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1、-CONTROLLABILITYOFNEUTRALFUNCTIONALDIFFERENTIALEQUATIONSWITHINFINITEDELAYAbstractInthisarticle,wegivesufficientconditionsforcontrollabilityofsomepartialneutralfunctionaldifferentialequationswithinfinitedelay.Wesupposethatthelinearpartisnotnecessarilydenselydefinedbutsatis
2、fiestheresolventestimatesoftheHille-Yosidatheorem.Theresultsareobtainedusingtheintegratedsemigroupstheory.Anapplicationisgiventoillustrateourabstractresult.KeywordsControllability;integratedsemigroup;integralsolution;infinitydelay1IntroductionInthisarticle,weestablisha
3、resultaboutcontrollabilitytothefollowingclassofpartialneutralfunctionaldifferentialequationswithinfinitedelay:(1)wherethestatevariabletakesvaluesinaBanachspaceandthecontrolisgivenin,theBanachspaceofadmissiblecontrolfunctionswithUaBanachspace.Cisaboundedlinearoperatorfr
4、omUintoE,A:D(A)⊆E→EisalinearoperatoronE,Bisthephasespaceoffunctionsmapping(−∞,0]intoE,whichwillbespecifiedlater,DisaboundedlinearoperatorfromBintoEdefinedbyisaboundedlinearoperatorfromBintoEandforeachx:(−∞,T]→E,T>0,andt∈[0,T],xtrepresents,asusual,themappingfrom(−∞,0]in
5、toEdefinedbyFisanE-valuednonlinearcontinuousmappingon.TheproblemofcontrollabilityoflinearandnonlinearsystemsrepresentedbyODEinfinitdimensionalspacewasextensivelystudied.ManyauthorsextendedthecontrollabilityconcepttoinfinitedimensionalsystemsinBanachspacewithunboundedope
6、rators.Uptonow,therearealotofworksonthistopic,see,forexample,[4,7,10,21].Therearemanysystemsthatcanbewrittenasabstractneutralevolutionequationswithinfinitedelaytostudy[23].Inrecentyears,thetheoryofneutralfunctionaldifferentialequationswithinfinitedelayininfinitedimension
7、wasdevelopedanditisstillafieldofresearch(see,forinstance,[2,9,14,15]andthereferencestherein).Meanwhile,thecontrollabilityproblemofsuchsystemswasalsodiscussedbymanymathematicians,see,forexample,[5,8].TheobjectiveofthisarticleistodiscussthecontrollabilityforEq.(1),where
8、thelinearpartissupposedtobenon-denselydefinedbutsatisfiestheresolventestimatesoftheHille-Yosidatheorem.Weshallassumeconditionsthatass
