theexponentialdecay,Starshaped,Starcomplemented,CubicKlein

    Gordonequations

2000MRSubjectClassification35B37,35L05

    1Introduction

    Inthispaper,wecontinuetostudytheexactboundarycontrollabilityproblemfornonlinear waveequations.The1ocalexactboundarycontrollabilityforsemi

    linearandquasi.1inearwave

    equationswasbuiltin[1a].Theaimofthispaperistostudytheglobalexactboundary controllabilityforsemilinearwaveequations.Herebylocalwemeanthattheinitialand finaldataaresmalljnsomesuitableSobolevspaces,whilebyglobalwemeanthereisno smallnessrestrictionontheinitialandfinaldata.Wefirstprovethedissipativeenergyestimate forthesemilinearwaveequation,whichreducestheproblemoftheglobalexactboundary controllabilitytoalocalone.ThenweapplytheconstructivemethodintroducedinI13lto establishthelocalexactboundarycontrollability.

    Tobestillustrateouridea.wetakethecubicsemilinearwaveequationinthreespace

    ManuscriptreceivedOctober29,2008.RevisedFebruary21,2009.PublishedonlineDecember11,2009.

    KeyLaboratoryofMathematicsforNonlinearSciences,SchoolofMathematicalSciencesFudanUniversity,

    Shanghai200433,China.E-mail:yizhou~fudan.ac.ca

    SchoolofMathematica1Sciences.FudanUniversity,Shanghai200433.C:hina. Email:xuweipartial@hotmail.COITIleizhn@yahoo.com

    ProiectsupportedbytheNationalNaturalScienceFoundationofChina(No.10728101),the973Projectof

    theMinistryofScienceandTechnologyofChina,theDoctoralProgramFoundationoftheMinistryofEd

    ucationofChina,the"111"ProiectandthePostdoctoralScienceFoundationofChina(No.20070410160).

    dimensionsforexample

    where

    R3.

E]u+u.=0,0<t<T,x?

    Y.Zhou,W,XuandZ.Lei

    

    ?,?=3,Aisapositiveconstantandflisaboundedopensubsetof

    Considertheinitialstate

    andthefinalstate

    u(0,):fo(),ut(0,)=^(z),x?Q

    u(T,)=go(x),u(,z)=91(z),?Q

    Lets2,fo,90?Hs(Q)and11,gl?H(Q),whereH.()isthestandardSobolevspace

    oforders.For0tTandx?Q,weimposeanyofthefollowingboundaryconditionson equation(1.1):

    h(t,)ofDirichlettype

    Ou

    :

    (t,z).fNeumanntype,

    ?).fthethirdt.ype.

    +.fthedissipatiVetype

    wherebandaregivenpositiveconstants.Herewecanuseanyotherkindofboundary conditionaslongasthecorrespondinginitialboundaryvalueproblemiswellposed.

    Thentheproblemoftheexactboundarycontrollabilityfortheequation(1.1)isstatedas follows:GivenT>0,isitpossibletofindacorrespondingboundarycontrolh(t,x)drivingthe

    equation(1.1)withtheinitialstate(,O,f1)tothedesiredstate(go,g1)attime? Inthispaper,wewillestablishtheglobalexactboundarycontrollabilityforthecubicsemi- linearwaveequation.

    Preciselyweprovethefollowingtheorem.

    Theorem1.1Supposefo,go?H.(Q),fl,gl?H(2),s2.Thereexistsas

    cientlylargepositiveconstantTodependingonlyontheSobolevnormofthedatalIfollHs(Q), Illl1日一(n),lIg0l1(Q),Igll1.

    (Q)and0boundarycontrolfunctionhsuchthatthecubic

    semi.1inearwaveequation(1.1)withtheinitiafstate(1.2)andoneoftheboundaryconditions (1.4)admitsauniquesolutiononthedomain(0,T)×Qwhichuee8thedesiredstate(1?3),

    providedthatthetimeT>To.

