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Exponential

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ExponentialExpone

    Exponential

Vo1.16No.3JournalofSouthwest~aotongUnwe~ity(EnglishEdition)Ju1.2008

    ;ArticleID:1005-2429(2008)03-0295-03 ;ExponentialStabilizationforCoSemigroups

    ;UnderCompactPerturbation

    ;LIULixin(刘立新)’

    ;DepartmentofMathematwsandPhysics,PanzhihuaUniversity,Panzhihua617000,Chin

    a

    ;Abstract

    ;Itisprovedthatasystemundercompactperturbationcannotbeuniformlyexponentiallysta

    bleforallisometricc0semigroupor

    ;ac0

    groupwithpolynomialgrowthfornegativetimeinaBanachspace.Theresultsextendandimp

    rovethecorrespondingresultsof

    ;previousliterature.

    ;KeywordsCompactperturbation;Exponentialstabilization;Isometricc0

    semigroup;Polynomialgrowthfornegativetime ;Introduction

    ;Thecontrollabilityandstabilizationoftheinfi

    ;nitedimensionalsystemsunderfeedbackcontrolare ;veryimportantandhavebeeninvestigatedextensively ;bymanyresearchers~.

    ;Letusconsidertheinfinite

    ;dimensionallinearcontrolsystem

    ;dx

    ;=

    ;Ax(f)+Bu(t),(1)

    ;where(t)?X,/,t(t)?U,thestatespaceXandthe

    ;controlspaceUareBanachspaces,(A,D(A))isthe ;infinitesimalgeneratorofaCosemigroup{S(t)}

    ;onX,D(A)denotesthedomainofA,andBisa ;compactoperatorfromUintoX.Fortheconceptand ;moredetailsaboutCosemigroup,onecanreferto

    ;Refs.[1O,l1].

    ;Zhueta1.[provedthatforanisometricCo

    ;semigroup{S(t)}withgenerator(A,D(A))and ;Received2007-06

    ;FotmdationitemProjectofSichuanProvincialScienceand ;TechnologyDepartment(No.2007J13-006) ;BiographyLixin(1967),associateprofessor.}Iis

    ;researchinterestisinappliedfunctionalanalysis ;%Correspondingauthor.Te1.:+86-8123371003:Fax:+86

    ;812-33710o0;Email:pzhliulixin@yahoo.tom.ca ;acompactoperatorB,theCosemigroup{SB(t)}

    ;generatedby(A+B,D(A))cannotbeuniformly ;exponentiallystableinaninfinitedimensionalHilbert

    ;space.TheyalsoprovedthatforaCogroup

    ;{S(t)}withboundednessfornegativelimeanda ;compactlinearoperatorB,theCosemigroup

    ;{S8(t)}f?0generatedby(A+B,D(A))cannotbe

    ;uniformlyexponentiallystableinaninfinite--dimen-- ;sionalreflexiveBanachspace.

    ;Inthispaper,westudythenonuniformlyexpo

    ;nenfialstabilizationforanisometricCosemigroupor

    ;aCogroupwithpolynomialgrowthfornegativetime ;undercompactperturbationofitsgeneratorandgener- ;alizethecorrespondingnonuniformlyexponentialsta- ;bilizationresultsintheBanachspace.Somevariables ;canbereferredtoinRef.[12].

    ;1MainResults

    ;Theorem1LetXbeaninfinitedimensional

    ;reflexiveBanachspace,{S(t)}anisometricCo

    ;semigroupwiththegenerator(A,D(A))onX,and ;BacompactoperatoronX.Then,theCosemigroup

    ;{S8(t)},?,generatedby(A+B,D(A)),cannot

    ;beuniformlyexponentiallystable.

    ;ProofLetW={?X:{txt{=1}beaunit

    ;

    ;296JournalofSouthwestJiaotongUniversity(EnglishEdition)

    ;sphereinX.SinceXisaninfinitedimensional

    ;Banachspace,thereexistsYj?W,J=1,2,…,such

    ;thatIlY—YjII?1foralli#j.

    ;FromthecompactnessofoperatorB,wecon

    ;cludethatBiscompact,R(B),whichistherange ;ofoperatorB,isseparable,andconsequently, ;{():?X}isseparableforeach?0.

    ;Hence,

    ;V=span{S():?X,I>0}

    ;isalsoseparable,orequivalently,thereexistsa ;countablesubset{y,v2,…,Vm,…}whichisdensein

    ;V.Fromtheboundednessof{Yj}1andthe

    ;diagonalmethod,itfollowsthatthereisasubse

    ;quence{Yj}:1of{Yj}1suchthat{Vm()}1is

    ;convergentforanyVm.Hence,fromIlYjIl=1andthe

;densenessof{y=k,y,…,y,…}inV,itfollows

    ;that{()}1isconvergentforany?V.By

    ;thereflexivityofX,wecandeducethatthereexists ;?XsuchthatforeachI>0,

    (,z..).(2) ;IIxll=1,IIBS(T)xll

    ;Supposethat{(t)}f?oisuniformlyexponen—

    ;tiallystable.Byformula(2),wecandeducethat ;thereexistsasubsequence{}:1of{}1such

    ;mat

    ;f1l[SB(T)BS(m一丁)d=0.}?

