Almost periodic solutions of differential equations in Banach spaces PDF
Preview Almost periodic solutions of differential equations in Banach spaces
AlmosPte riodSiocl utioofnD si ffereEnqtuiaatli ons inB anacShp aces StabilaintdCy o ntroTlh:e oryM,e thodasn dA pplications A serioefsb ookasn dm onograpohnts h et heoorfys tabialnidtc yo ntrol EditbedyA .A.M artynyuk, IonfsM teicthuatnei cs, UkKriaeivan,ne d V .L akshmikantham, HoridIan stiotfuT teceh noloUgSyA, Volum1e TheoroyfI ntegro-DiffEeqrueanttiioanls V. LakshmikanatnhdaM. m RamaM ohanRaa o Volum2e StabilAintayl ysNiosn:l inMeeacrh aniEcqsu ations A.AM.a rtynyuk Volum3e StabiloifMt oyt ioonfN onautonomSoyusst em(sM ethoodfL imitiEnqgu ations) J.K atoA,. AM.a rtynyaunkdA .AS.h estakov Volum4e ContrTohle orayn di tAsp plications E.OR.o xin VolumeS AdvanciensN onlinDeyanra mics editbeydS .S ivasundaarnadAm . AM.a rtynyuk Volum6e SolviDnigf ferePnrtoiballe bmysM ultisItneipt ainadlB oundaVrayl uMee thods L.B rugnaannod D .T rigiante Volum7e DynamiocfsM achinweist Vha riaMbalses L.C veticanin VolumeS OptimizaotfiL oinn eCaorn trSoyls temAsn:a lytiMceatlh odasn dC omputatiAolngaolr ithms F.A.A lieavn dv. B.La rin Volum9e Dynamiacnsd C ontrol editbeydG .L eitmaF.nEn.,U dwadiaan dA .V. Kryazhimskii Volum1e0 VolteErqruaa tiaonndsA pplications editbeydC. Corduneaanndu I . SW.a ndberg Volum1e1 NonlinPeraorb leimnAs v iatiaonndA erospace editbeydS .S ivasundaram Volum1e2 StabilizoafPt rioognr ammMeodt ion E.YaS.m irnov Pleassee et heb acko ft hibso okf oro thetri tlients h eS tabilaintdCy o ntrolT:h eorMye,t hods andA pplicatsieornise s. AlmosPte riodSiocl utioofnD si fferentEiqaula tions inB anacShp aces YoshiyHuiknio DepartmenotfM athemataincdsI nformatics ChibUan iversCihtyi,b aJ,a pan ToshiNkaii to DepartmeonfMt a thematics UniversoifEty l ectro-CommuniTcoakytoi,Jo anpsa,n NguyeVna nM inh DepartmenotfM athematics HanoUin iversoifSty c ienHcae,n oiV,i etnam JonSgo nS hin DepartmenotfM athematics KoreUan iversTiotyky,o ,J apan Londoann dN ewY ork CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2001 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140416 International Standard Book Number-13: 978-1-4822-6316-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Introducttoti hoeSn e ries 3 Preface Acknowledgements 5 Co-SEMIGROUPWESL, L POSEEDV OLUTIONE QUATIONS, SPECTRALT HEORY AND ALMOST PERIODICITOYF FUNCTIONS 7 1.1S.TR ONGLY CONTINUOUSSE MIGROUPSO F LINEARO PERATORS 7 1.1.D1efi.n itiaonndB asiPcr operties 7 1.1.C2om.p acSte migrouapnsdA nalytSitcr ongCloyn tinuous Semigroups 10 1.1.S3pe.c trMaalp pinTgh eorems 11 1.2E.VO LUTIONE QUATIONS 15 1.2.W1el.l -PoEsveodl utEiqouna tions 15 1.2.F2un.c tioDniaffle renEtqiuaalt iwonistF hi niDteel ay 18 1.2.E3qu.a tiownistI hn finiDteel ay 20 1.3S.PE CTRALT HEORYA ND ALMOST PERIODICITOYF BOUNDED UNIFORMLYC ONTINUOUSF UNCTIONS 24 1.3.S1pe.c troufma BoundeFdu nction 24 1.3.A2lm.o st Periodic Functions 26 1.3.S3pe.c troufma nA lmosPte riodFiucn ction 27 1.3.A4 S.p ectraClr iterfioorAn l mosPte riodiocfia t Fyu nction 28 2 SPECTRALC RITERIFAO R PERIODIACN D ALMOST PERIODIC SOLUTIONS 31 2.1E.VO LUTIONS EMIGROUPSA ND ALMOST PERIODIC SOLUTIONSO F PERIODIECQ UATIONS 31 2.1.E1vo.l utSieomni groups 31 v vi CONTENTS 2.1.A2l.mo sPte riodSiocl utiaonndsA pplications 35 2.2E.V OLUTIONS EMIGROUPSS,U MS OF COMMUTING OPERATORS AND SPECTRACLR ITERIFAO R ALMOST PERIODICITY 45 2.2.D1i.ffe rential dlOpdet-r .