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DifferentialCalculus
/D/x*p2R6r$]$NewtonandLeibniz,quiteindependentlyofoneanother,werelargelyresponsiblefordevelopingtheideasofintegralcalculustothepointwherehithertoinsurmountableproblemscouldbesolvedbymoreorlessroutinemethods.Thesuccessfulaccomplishmentsofthesemenwereprimarilyduetothefactthattheywereabletofusetogethertheintegralcalculuswiththesecondmainbranchofcalculus,differentialcalculus.

Inthisarticle,wegivesuﬃcientconditionsforcontrollabilityofsomepartialneutralfunctionaldiﬀerentialequationswithinﬁnitedelay.WesupposethatthelinearpartisnotnecessarilydenselydeﬁnedbutsatisﬁestheresolventestimatesoftheHille-Yosidatheorem.Theresultsareobtainedusingtheintegratedsemigroupstheory.Anapplicationisgiventoillustrateourabstractresult.
KeywordsControllability;integratedsemigroup;integralsolution;inﬁnitydelay


1Introduction

Inthisarticle,weestablisharesultaboutcontrollabilitytothefollowingclassofpartialneutralfunctionaldiﬀerentialequationswithinﬁnitedelay:
(1)
wherethestatevariabletakesvaluesinaBanachspaceandthecontrolisgivenin,theBanachspaceofadmissiblecontrolfunctionswithUaBanachspace.CisaboundedlinearoperatorfromUintoE,A:D(A)⊆E→EisalinearoperatoronE,Bisthephasespaceoffunctionsmapping(−∞,0]intoE,whichwillbespeciﬁedlater,DisaboundedlinearoperatorfromBintoEdeﬁnedby

isaboundedlinearoperatorfromBintoEandforeachx:(−∞,T]→E,T>0,andt∈[0,T],xtrepresents,asusual,themappingfrom(−∞,0]intoEdeﬁnedby

FisanE-valuednonlinearcontinuousmappingon.

TheproblemofcontrollabilityoflinearandnonlinearsystemsrepresentedbyODEinﬁnitdimensionalspacewasextensivelystudied.ManyauthorsextendedthecontrollabilityconcepttoinﬁnitedimensionalsystemsinBanachspacewithunboundedoperators.Uptonow,therearealotofworksonthistopic,see,forexample,[4,7,10,21].Therearemanysystemsthatcanbewrittenasabstractneutralevolutionequationswithinﬁnitedelaytostudy[23].Inrecentyears,thetheoryofneutralfunctionaldiﬀerentialequationswithinﬁnitedelayininﬁnite
dimensionwasdevelopedanditisstillaﬁeldofresearch(see,forinstance,[2,9,14,15]