微分学数学专业英语论文双语(含中文版)

DifferentialCalculusNewtonandLeibniz,quiteindependentlyofoneanother,werelargelyresponsiblefordevelopingtheideasofintegralcalculustothepointwherehithertoinsurmountableproblemscouldbesolvedbymoreorlessroutinemethods.Thesuccessfulaccomplishmentsofthesemenwereprimarilyduetothefactthattheywereabletofusetogethertheintegralcalculuswiththesecondmainbranchofcalculus,differentialcalculus.Inthisarticle,wegivesufficientconditionsforcontrollabilityofsomepartialneutralfunctionaldifferentialequationswithinfinitedelay.WesupposethatthelinearpartisnotnecessarilydenselydefinedbutsatisfiestheresolventestimatesoftheHille-Yosidatheorem.Theresultsareobtainedusingtheintegratedsemigroupstheory.Anapplicationisgiventoillustrateourabstractresult.KeywordsControllability;integratedsemigroup;integralsolution;infinitydelay1IntroductionInthisarticle,weestablisharesultaboutcontrollabilitytothefollowingclassofpartialneutralfunctionaldifferentialequationswithinfinitedelay:0,),()(0txxttFtCuADxtDxtt(1)wherethestatevariable(.)xtakesvaluesinaBanachspace).,(Eandthecontrol(.)uisgivenin0),,,0(2TUTL,theBanachspaceofadmissiblecontrolfunctionswithUaBanachspace.CisaboundedlinearoperatorfromUintoE,A:D(A)⊆E→EisalinearoperatoronE,Bisthephasespaceoffunctionsmapping(−∞,0]intoE,whichwillbespecifiedlater,DisaboundedlinearoperatorfromBintoEdefinedbyBDD,)0(00DisaboundedlinearoperatorfromBintoEandforeachx:(−∞,T]→E,T0,andt∈[0,T],xtrepresents,asusual,themappingfrom(−∞,0]intoEdefinedby]0,(),()(txxtFisanE-valuednonlinearcontinuousmappingon.TheproblemofcontrollabilityoflinearandnonlinearsystemsrepresentedbyODEinfinitdimensionalspacewasextensivelystudied.ManyauthorsextendedthecontrollabilityconcepttoinfinitedimensionalsystemsinBanachspacewithunboundedoperators.Uptonow,therearealotofworksonthistopic,see,forexample,[4,7,10,21].Therearemanysystemsthatcanbewrittenasabstractneutralevolutionequationswithinfinitedelaytostudy[23].Inrecentyears,thetheoryofneutralfunctionaldifferentialequationswithinfinitedelayininfinitedimensionwasdevelopedanditisstillafieldofresearch(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,supposethat).,(BBisa(semi)normedabstractlinearspaceoffunctionsmapping(−∞,0]intoE,andsatisfiesthefollowingfundamentalaxiomsthatwerefirstintroducedin[13]andwidelydiscussedin[16].(A)ThereexistapositiveconstantHandfunctionsK(.),M(.):,withKcontinuousandMlocallybounded,suchthat,foranyand0a,ifx:(−∞,σ+a]→E,Bxand(.)xiscontinuouson[σ,σ+a],then,foreverytin[σ,σ+a],thefollowingconditionshold:(i)Bxt,(ii)BtxHtx)(,whichisequivalenttoBH)0(oreveryB(iii)BxtMsxtKxttsB)()(sup)((A)Forthefunction(.)xin(A),t→xtisaB-valuedcontinuousfunctionfortin[σ,σ+a].(B)ThespaceBiscomplete.Throughoutthisarticle,wealsoassumethattheoperatorAsatisfiestheHille-Yosidacondition:(H1)Thereexistand，suchthat)(),(AandMNnAInn,:)()(sup（2）LetA0bethepartofoperatorAin)(ADdefinedby)(,,)(:)()(000ADxforAxxAADAxADxADItiswellknownthat)()(0ADADandtheoperator0Ageneratesastronglycontinuoussemigroup))((00ttTon)(AD.Recallthat[19]forall)(ADxand0t,onehas)()(000ADxdssTftandxtTxsdssTAt)(0)(00.Wealsorecallthat00))((ttTcoincideson)(ADwiththederivativeofthelocallyLipschitzintegratedsemigroup0))((ttSgeneratedbyAonE,whichis,accordingto[3,17,18],afamilyofboundedlinearoperatorsonE,thatsatisfies(i)S(0)=0,(ii)foranyy∈E,t→S(t)yisstronglycontinuouswithvaluesinE,(iii)sdrrsrtStSsS0))()(()()(forallt,s≥0,andforanyτ0thereexistsaconstantl(τ)0,suchthatstlsStS)()()(orallt,s∈[0,τ].TheC0-semigroup0))((ttSisexponentiallybounded,thatis,thereexisttwoconstantsMand,suchthatteMtS)(forallt≥0.Noticethatthecontrollabilityofaclassofnon-denselydefinedfunctionaldifferentialequationswasstudiedin[12]inthefinitedelaycase.、2MainResultsWestartwithintroducingthefollowingdefinition.Definition1LetT0andϕ∈B.Weconsiderthefollowingdefinition.Wesaythatafunctionx:=x(.,ϕ):(−∞,T)→E,0T≤+∞,isanintegralsolutionofEq.(1)if(i)xiscontinuouson[0,T),(ii)tADDxsds0)(fort∈[0,T),(iii)tsttdsxsFsCuDxsdsADDx00),()(fort∈[0,T),(iv))()(ttxforallt∈(−∞,0].Wededucefrom[1]and[22]thatintegralsolutionsofEq.(1)aregivenforϕ∈B,suchthat)(ADDbythefollowingsystem],0,(),()(),,0[,)),()(()(lim)(0tttxttdsxsFsCuBstSDtSDxtts、(3)Where1)(A