<Aat aonrd N otioonfsA dmissibility4 8 2.2.A2d.mis sibifloirAt bys traOcrtd inaDriyff erenEtqiuaalt ions 53 2.2.H3i.gh eOrr deDri fferenEtqiuaalt ions 55 2.2.A4b.st raFcutn ctioDniaffle renEtqiuaalt ions 62 2.2.E5xa.m plaensd A pplications 66 2.3D.EC OMPOSITIOTNH EOREMA ND PERIODICA,L MOST PERIODISCO LUTIONS 77 2.3.S1p.ec tDraelc omposition 79 2.3.S2p.ec tOrrailt eFroiraA lmosPte riodSiocl utions 85 2.3.W3h.en D oesB oundednYeisesl Udn iforCmo ntinuity? 89 2.3.P4e.ri odSiocl utioofnP sa rtiFauln ctioDniaffle renEtqiuaalt ions 91 2.3.A5l.mo sPte riodSiocl utioofnP sa rtiFauln ctioDniaffle rential Equations 95 2.4F.IX EDP OINTT HEOREMSA ND FREDHOLMO PERATORS 109 2.4.F1i.xe Pdo inTth eorems 109 2.4.D2e.co mposiotfiS oonl ution Operators 110 2.4.P3e.ri odSiocl utiaonndsF ixePdo inTth eorems 113 2.4.E4xi.s teonfcP ee riodSiocl utioBnosu:n dePde rturbations 116 2.4.E5xi.s teonfcP ee riodSiocl utioCnosm:p acPte rturbations 120 2.4.U6n.iq uenoefsP se riodSiocl utiIo ns 125 2.4.U7ni.q uenoefsP se riodSiocl utiIoIn s 127 2.4.A8n. Ex ample 129 2.4.P9er.i odSiocl utiionnE sq uatiownist Ihn finiDteel ay 130 2.5B.OU NDEDNESSA ND ALMOST PERIODICIITND YI SCRETE SYSTEMS 132 2.5.S1pe.c troufmB oundeSde quencaensdD ecomposition 133 2.5.A2l.mo sPte riodSiocl utioofnD si screStyes tems 137 2.5.A3pp.l icattioEo vnosl utEiqouna tions 139 2.6B.OU NDEDNESSA ND ALMOST PERIODISCO LUTIONSO F SEMILINEAERQ UATIONS 143 2.6.E1v.ol utSieoming roupasn dS emilinEevaorl utEiqouna tions 143 2.6.B2ou.n deadn dP eriodSiocl utitoonA sb straFcutn ctional DifferenEtqiuaalt iownistF hi niDteel ay 151 2.7A.LM OST PERIODISCO LUTIONSO F NONLINEARE VOLUTION EQUATIONS 153 2.7.N1on.l inEevaorl utSieoming roupisnA P(Ll) 153 2.7.A2l.mo sPte riodSiocl utioofnD si ssipaEtqiuvaet ions 157 2.7.A3n. Ex ample 160 2.8N.O TES 161 3 STABILITMYE THODS FOR SEMILINEAERV OLUTION EQUATIONSA ND NONLINEARE VOLUTIONE QUATIONS 163 3.1S.K EW PRODUCTF LOWS OF PROCESSESA ND QUASI-PROCESSAENSD STABILITOYF INTEGRALS 163 CONTENTS vii 3.2E.XI STENCTEH EOREMSO F ALMOST PERIODIICN TEGRALS 168 3.2.A1sy.m ptoAtlimco sPte riodiacnidtA yl mosPte riodicity 168 3.2.U2ni.f onnA symptoSttiacb ilaintdEy x isteonfcA el mosPte riodic Integrals 171 3.2.S3ep.ar atiCoonn ditainodnE xisteonfcA el mosPte riodIinct egra1ls7 2 3.2.R4el.a tionbsehtiwpe etnh eU nifonnA symptoSttiacb ilaintdy the SeparatCioonnd ition 175 3.2.E5xi.s teonfca en A lmosPte riodIinct egorfaA ll most Quasi-Processes 176 3.3P.RO CESSESA ND QUASI-PROCESSES GENERABTYE D ABSTRACTF UNCTIONALD IFFERENTIAELQ UATIONS 176 3.3.A1bs.t raFcutn ctioDniaffle renEtqiuaalt iwonistI hn finiDteel ay 176 3.3.P2ro.c esasnedsQ uasi-ProcGeesnseersa tbeyAd b straFcutn ctional DifferenEtqiuaalt iownistI hn finiDteel ay 180 3.3.S3ta.b ilPirtoyp ertfioeArsb straFcutn ctioDniaflf erential EquatiownistI hn finiDteel ay 185 3.4E.QU IVALENTR ELATIONSHIPBSE TWEEN BC-STABILITIES AND p-STABILITIES 190 3.4.B1C-.S tabiliinAt bisetsr aFcutn ctioDniaffle renEtqiuaalt ions witIhn finiDteel ay 190 3.4.E2qu.i valReenlta tionbsehtiwpe eBnC -UnifoAnns ymptotic Stabilaintdpy - UnifoArsmy mptoSttiacb ility 192 3.4.E3qu.i valReenlta tionBsehtiwpe eBnC -TotSatla bilaintdy p-TotSatla bility 195 3.4.E4qu.i valReenlta tionsohfSi tpasb ilfiotriL eisne aArb