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1、DifferentialCalculusNewtonandLeibniz,quiteindependentlyofoneanother,werelargelyresponsibleiordevelopingtheideasofintegralcalculustothepointwherehithertoinsunnountableproblemscouldbesolvedbymoreorlessroutinemethods.Thesuccessfiilaccomplishmentsofthesemenwereprimarilyduetothefactthattheywereabletofu
2、setogethertheintegralcalculuswiththesecondmainbranchofcalculus,differentialcalculus・Inthisarticle,wegivesufficientconditionsforcontrollabilityofsomepartialneutralfunctionaldifferentialequationswithinfinitedelay.Wcsupposethatthelinearpartisnotnecessarilydenselydefinedbutsatisfiestheresolventestimat
3、esoftheHillc-Yosidatheorem.Theresultsarcobtainedusingtheintegratedsemigroupstheory.Anapplicationisgiventoillustrateourabstractresult.KeywordsControllability;integratedsemigroup;integralsolution;infinitydelay1IntroductionInthisarticle,weestablisharesultaboutcontrollabilitytothefollowingclassofparti
4、alneutralfunctionaldifferentialequationswithinfinitedelay:—Dxt=ADxt+Cu⑴+F(t,兀/)#、cc、09theBanachspaceofadmissiblecontrolfunctionswithUaBanachspace・CisaboundedlinearoperatorfromUintoE,A:D(A)匸E—>Eisalinearope
5、ratoronE,Bisthephasespaceoffunctionsmapping(-00,0]intoE,whichwillbespecifiedlater,DisaboundedlinearoperatorfromBintoEdefinedbyD(p=(p(0)-DQ(p,(peBZ)oisaboundedlinearoperatorfromBintoEandforeachx:(-co,T]—>E,T>0,andt丘[0,T],xtrepresents,asusual,themappingfrom(-00,0]intoEdefinedbyXt(0)=x(t+&),0G(-oo,0]
6、FisanE-valuednonlinearcontinuousmappingon9i+XB・TheproblemofcontrollabilityoflinearandnonlinearsystemsrepresentedbyODEinfinitdimensionalspacewasextensivelystudied.ManyauthorsextendedthecontrollabilityconcepttoinfinitedimensionalsystemsinBanachspacewithunboundedoperators・Uptonow,therearealotofworkso
7、nthistopic,see,forexample,[4,7,10,21]・Therearemanysystemsthatcanbewrittenasabstractneutralevolutionequationswithinfinitedelaytostudy[23]・Inrecentyears,thetheoryofneutralfunctionaldifferentialequationswithinfinite
