Asymptotic stability equals exponential stability,

arXiv:math/9812137v3[math.OC]19May1999Asymptoticstabilityequalsexponentialstability,andISSequalsfiniteenergygain—ifyoutwistyoureyesLarsGr¨une⋆FachbereichMathematik,J.W.Goethe-Universit¨at,Postfach111932,D-60054Frankfurta.M.,Germany,E-Mail:[e-mail]⋆⋆DepartmentofMathematics,RutgersUniversity,NewBrunswick,NJ08903,USA,E-Mail:[e-mail]¨urTechnomathematik,Universit¨atBremen,D-28344Bremen,Germany,E-Mail:[e-mail]howthatuniformlyglobalasymptoticstabilityforafamilyofordinarydifferentialequationsisequivalenttouniformlyglobalexponentialstabilityunderasuitablenonlinearchangeofvariables.Thesameisshownforinput-to-statestabilityandinput-to-stateexponentialstability,andforinput-to-stateexponentialstabilityandanonlinearH∞estimate.Keywords:asymptoticstability,exponentialstability,input-to-statestability,nonlinearH∞1IntroductionLyapunov’snotionof(global)asymptoticstabilityofanequilibriumisakeyconceptinthequalitativetheoryofdifferentialequationsandnonlinearcontrol.Ingeneral,afarstrongerpropertyisthatofexponentialstability,whichrequiresdecayestimatesofthetype“kx(t)k≤⋆ThispaperhasbeenwrittenwhilethefirstauthorwasvisitingtheDipartimentodiMatematica,Universit´adiRoma“LaSapienza”,Italy,supportedbyDFG-GrantGR1569/2-1.⋆⋆SupportedinpartbyUSAirForceGrantF49620-98-1-0242PreprintsubmittedtoElsevierPreprint1February2008ce−λtkx(0)k.”(Seeforinstance[16]fordetaileddiscussionsofthecomparativerolesofasymp-toticandexponentialstabilityincontroltheory.)Inthispaper,weshowthat,fordifferentialequationsevolvinginfinite-dimensionalEuclideanspacesRn(atleastinspacesofdimensions6=4,5)thetwonotionsareoneandthesameundercoordinatechanges.Ofcourse,onemustdefine“coordinatechange”withcare,sinceunderdiffeomorphismsthecharacterofthelinearizationattheequilibrium(whichwetaketobetheorigin)isinvariant.However,if,inthespiritofbothstructuralstabilityandtheclassicalHartman-GrobmanTheorem(which,cf.[23],givesinessencealocalversionofourresultinthespecialhyperboliccase),werelaxtherequirementthatthechangeofvariablesbesmoothattheorigin,thenallobstructionsdisappear.Thus,weaskthattransformationsbeinfinitelydifferentiableexceptpossiblyattheorigin,wheretheyarejustcontinuouslydifferentiable.Theirrespectiveinversesarecontinuousglobally,andinfinitelydifferentiableawayfromtheorigin.Closelyrelatedtoourworkisthefactthatallasymptoticallystablelinearsystemsareequivalent(inthesensejustdiscussed)to˙x=−x;seee.g.[1].Thebasicideaoftheproofin[1]isbaseduponprojectionsonthelevelsetsofLyapunovfunctions,whichinthelinearcaseofcoursebetakentobequadratic(andhencehaveellipsoidsaslevelsets).Itisnaturaltousetheseideasalsointhegeneralnonlinearcase,andWilson’spaper[36],oftencitedincontroltheory,remarkedthatlevelsetsofLyapunovfunctionsarealwayshomotopicallyequivalenttospheres.Indeed,itispossibletoobtain,ingreatgenerality,achangeofcoordinatesrenderingthesysteminnormalform˙x=−x(andhenceexponentiallystable),andseveralpartialversionsofthisfacthaveappearedintheliterature,especiallyinthecontextofgeneralizednotionsofhomogeneityfornonlinearsystems;seeforinstance[6,25,15,27,24].Itisperhapssurprisingthat,atleastforunperturbedsystems,thefullresultseemsnottohavebeenobservedbefore,astheproofisafairlyeasyapplicationofresultsfromdifferentialtopology.(Thoseresultsarenontrivial,andarerelatedtothegeneralizedPoincar´econjectureandcobordismtheory;infact,thereasonthatweonlymakeanassertionfor6=4,5iscloselyrelatedtothefactthattheoriginalPoincar´econjectureisstillopen.)Note,however,thatithasbeencommonpracticeinthepaperstreatingthenonlinearcasetousetheflowgeneratedbytheoriginalsystemtodefineanequivalencetransformation,therebyreducingtheregularityofthetransformationtothatofthesystem.Hereweusetheflowgeneratedbythe(normalized)Lyapunovfunctionitself,whichyieldsmoreregulartransformations.Inaddition,andmostimportantly,ourpoofalsoallowsforthetreatmentofperturbedsystems(forwhichthereductionto˙x=−xmakesnosense).Lyapunov’snotionistheappropriategeneralizationofexponentialstabilitytononlineardifferentialequations.Forsystemswithinputs,thenotionofinputtostatestability(ISS)introducedin[29]anddevelopedfurtherin[5,9,13,14,17,18,26,28,32,33]andotherreferences,hasbeenproposedasanonlineargeneralizationoftherequirementoffiniteL2gainor,asoftenalsotermedbecauseofthespectralcharacterizationsvalidforlinearsystems,“finitenonlinearH∞gain”(forwhichseee.g.[2,11,12,34]).Wealsoshowinthispaperthatundercoordinatechanges(nowinbothstateandinputvariables),thetwoproperties(ISSandfiniteH∞gain)coincide(again,assumingdimension6=4,5).2Wedonotwishtospeculateabouttheimplicationsofthematerialpresentedhere.Obviously,thereareno“practical”consequences,sincefindingatransformationintoanexponentiallystablesystemisnoeasierthanestablishingstability(viaaLyapunovfunction).PerhapstheseremarkswillbeofsomeuseinthefurthertheoreticaldevelopmentofISSandotherstabilityquestions.Inanycase,theyservetofurtherjustifythenaturalityofLyapunov’sideasandofconceptsderivedfromhiswork.2SetupWeconsiderthefamilyofdiff