微分学数学专业英语论文双语含中文版.docx
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微分学数学专业英语论文双语含中文版.docx
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微分学数学专业英语论文双语含中文版
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微分学数学专业英语论文双语(含中文版)
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DifferentialCalculus
/D/x*p2R6r$]$NewtonandLeibniz,quiteindependentlyofoneanother,werelargelyresponsiblefordevelopingtheideasofintegralcalculustothepointwherehithertoinsurmountableproblemscouldbesolvedbymoreorlessroutinemethods.Thesuccessfulaccomplishmentsofthesemenwereprimarilyduetothefactthattheywereabletofusetogethertheintegralcalculuswiththesecondmainbranchofcalculus,differentialcalculus.
Inthisarticle,wegivesufficientconditionsforcontrollabilityofsomepartialneutralfunctionaldifferentialequationswithinfinitedelay.WesupposethatthelinearpartisnotnecessarilydenselydefinedbutsatisfiestheresolventestimatesoftheHille-Yosidatheorem.Theresultsareobtainedusingtheintegratedsemigroupstheory.Anapplicationisgiventoillustrateourabstractresult.
KeywordsControllability;integratedsemigroup;integralsolution;infinitydelay
1Introduction
Inthisarticle,weestablisharesultaboutcontrollabilitytothefollowingclassofpartialneutralfunctionaldifferentialequationswithinfinitedelay:
(1)
wherethestatevariabletakesvaluesinaBanachspaceandthecontrolisgivenin,theBanachspaceofadmissiblecontrolfunctionswithUaBanachspace.CisaboundedlinearoperatorfromUintoE,A:
D(A)⊆E→EisalinearoperatoronE,Bisthephasespaceoffunctionsmapping(−∞,0]intoE,whichwillbespecifiedlater,DisaboundedlinearoperatorfromBintoEdefinedby
isaboundedlinearoperatorfromBintoEandforeachx:
(−∞,T]→E,T>0,andt∈[0,T],xtrepresents,asusual,themappingfrom(−∞,0]intoEdefinedby
FisanE-valuednonlinearcontinuousmappingon.
TheproblemofcontrollabilityoflinearandnonlinearsystemsrepresentedbyODEinfinitdimensionalspacewasextensivelystudied.ManyauthorsextendedthecontrollabilityconcepttoinfinitedimensionalsystemsinBanachspacewithunboundedoperators.Uptonow,therearealotofworksonthistopic,see,forexample,[4,7,10,21].Therearemanysystemsthatcanbewrittenasabstractneutralevolutionequationswithinfinitedelaytostudy[23].Inrecentyears,thetheoryofneutralfunctionaldifferentialequationswithinfinitedelayininfinite
dimensionwasdevelopedanditisstillafieldofresearch(see,forinstance,[2,9,14,15]andthereferencestherein).Meanwhile,thecontrollabilityproblemofsuchsystemswasalsodiscussedbymanymathematicians,see,forexample,[5,8].TheobjectiveofthisarticleistodiscussthecontrollabilityforEq.
(1),wherethelinearpartissupposedtobenon-denselydefinedbutsatisfiestheresolventestimatesoftheHille-Yosidatheorem.Weshallassumeconditionsthatassureglobalexistenceandgivethesufficientconditionsforcontrollabilityofsomepartialneutralfunctionaldifferentialequationswithinfinitedelay.TheresultsareobtainedusingtheintegratedsemigroupstheoryandBanachfixedpointtheorem.Besides,wemakeuseofthenotionofintegralsolutionandwedonotusetheanalyticsemigroupstheory.
TreatingequationswithinfinitedelaysuchasEq.
(1),weneedtointroducethephasespaceB.Toavoidrepetitionsandunderstandtheinterestingpropertiesofthephasespace,supposethatisa(semi)normedabstractlinearspaceoffunctionsmapping(−∞,0]intoE,andsatisfiesthefollowingfundamentalaxiomsthatwerefirstintroducedin[13]andwidelydiscussedin[16].
ThereexistapositiveconstantHandfunctionsK(.),M(.):
withKcontinuousandMlocallybounded,suchthat,foranyand,ifx:
(−∞,σ+a]→E,andiscontinuouson[σ,σ+a],then,foreverytin[σ,σ+a],thefollowingconditionshold:
(i),
(ii),whichisequivalenttoorevery
(iii)
(A)Forthefunctionin(A),t→xtisaB-valuedcontinuousfunctionfortin[σ,σ+a].
ThespaceBiscomplete.
Throughoutthisarticle,wealsoassumethattheoperatorAsatisfiestheHille-Yosidacondition:
(H1)Thereexistand,suchthatand
(2)
LetA0bethepartofoperatorAindefinedby
Itiswellknownthatandtheoperatorgeneratesastronglycontinuoussemigroupon.
Recallthat[19]foralland,onehasand.
WealsorecallthatcoincidesonwiththederivativeofthelocallyLipschitzintegratedsemigroupgeneratedbyAonE,whichis,accordingto[3,17,18],afamilyofboundedlinearoperatorsonE,thatsatisfies
S(0)=0,
foranyy∈E,t→S(t)yisstronglycontinuouswithvaluesinE,
forallt,s≥0,andforanyτ>0thereexistsaconstantl(τ)>0,suchthat
orallt,s∈[0,τ].
TheC0-semigroupisexponentiallybounded,thatis,thereexisttwoconstantsand,suchthatforallt≥0.
Noticethatthecontrollabilityofaclassofnon-denselydefinedfunctionaldifferentialequationswasstudiedin[12]inthefinitedelaycase.、
2MainResults
Westartwithintroducingthefollowingdefinition.
Definition1LetT>0andϕ∈B.Weconsiderthefollowingdefinition.