    B.Dehman,G.LebeauandE.Zuazuaobtainedsimilarresultsontheexactinternalcon

    trollabilityprobleminf21.NotethatthecontrollabilitytimeTdependsontheinitialandfina1 data.WhetherTmaybeindependentoftheinitialandfinaldataiscertainlyoneofthemain openproblemsinthecontextofcontrollabilityofnonlinearPDE.

    Thereareanextremelylargenumberofpublicationsontheexactboundarycontrollability problem.Someclassicalreferencescanbefoundinf7,91.Forsemi

    linearhyperbolicequations,

    E.Zuazuaf14]introducedavariantoftheHilbertuniquenessmethodtostudythecontrol problemforsemilinearwaveequationsYAy+f(Y1:0innspacedimensionswithboth

    GlobalControllabilityforWaveandKleinGordonEquations37

    DirichletandNeumannboundaryconditions,wheref?

    W1OCoo()isalocallyLipschitzfunction.

    Theauthorgottheexactcontrollabi1itywhenthenonlinearityisasymptoticallylinearandlocal

    controllabilityresultsforalargeclassofnonlinearitiesundersomenaturalgrowthassumptions

    onthenonlinearitiesin[14].E.Zuazuaalsostudiedtheexactcontrollabilityforsemilinear

    waveequationstYz+f(Y)=hwithDirichletboundaryconditioninonespacedimension in[16],wheretheauthorestablishedtheexactcontrollabilityin(2)×L(Q)withcontrols

    h?L(2×(0,))supportedinanyopenandnonemptysubsetofQif0as

    s}..byHUMandafixedpointtechnique.whichcanbealsoappliedtothewaveequation withNeumanntypeboundarycondition.Itwasalsoshownin[16thatiffbehaveslike

    

    8logP(1+Is1)withP>2aslsI_?,thesystemisnotexactlycontrollableinanytimeT.

    In4],x.Fu,J.YongandX.Zhangobtainedaglobalexactcontrollabilityresultforaclassof multidimensionalsemi

    -linearhyperbolicequationswithsuper?-linearnonlinearityandvariable

    coefficients,viaanobservabilityestimateforthelinearhyperbolicequationwithanunbounded

    potentia1.whichisobtainedbyapointwiseestimateandaglobalCarlemanestimateforthe

    hyperbolicdifferentialoperatorsandanalysisontheregularityoftheoptimalsolutiontoan auxiliaryoptimalcontrolproblem.I.LasieckaandR.Triggiani[6]studiedthe(globa1)exact controllabilityforthesemilinearwaveequationUtt一?=.(),wherefisanabsolutely

    continuousfunctionwithfirstderivativefa.e.beinguniformlyboundedIfICa_v_1eand obtainedtheexactcontrollabilityresultsonanystatespaceH=H3(Q)×H-1(Q)usingthe

    controlspaceH3([0,,2(aQ)),071,?1,aswellasthespecialcase7=1in[6].

    Thenletusshowourstrategyofestablishingtheglobalexactcontrollabilityforthecubic semilinearwaveequation.

    WithoutlOSSofgenerality,weassumeQCCBl,whereBlistheunitballcenteredatthe originwiththeboundaryaB1.Wecanalwaysextendthefunctionsf0,fl,go,gltof0,fl,g)o,g)

    l

    Sl1hthalt

    supp(f0,fl,,)CCB1

    II.1llHs(B)Cs1Ifl1lHs)(2),

    lIllHs(B)CsIIgillHs)(Q)

    forsomeconstant>0andalls0(forexample,see[3]).Inwhat~llows,wewilluse sameextensionforseveraltimesandalwaysdenotetheextensionoperatorby){_1.

