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1、DifferentialCalculus/D/x*p2R6r$]$NewtonandLeibniz,quiteindependentlyofoneanother,werelargelyresponsiblefordevelopingtheideasofintegralcalculustothepointwherehithertoinsurmountableproblemscouldbesolvedbymoreorlessroutinemethods.Thesuccessfulaccomplishmentsofthesemenwereprimarilyd
2、uetothefactthattheywereabletofusetogethertheintegralcalculuswiththesecondmainbranchofcalculus,differentialcalculus.Inthisarticle,wegivesufficientconditionsforcontrollabilityofsomepartialneutralfunctionaldifferentialequationswithinfinitedelay.Wesupposethatthelinearpartisnotnecessaril
3、ydenselydefinedbutsatisfiestheresolventestimatesoftheHille-Yosidatheorem.Theresultsareobtainedusingtheintegratedsemigroupstheory.Anapplicationisgiventoillustrateourabstractresult.KeywordsControllability;integratedsemigroup;integralsolution;infinitydelay1IntroductionInthisarticle,we
4、establisharesultaboutcontrollabilitytothefollowingclassofpartialneutralfunctionaldifferentialequationswithinfinitedelay:(1)wherethestatevariabletakesvaluesinaBanachspaceandthecontrolisgivenin,theBanachspaceofadmissiblecontrolfunctionswithUaBanachspace.Cisaboundedlinearoperatorfrom
5、UintoE,A:D(A)⊆E→EisalinearoperatoronE,Bisthephasespaceoffunctionsmapping(−∞,0]intoE,whichwillbespecifiedlater,DisaboundedlinearoperatorfromBintoEdefinedbyisaboundedlinearoperatorfromBintoEandforeachx:(−∞,T]→E,T>0,andt∈[0,T],xtrepresents,asusual,themappingfrom(−∞,0]intoEdefinedbyFis
6、anE-valuednonlinearcontinuousmappingon.TheproblemofcontrollabilityoflinearandnonlinearsystemsrepresentedbyODEinfinitdimensionalspacewasextensivelystudied.ManyauthorsextendedthecontrollabilityconcepttoinfinitedimensionalsystemsinBanachspacewithunboundedoperators.Uptonow,therearealo
7、tofworksonthistopic,see,forexample,[4,7,10,21].Therearemanysystemsthatcanbewrittenasabstractneutralevolutionequationswithinfinitedelaytostudy[23].Inrecentyears,thetheoryofneutralfunctionaldifferentialequationswithinfinitedelayininfinitedimensionwasdevelopedanditisstillafieldofresearc
8、h(see,forinstance,[2,9,14,15]andthereferencesth