Wesaythatafunctionx:
=x(.,ϕ):
(−∞,T)→E,0 (1)if xiscontinuouson[0,T), fort∈[0,T), fort∈[0,T), forallt∈(−∞,0]. Wededucefrom[1]and[22]thatintegralsolutionsofEq. (1)aregivenforϕ∈B,suchthatbythefollowingsystem 、(3) Where. Toobtainglobalexistenceanduniqueness,wesupposedasin[1]that (H2). (H3)iscontinuousandthereexists>0,suchthat forϕ1,ϕ2∈Bandt≥0.(4) UsingTheorem7in[1],weobtainthefollowingresult. Theorem1Assumethat(H1),(H2),and(H3)hold.Letϕ∈BsuchthatDϕ∈D(A).Then,thereexistsauniqueintegralsolutionx(.,ϕ)ofEq. (1),definedon(−∞,+∞). Definition2Undertheaboveconditions,Eq. (1)issaidtobecontrollableontheintervalJ=[0,δ],δ>0,ifforeveryinitialfunctionϕ∈BwithDϕ∈D(A)andforanye1∈D(A),thereexistsacontrolu∈L2(J,U),suchthatthesolutionx(.)ofEq. (1)satisfies. Theorem2Supposethat(H1),(H2),and(H3)hold.Letx(.)betheintegralsolutionofEq. (1)on(−∞,δ),δ>0,andassumethat(see[20])thelinearoperatorWfromUintoD(A)definedby ,(5) nducesaninvertibleoperatoron,suchthatthereexistpositiveconstantsandsatisfyingand,then,Eq. (1)iscontrollableonJprovidedthat ,(6) Where . ProofFollowing[1],whentheintegralsolutionx(.)ofEq. (1)existson(−∞,δ),δ>0,itisgivenforallt∈[0,δ]by Or Then,anarbitraryintegralsolutionx(.)ofEq. (1)on(−∞,δ),δ>0,satisfiesx(δ)=e1ifandonlyif Thisimpliesthat,byuseof(5),itsufficestotake,forallt∈J, inordertohavex(δ)=e1.Hence,wemusttakethecontrolasabove,andconsequently,theproofisreducedtotheexistenceoftheintegralsolutiongivenforallt∈[0,δ]by Withoutlossofgenerality,supposethat≥0.Usingsimilarargumentsasin[1],wecanseehat,forevery,andt∈[0,δ], AsKiscontinuousand,wecanchooseδ>0smallenough,suchthat . Then,Pisastrictcontractionin,andthefixedpointofPgivestheuniqueintegralolutionx(.,ϕ)on(−∞,δ]thatverifiesx(δ)=e1. Remark1SupposethatalllinearoperatorsWfromUintoD(A)definedby 0≤a0,induceinvertibleoperatorson,suchthatthereexistpositiveconstantsN1andN2satisfyingand,taking,Nlargeenoughandfollowing[1].Asimilarargumentastheaboveproofcanbeusedinductivelyin,toseethatEq. (1)iscontrollableon[0,T]forallT>0. AcknowledgementsTheauthorswouldliketothankProf.KhalilEzzinbiandProf.PierreMagalforthefruitfuldiscussions. References [1]AdimyM,BouzahirH,EzzinbiK.Existenceandstabilityforsomepartialneutralfunctionaldifferentialequationswithinfinitedelay.JMathAnalAppl,2004,294: 438–461 [2]AdimyM,EzzinbiK.Aclassoflinearpartialneutralfunctionaldifferentialequationswithnondensedomain.JDifEq,1998,147: 285–332 [3]ArendtW.Resolventpositiveoperatorsandintegratedsemigroups.ProcLondonMathSoc,1987,54(3): 321–349 [4]AtmaniaR,MazouziS.Controllabilityofsemilinearintegrodifferentialequationswithnonlocalconditions.ElectronicJofDiffEq,2005,2005: 1–9 [5]BalachandranK,AnandhiER.ControllabilityofneutralintegrodifferentialinfinitedelaysystemsinBanachspaces.TaiwaneseJMath,2004,8: 689–702 [6]BalasubramaniamP,NtouyasSK.Controllabilityforneutralstochasticfunctionaldifferentialinclusionswithinfinitedelayinabstractspace.JMathAnalAppl,2006,324 (1): 161–176 、 [7]BalachandranK,BalasubramaniamP,DauerJP.Localnullcontrollabilityofnonlinearfunctionaldiffer-entialsystemsinBanachspace.JOptimTheoryAppl,1996,88: 61–75 [8]BalasubramaniamP,LoganathanC.ControllabilityoffunctionaldifferentialequationswithunboundeddelayinBanachspace.JIndianMathSoc,2001,68: 191–203 [9]BouzahirH.Onneutralfunctionaldifferentialequations.FixedPointTheory,2005,5: 11–21 Thestudyofdifferentialequationsisonepartofmathematicsthat,perhapsmorethananyother,hasbeendirectlyinspiredbymechanics,astronomy,andmathematicalphysics.Itshistorybeganinthe17thcenturywhenNewton,Leibniz,andtheBernoullissolvedsomesimpledifferentialequationarisingfromproblemsingeometryandmechanics.Thereearlydiscoveries,beginningabout1690,graduallyledtothedevelopmentofalotof“specialtricks”forsolvingcertainspecialkindsofdifferentialequations.Althoughthesespecialtricksareapplicableinmechanicsandgeometry,sotheirstudyisofpracticalimportance. 微分方程 牛顿和莱布尼茨,完全相互独立,主要负责开发积分学思想的地步,迄今无法解决的问题可以解决更多或更少的常规方法。 这些成功的人主要是由于他们能够将积分学和微分融合在一
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