stract FunctioDniaflf ereEnqtuiaatli ownistI hn finiDteel ay 198 3.5E.XI STENCOEF ALMOST PERIODISCO LUTIONS 202 3.5.A1lm.o sPte rioAdbisct raFcutn ctional DiffEeqrueatnitoinasl witIhn finite Delay 202 3.5.E2xi.s teTnhceeo reomfsA lmost PerSioolduitci foonrNs o nlinear Systems 203 3.5.E3xi.s teTnhceeo reomfsA lmosPte riodSiocl utifoonLrsi near Systems 204 3.6A.PP LICATIONS 207 3.6.D1am.p edW aveE quation 207 3.6.I2nt.e grodiffeErqeunattiiawolin t Dhi ffusion 210 3.6.P3ar.t iFauln ctioDniaffle renEtqiuaalt ion 214 3.7N.OT ES 217 4 APPENDICES 221 4.1F.RE DHOLM OPERATORSA ND CLOSEDR ANGE THEOREMS 221 4.2E.SS ENTIASLP ECTRUMA ND MEASURESO F NONCOMPACTNESS2 24 4.3S.UM S OF COMMUTING OPERATORS 231 4.4L.IP SCHITOZP ERATORS 232 REFERENCES 235 INDEX 249 TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Introdutcott ihoeSn e ries Thep robleomfsm odems ocieatryeb othc ompleaxn di nterdisacriypD.le isnpitthee apparednitv ersoifpt ryo bletmoso,l dse velopienod n ec onteaxrteo ftena daptatbola en entirdeilffye resnitt uatFioorne .x amplceo,n sidtehreL yapunovw'esl kln owns econd methoTdh.i isn teresatnidfrn ugi tfutle chnihqausge a ineidn creassiinggn ificaanncdhe a s givean d ecisive ifmoprme otduesm d evelopmoefnt th es tabitlhietoyr oyfd ifferential equatiAo nmsa.n ifest advoaftn htimaseg teh oidst haittd oenso td emantdh ek nowledge ofs olutiaonndts h erefhoarsge r eapto weirn a pplicaIttii osnn o.w w elrle cognitzheadt thec onceopftL yapunov-fulnicktei oannsdt het heoorfyd ifferenatnidia nlt egirnaelq ual iticeasn b eu tilitzoei dn vestiqguaatlei taatnidvq eu antitaptriovpee rtoifne osn linear dynamiscy stemLsy.a punov-funlcitkieo nsse rvaesv ehictloet sr ansftohregm i vecno m plicadtyenda miscy steimnst ao r elatisviemlpyl seyrs teamn dt herefiotir ses ufficient tos tudtyh ep ropertoifte hsi ssi mpldeyrn amiscy steImt.i sa lsboe inrge alitzheadtt h e samev ersattioloelc sa nb ea daptteodd iscuesnst irdeilffye rennotn linseyasrt emasn,d thaotth ert oolssu,c ha st hev ariatoifop na rametaenrdst hem ethodo fu ppearn dl ower solutipornosv iedqeu alelffye ctimveet hodtsod eawli tphr obleomfsa similnaart ure. Moreoveirn,t eresnteiwni gd eahsa veb eeni ntroduwcheidc who ulsde emt oh olgdr eat potential. Contrtohle oroyn,t heo thehra ndi,st habtr ancohfa pplication-moartiheenmtaetdi cs thadte alwsi tthh eb asipcr inciupnldeesr lytihneag n alyasnidsd esigonfc ontrsoyls tems. Toc ontraonlo bjeicmtp litehsei nfluenocfei tbse havisoora st oa ccomplai dsehs ired goalI.n o rdetro i mplemetnhti isn fluencper,a ctitibounieldrdes v ictehsa itn corporate variomuast hematitceaclh niqTuheess .t udoyf t hesdee vicaensd t heiirn teracwtiitohn the objebceti ncgo ntrolilste hde s ubjeocftc ontrtohle orTyh.e rhea veb eenr,o ughly speakitnwgom, a inl inoefsw orki nc ontrtohle owrhyi cahr ceo mplementOanreiy s.b ased ont hei detah aat g oodm odeolf t heo bjetcotb ec ontrolilsae vda ilaabnldte h awte wish too ptimiiztbese haviaonrdt, h eo theirsb aseodn t hec onstraiimnptoss ebdyu ncertainty aboutth em odeli nw hicthh eo bjeocpte ratTehse.c ontrtoolo iln t hel attiestr h eu se off eedbaicnok r detroc orrefcotrd eviatifroonms t hed esirbeedh aviMoart.h ematically, stabitlhietoyr dyy,n amiscy steamnsd fu nctionaanla lyhsaivseh ada stroinngfl uenocne thiasp proach.
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