An operator theory of linear functional differential equations

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1、JOURNALOFDIFFERENTIAI.!-QIIATIOKS27,274-297(1978)AnOperatorTheoryofLinearFunctionalDifferentialEquations4Anoperatortheory,basedonconvolutionringsandmodules,isdevelopedforvariousclassesoffirst-ordervectorfunctionaldifferentialequationsoftheretardedtypewithfiniteandinfinitedelays.Theexistenceand

2、constructionofcompletesolutionsisapproachedinanovelmannerbyincorporatinginitialdataintotheoperatorframework.Resultsarethenobtainrdonexponentialandasymptoticstability.Theoperatorframeworkisalsoappliedtothestud>-ofstatefeedback.Constructiveresultsonspectrumassignabilityandstabilizabilitybyfeedba

3、ckaregiven.I.INTRODUCTIONInthefirstpartofthispaperweconsidertheclassoffirst-orderlinearfunctionaldifferentialequationsoftheretardedtypegivenh>-In(1.1)a,bandtheh,,J,arepositiverealnumbers,the-Gti(resp.11,)arcTZxn(resp.II:im)matricesovertherealsW,and,4(7)(resp.B(r))isanxn(nxm)matrixofLebesguemea

4、surableandintegrablefunctionswithsupportcontainedin[-a,O](resp.[--h,01).Finally,x(t)E[wistheinstan-taneousstateandu(t)t;IL!istheinputorcontrol.Previousworkdealingwiththeexistenceanduniquenessofsolutionsof(1.I)isbasedonthedefinitionofaninfinite-dimensionalstatespacewhoseelementsareW-valuedfunct

5、ions&finedon[-/I,01,whereh=max{a,hi,...,o,.j.For*ThisworkwassupportedbytheU.S.ArmyResearchOffice,ResearchTrianglePark,K.C.,underGrantI).~XC;?9-76-(;-0048.2740022-0396/78/0272-0274$02.00/OCopyrighti1978by.cademicPress,Inc.Allrightsofrcproductmninanyformwsinid.OPERhTORTHEORYOFLI?XARFUSCTIONALDI

6、FFERENTT.tLEQUATIONS215example,Hale[l]considerstheBanachspace%([--h,01;LQl)consistingofR-valuedcontinuousfunctionson[--h,0]withtheusualsupnorm.Withinitialfunctionx(t),--h

7、lycontinuoussemigroupofoperatorsdefinedonq-/z,01;W).IntheworkofDelfour[2]andDelfourandMitter[335],equationsofform(1.1)arecharacterizedbyanequationofevolutionintheproductspacel,R:XLJ([--h,01;KY),whereL([--h,01;Rl)isthespaceofR-valuedp-in