    Weshallconstructasolutionof(1.1)withinitialdataf0,flandfinaldatag)o,g)lon

    domain(0,T)×B1forsufficientlylargeT.ThentherestrictionofthesolutiontoaQyi the

    the

    elds

    thedesiredboundarycontrolfunction.Tothisend,wefirstevolvetheequation(1.1)withthe initialdataf0,/iandtheboundarycondition+4-=0onthedomain[0,xBland

    provetheglobalexistenceandanexponentialdecayofenergyestimateforthesolutionofthis problem.Consequently,u(T1,x)andt(,x)willbesufficientlysmallinappropriateSobolev spacesifislargeenough.Similarly,weevolveequation(1.1)withthefinaldata,and

backwardwiththeboundarycondition架一雨Ou+u=0onthedomain[TT2,T]×

    B1.Then

    u(TT2,x)andt(T,)willbesmallenoughinsuitableSobolevspacesprovidedthat islargeenough.BytakingT>T1+,wefindthatitsufficestoconstructasolutiononthe domain,TT2]×Qwithinitialcondition"(,x),ztt(T1,x)andfinalconditionu(T)T2,z),

    t(,z).Notingthat(T1,),t(T1,x)andu(TT2,),ut(TT2,x)aresmall,wehave

    d

    s8HH

    < <

    ssHH

    )

    y.Zhou.w.XuandZ.Lei

    reducedtheglobalexactcontrollabilityproblemtoalocalone,whichwasstudiedbyZhouand Lei[1a].

    Nowwetakealookatthecontrolproblemofequation(1.1)inthestarcomplemented

    region.Forthisproblem,weattempttoobtainacontrolfunctiononlyappliedonarelatively opensubsetoftheboundary.D.Russell[10JandG.Chen1studiedsuchproblemforthe

    1inearwaveequationwiththehelpofthedecayestimateforthesolutionofthewaveequation onanexteriordomainduetoC.S.Morawetz.

    Firstwegivesomeusefuldefinitions.

    Definition1.1(see3])LetQbeanopensubsetof..Wesay2isstarshapedifthere

    existsapointx?suchthatforallx?thelinesegment{?l?=(1t)x+tx,V0

    t1)liesinQ.Wealsocallitstarshapedwithrespecttox.

    Definition1.2(see[10])Thepair(2,F)isstar-complementedifFisarelativelyopen subset0aQandthereis0pointx?-0.withthepropertythateachpointx?OftFcanbe

    connectedtoxbyalinesegmentwhich,exceptrxitself,liesentirelyoutside2.

    HereQmeanstheclosureof2andft.meansthecomplementof.

    Assumethatthereexistsastarcomplementedpair(Ft,F)fortheregionQandaisa

    regular.piecewise..manifoldofdimensiontwo.Whattheword"regular"meanswillbe

giveninSection4.Letrl:a2一工1.

    Nowweintroducetheboundarycontrolconditions,anyofwhichweimposeontheequation

    (1.1):.

    u(t,x)=0,0tT,?Pl(1.7)

    and

    ofDirichlettype,

    ofNeumanntype

    ofthethirdtype

    )ofthedissipativetype

    wherecandaregivenpositiveconstants,for0tTandx?F.

    Brieflyweestablishthefollowingtheoremforthecontrolproblemofthecubicsemilinear

    waveequation(1.1)inthestarcomplementedregion.

    Theorem1.2Assumethattheboundedregion2isstar-complemented.Foranyfogo?

    H.(Q),fl,gl?日一(Q)s2withthepropertythatfo:go=0onF1forIIs1

    and11=.91=0onF1forIls2,thereexistsasufficientlylargeconstantTo>0 dependingonlyontheSobolevnor'lTtofthedatall,OllHs(2),JIf1IIHs

    (2),Ill9ollHs(),lIgllIHs(Q)

    andaboundarycontrolhonlyappliedonFsuchthatthecubicsemi?linearwaveequation(1.1)

    withtheinitialdata(1.2),theboundarycondition(1.7)andoneoftheconditions(1.8)admitsa

    uniquesolutiononthedomain(0,T)×

    Qsatisfyingthedesireddata(1.3),providedthatT>To. Remark1.1ThesolutioninTheorem1.1or1.2belongs J=0

    theboundarycontrolobtainedbythewayofourconstructionisnotunique d

    n

    a

    ,l,

    ,l,Q

    ,,

J

    

    SH

    O

    ,,

    ,

    n

    o

    t

    GlobalControllabilityforWaveandKleinGordonEquations39

    IheproofofTheorem1.2issimilartothatofTheorem1.1toalargeextent.howeverit1S morecomplicated.

