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1、实用标准文案CONTROLLABILITYOFNEUTRALFUNCTIONALDIFFERENTIALEQUATIONSWITHINFINITEDELAYAbstractInthisarticle,wegivesufficientconditionsforcontrollabilityofsomepartialneutralfunctionaldifferentialequationswithinfinitedelay.Wesupposethatthelinearpartisnotnecessarilydenselydefinedbutsa
2、tisfiestheresolventestimatesoftheHille-Yosidatheorem.Theresultsareobtainedusingtheintegratedsemigroupstheory.Anapplicationisgiventoillustrateourabstractresult.KeywordsControllability;integratedsemigroup;integralsolution;infinitydelay1IntroductionInthisarticle,weestablish
3、aresultaboutcontrollabilitytothefollowingclassofpartialneutralfunctionaldifferentialequationswithinfinitedelay:(1)wherethestatevariabletakesvaluesinaBanachspaceandthecontrolisgivenin,theBanachspaceofadmissiblecontrolfunctionswithUaBanachspace.Cisaboundedlinearoperatorfro
4、mUintoE,A:D(A)⊆E→EisalinearoperatoronE,Bisthephasespaceoffunctionsmapping(−∞,0]intoE,whichwillbespecifiedlater,DisaboundedlinearoperatorfromBintoEdefinedbyisaboundedlinearoperatorfromBintoEandforeachx:(−∞,T]→E,T>0,andt∈[0,T],xtrepresents,asusual,themappingfrom(−∞,0]intoE
5、definedbyFisanE-valuednonlinearcontinuousmappingon.TheproblemofcontrollabilityoflinearandnonlinearsystemsrepresentedbyODEinfinitdimensionalspacewasextensivelystudied.ManyauthorsextendedthecontrollabilityconcepttoinfinitedimensionalsystemsinBanachspacewithunboundedoperator
6、s.Uptonow,therearealotofworksonthistopic,see,forexample,[4,7,10,21].Therearemanysystemsthatcanbewrittenasabstractneutralevolutionequationswithinfinitedelaytostudy[23].Inrecentyears,thetheoryofneutralfunctionaldifferentialequationswithinfinitedelayininfinitedimensionwasdeve
7、lopedanditisstillafieldofresearch(see,forinstance,[2,9,14,15]andthereferencestherein).Meanwhile,thecontrollabilityproblemofsuchsystemswasalsodiscussedbymanymathematicians,see,forexample,[5,8].TheobjectiveofthisarticleistodiscussthecontrollabilityforEq.(1),wherethelinear
8、partissupposedtobenon-denselydefinedbutsatisfiestheresolventestimatesoftheHille-Yosidatheorem.Weshallassumeconditionsth