    ;FromtheCo—semigrouptheory[.’?.wehave

    ;(f)=S(t)+f(t)BS(t—r)xdr,30’

    ;f?0.?X.

    ;Then

    ;

    ;limI}()11?im(}}()II+—?”‘—?

    ;J0IIS(m)BS(m一丁)Ild~),

    ;whichisacontradictionwiththeisometryof ;{S(t)}.

    ;ThiscompletestheproofofTheorem1. ;Fromthedualtheoryoflinearoperators,inthe ;nonreflexiveBanachspaces,theadjoint{S(t)} ;ofCosernigroup{S(t)}maynotbestronglycon

    ;tinuous,andtheadjointAoftheinfinitesimalgenera

    ;t0rAmaynotbedenselydefinedinX.Fortunately. ;wecouldreplacethedualXandtheadjointsemig

    ;roup{S(t)}withthePhillipsdualandthe ;Phillipsadjointsernigroup{S.(t)},respectively. ;Moreprecisely,wehavethefollowingresult. ;Theorem2Eu]LetAbetheinfinitesimalgen

    ;eratorofCosemigroup{S(t)}with

    ;llS(t)II?,f?0,(3)

    ;whereMisaconstant,andlet=D(A)cXbe ;thePhillipsdualofX.Thentherestriction ;{S.(t)}of{S(t)}isastronglycontinuous ;semigroupinwiththeinfinitesimalgeneratorA.de

    ;finedby

    ;A.X=A,D(A.)={?D(A):A?}

    ;and

    ;=

    ;{?X:(t)isstronglycontinuousfor

    ;f?0}.

    ;Furthermore,

    ;.

;I(,)I?

    ;IIxlI?sup..

    ;I(,)I.

    ;xe.IfxIl=1

    ;Theorem3LetlinearoperatorBbecompact

     ;and(A,D(A))betheinfinitesimalgeneratorofCo

    ;group{S(t)}fRinaBanachspaceX.Then ;SB(t)BS(t)isnorlTlcontinuousfort>0. ;ProofLetUo>11andWoI>0besuchnumbers ;thatIIS(t)Il?MoeW0,f?0.Letf?0befixed.Itfol—

    ;lOWSfromtheboundednessof

    ;{S(-(t+h)):IIxlI?1,IhI?f}

    ;andthecompactnessofBthat

    ;{BS((t+h)):IIxlI?1,IhI?f}

    ;issequentiallycompactinX.Thus,forany8>0,

    ;thereexists{1,x2,…,}CXsuchthat

    ;(e-28)

    ;{BS(-(t+h)):IIxll?1,IhI?f}.

    ;Thatis,forany?XwithIIxlI?1andIhI?f,there

    ;existsk?{1,2,…,n}suchthat

    (+Jfz))x-x~If<1e8, ;[tBS(

    ;whichyields

    ;ll(S(t+h)-S(t))(BS(_(t+h)))ll?

    ;ll(SB(t+h)ll+Il((t)ll×

    ;

    ;LIULixin/ExponentialStabilizationforCoSemigroupsUnderCompactPerturbatio

    n

    ;fIBS((t+.1z))—II?.

    ;Because{SB(t)}isstronglycontinuous, ;thereexists1E(0,t]suchthat

    ;lIs.(t+.1z)SB(f)II<,IhI<,

    ;k=1,2,…,n.(5)

    ;Itfollowsfromformulas(4)and(5)that ;lIs.(t+h)-S.(t)BS(-(t+h)xll<8, ;297

    ;lhf<,lIxlf?1.

    ;Ontheotherhand,Biscompact,andthePhil

    ;lipsadjointgroups{(t)}Rand{(t)}fRare ;stronglycontinuous.Similarlyasabove,thereexists

    ;2E(0,t]suchthat

    ;.(_(f+.1z)~-so(f))(f)II<,

    ;lhl<,xE,lIxll?1.

    ;Thus,forxEXwithlIxll=1,wehave ;lIS.(f)B(((t+fz))-s(-f))xll?

;P..I(X,S.(t)B(S(--(t”1--h))--(_t))X)I?

    ;EXe,lII1

    ;Msup..I((S.(--(t-I-h))__(t))B;(_(t-I-.1z)),x)I<8, ;Exe,?lI1

    ;andhence,

    ;lIs.(t-I-h)BS(__(t+h))S.(t)BS(-t))xll?

    ;ll(t-I-h)S.(t))BS(-(t-I-h))xll+

    S(-t))xll<28. ;llSB(t)B(S((t-I-h))x

    ;Therefore,S(t)(t)isnormcontinuousfor

    ;t>O.ThisendstheproofofTheorem3. ;Definition1TheCogroup{S(f)}fRhasa

    ;polynomialgrowthfornegativetimeifthereexist ;constantsM?1andvI>0suchthat

    ;lIs(t)ll?(1+ltl),t<O.

    ;UsingTheorems2and3,wecanprovethefol

    ;lowingtheoreminthesamewayasinRef.[1]. ;Theorem4LetXbeaninfinitedimensional

    groupwitha ;Banachspace,and{S(t)}fRbeCo

    ;polynomialgrowthfornegativetime.Thenforany ;compactlinearoperatorBinX,theCosemigroup

    ;{SB(t)}generatedby(A-I-B,D(A))isnotuni- ;formlyexponentiallystable.

    ;[1]

    ;[2]

    ;[3]

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