    Therestofthispaperisorganizedasfollows.InSection2,weprovetheglobalexistence andanexponentiallydissipativeenergyestimateforthesolutionoftheequation(1.1)withthe

    initialdata,/1andboundarycondition0U+OU+=0onthedomain[0,+..)×皿1.Then

    weproveTheorem1.1inSection3andTheorem1.2inSection4respectively.InSection5.we provesimilarresultsoftheglobalexactboundarycontrollabilityforthecubicKleinGordon

    equation.

    2TheGlobalExistenceandExponentiallyDissipativeEnergy EstimatesfortheCubicSemi-linearWaveEquation

    Inthissection,wewillstudytheglobalexistenceofthestrongsolutiontothemixedinitial

    boundaryvalueproblem:

    K]u+r".=0,t0,?Bl,

    t=0:u:fo,t=fl,?B1,

    ++_o,?

    Theproofreliesonalocalexistencetheoryandanexponentiallydissipativeapriorienergy

    estimate.

    WefirstestablishthefollowingIOCalexistenceforsystem(2.1). Lemma2.1AssumeS2,f0?H(1),fl?H)(B1)andsupp(fo,f1)cc151.There

    existpositiveconstantsTandMdependingonlyonlIfolIHs(1)andll,ls(B1)suchthat systemf2.1)admitsauniquesolutionu(t,)onthedomain[0,T】×B1satisfying

    81

    og~(t,?)lI(B)+?I10~u(t,')11~/(B)M.,V0t

    /=0

    (2.2)

    ProofInwhatfollows,wewilluseCtodenoteagenericpositiveconstantwhichmayvary

    fromlinetoline(unlessotherwisestated).

    Theconditionsupp(f0,f1)CCB1impliessupp(O[u(O,'))CCB1foranyl0.Theneven

    ifweapply(0ls1)tosystem(2.1),thecompatibleconditionstillholds. Nowweprovethelocalexistenceofsolutiontosystem(2.1)byenergyestimatesandthe

    standardcontractionmappingtheorem.

    ForanyV?D5,where

    J[)={:[0,T】×B1I"(0,?)=f0,vt(0,')=,1,

    .

    sup

    (1l0~(?,')llz(B)+?/=0

    lI6(t,.)ll(B)).)

    wedefineamapn:Vwheresatisfiesthemixedinitialboundaryvalueproblem Du+Av0=0,t0,X?B1,

    t=0::f0Ut=fl,?B1,

    ++^o.tz?

    (2.3)

    YZhou.w.XuandZ.Lei

    Applying(0ls1)tosystem(2.3)andtakingtheL.innerproductoftheresulting

    equationwith"u,

    weobtaintheenergyestimate:

    s-1

    2

    1

d

    d

    t,..Oz+慨皿)+fJB)J.(aB))

    I+s-1)

    .(2.4)

    whereweusedtheboundaryconditions+"++f"=0forall0ls1

    Consequently,weget

    ~*

    2

    l

    dt

    d

    ,..

    Ol+1112.()+JIll()+_llI(aB))AEjj0~(.)+1?llL().(2.5) ?=Uf=0

    ByH61derinequality,foranyll+12+l3:l,wehave "..+"IlL(B)lllie)ll.IIe(B)ll.cle(B)ll+lJ(B) BySobolevembeddingtheoremthatH(B1)L.(B1)in.andtheinequality(2.5),wefind

    Let

    M=4(1lO:(0,?))+s1?

    ZO

    +?S--1))cMs(2.6)

    andintegratetheinequality(2.6)withrespecttotimetover[0,to].Hencethereexistsapositive

    constantdependingonlyonMsuchthatforany0to, s1

    o2~(t.,-)(+?IIo~(to

    Z:1

    Promtheequalityu(to

    constant<such

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