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1、CONTROLLABILITYOFNEUTRALFUNCTIONALDIFFERENTIALEQUATIONSWITHINFINITEDELAYAbstractInthisarticle,wegivesufficientconditionsforcontrollabilityofsomepartialneutralfunctionaldifferentialequationswithinfinitedelay.Wesupposethatthelinearpartisnotnecessarilydenselydefinedbutsatisfiesth
2、eresolventestimatesoftheHille-Yosidatheorem.Theresultsareobtainedusingtheintegratedsemigroupstheory.Anapplicationisgiventoillustrateourabstractresult.KeywordsControllability;integratedsemigroup;integralsolution;infinitydelay1IntroductionInthisarticle,weestablisharesultabo
3、utcontrollabilitytothefollowingclassofpartialneutralfunctionaldifferentialequationswithinfinitedelay:(1)wherethestatevariabletakesvaluesinaBanachspaceandthecontrolisgivenin,theBanachspaceofadmissiblecontrolfunctionswithUaBanachspace.CisaboundedlinearoperatorfromUintoE,A:D(
4、A)⊆E→EisalinearoperatoronE,Bisthephasespaceoffunctionsmapping(−∞,0]intoE,whichwillbespecifiedlater,DisaboundedlinearoperatorfromBintoEdefinedbyisaboundedlinearoperatorfromBintoEandforeachx:(−∞,T]→E,T>0,andt∈[0,T],xtrepresents,asusual,themappingfrom(−∞,0]intoEdefinedbyFisanE
5、-valuednonlinearcontinuousmappingon.TheproblemofcontrollabilityoflinearandnonlinearsystemsrepresentedbyODEinfinitdimensionalspacewasextensivelystudied.ManyauthorsextendedthecontrollabilityconcepttoinfinitedimensionalsystemsinBanachspacewithunboundedoperators.Uptonow,therea
6、realotofworksonthistopic,see,forexample,[4,7,10,21].Therearemanysystemsthatcanbewrittenasabstractneutralevolutionequationswithinfinitedelaytostudy[23].Inrecentyears,thetheoryofneutralfunctionaldifferentialequationswithinfinitedelayininfinitedimensionwasdevelopedanditisstilla
7、fieldofresearch(see,forinstance,[2,9,14,15]andthereferencestherein).Meanwhile,thecontrollabilityproblemofsuchsystemswasalsodiscussedbymanymathematicians,see,forexample,[5,8].TheobjectiveofthisarticleistodiscussthecontrollabilityforEq.(1),wherethelinearpartissupposedtobeno
8、n-denselydefinedbutsatisfiestheresolventestimatesoftheHille-Yosidatheorem.Weshallassumeconditionsthatassu
