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Stability of Dynamical Systems MONOGRAPH SERIES ON NONLINEAR SCIENCE AND COMPLEXITY SERIESEDITORS AlbertC.J.Luo SouthernIllinoisUniversity,Edwardsville,USA GeorgeZaslavsky NewYorkUniversity,NewYork,USA ADVISORYBOARD ValentinAfraimovich, SanLuisPotosiUniversity,SanLuisPotosi,Mexico MauriceCourbage, UniversitéParis7,Paris,France Ben-JacobEshel, SchoolofPhysicsandAstronomy,TelAvivUniversity, TelAviv,Israel BernoldFiedler, FreieUniversitätBerlin,Berlin,Germany JamesA.Glazier, IndianaUniversity,Bloomington,USA NailIbragimov, IHN,BlekingeInstituteofTechnology,Karlskrona,Sweden AnatolyNeishtadt, SpaceResearchInstituteRussianAcademyofSciences, Moscow,Russia LeonidShilnikov, ResearchInstituteforAppliedMathematics&Cybernetics, NizhnyNovgorod,Russia MichaelShlesinger, OfficeofNavalResearch,Arlington,USA DietrichStauffer, UniversityofCologne,Köln,Germany Jian-QiaoSun, UniversityofDelaware,Newark,USA DimitryTreschev, MoscowStateUniversity,Moscow,Russia VladimirV.Uchaikin, UlyanovskStateUniversity,Ulyanovsk,Russia AngeloVulpiani, UniversityLaSapienza,Roma,Italy PeiYu, TheUniversityofWesternOntario,London,OntarioN6A5B7,Canada Stability of Dynamical Systems XIAOXINLIAO HuazhongUniversityofScienceandTechnology Wuhan430074 China LIQIUWANG TheUniversityofHongKong HongKong HongKong PEIYU TheUniversityofWesternOntario London,Ontario Canada AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•OXFORD PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO Elsevier Radarweg29,POBox211,1000AEAmsterdam,TheNetherlands TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK Firstedition2007 Copyright©2007ElsevierB.V.Allrightsreserved Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinanyform orbyanymeanselectronic,mechanical,photocopying,recordingorotherwisewithoutthepriorwritten permissionofthepublisher PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRightsDepartmentinOxford, UK:phone(+44)(0)1865843830;fax(+44)(0)1865853333;email:[email protected] nativelyyoucansubmityourrequestonlinebyvisitingtheElsevierwebsiteathttp://elsevier.com/locate/ permissions,andselectingObtainingpermissiontouseElseviermaterial Notice Noresponsibilityisassumedbythepublisherforanyinjuryand/ordamagetopersonsorpropertyas amatterofproductsliability,negligenceorotherwise,orfromanyuseoroperationofanymethods, products,instructionsorideascontainedinthematerialherein.Becauseofrapidadvancesinthemedical sciences,inparticular,independentverificationofdiagnosesanddrugdosagesshouldbemade LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-444-53110-0 ISSN:1574-6917 ForinformationonallElsevierpublications visitourwebsiteatbooks.elsevier.com PrintedandboundinTheNetherlands 06 07 08 09 10 10 9 8 7 6 5 4 3 2 1 Preface The main purpose of developing stability theory is to examine the dynamic re- sponses of a system to disturbances as time approaches infinity. It has been and still is the subject of intense investigations due to its intrinsic interest and its relevance to all practical systems in engineering, finance, natural science and social science. Lyapunov stability theory, one celebrated theory, is the founda- tionofstabilityanalysesfordynamicsystemsthataremathematicallydescribed byordinarydifferentialequations(ODE).Inspiredbynumerousapplicationsand new emerging fields, it has been significantly developed and extended to systems that are modeled using difference equations (DE), differential-difference equations (DDE), functional differential equations (FDE), integral-differential equations (IDE), partial differential equations (PDE), and stochastic differential equations(SDE).Forinstance,interestinautomaticcontrolstartinginthe1950s hasgeneratedthetheoriesofglobalandabsolutestability.Theorydescribingthe co-existence and sustainability of ecological systems originated from interest in bio-systemsanalysisbeginninginthe1970s.Interestinartificialneuralnetworks beginninginthe1980shasstimulatedthesolutionofnonlinearequationsbyusing electroniccircuitanalogues. Theevolutionofstabilitytheoryhasbeenveryrapidandextensive.Majorde- velopmentsarescatteredthroughoutanarrayofscientificjournals,makingitoften difficulttodiscoverwhattherealadvancesare,especiallyforaresearchernewto thefieldorapractitionerusingtheresultsinvariousapplicableareas.Therefore, it appears necessary to have monographs on topics of current interest to both researchers and practitioners in the field. The present monograph is intended to provide some state-of-the-art expositions of major advances in fundamentalsta- bility theories and methods for dynamical systems of ODE and DDE types and in limit cycle, normal form and Hopf bifurcation control of nonlinear dynamic systems. The present monograph comes mainly from our research results and teach- ing of graduate students in the stability of dynamical systems. Chapter 1 is the introduction where we define various stabilities mathematically, illustrate their relations using examples and discuss the main mathematical tools for stability analyses(e.g.,Lyapunovfunctions,K-classfunctions,Diniderivatives,differen- tialandintegralinequalitiesandmatrices).InChapter2,were-visitthestability oflinearsystemswithconstantcoefficients,presentanewmethodforsolvingthe Lyapunovmatrixequationanddiscussourgeometricalmethodforstabilityanaly- ses.Chapter3describesthestabilityoflinearsystemswithvariablecoefficients. v vi Preface Particularly, we first develop relations between the stabilities of homogeneous and nonhomogeneous systems, and relations between Cauchy matrix properties andvariousstabilities.Wethendiscusstherobuststability,analyticalexpressions of Cauchy matrix solutions for some linear systems and the Floquet–Lyapunov theoryforlinearsystemswithperiodiccoefficients.Finally,wepresentthetrun- cated Cauchy matrix and partial variable stability. In Chapter 4, we present the LyapunovstabilitytheorybyusingamodernapproachthatemploystheK-class functionandDiniderivative.Thenecessaryandsufficientconditionsaresystem- aticallydevelopedforstability,uniformstability,uniformlyasymptoticstability, exponentialstability,andinstability.WepresentclassicalLyapunovtheoremsof stability and their inverse theorems together to illustrate the universality of the Lyapunov direct method. Also developed in Chapter 4 are some new sufficient conditionsforstability,asymptoticstability,andinstability.Thischapterendswith abriefsummaryconstructingLyapunovfunctions.Chapter5presentsamajorex- tension and development of the Lyapunov direct method, including the LaSalle invariant principle, theory of comparability, robust stability, practical stability, Lipschitz stability, asymptotic equivalence, conditional stability, partial variable stability, stability and the boundedness of sets. In Chapter 5, we also apply the LyapunovfunctiontostudytheclassicalLagrangestability,Lagrangeasymptotic stabilityandLagrangeexponentialstability. Chapter6isdevotedtothestabilityofnonlinearsystemswithseparablevari- ables. The topics covered include linear and nonlinear Lyapunov functions, the globalstabilityofautonomousandnonautonomoussystems,transformationinto systemswithseparablevariables,andpartialvariablestability.Thischapterpro- vides the methods and tools for examining the absolute stability of nonlinear controlsystemsinChapter9andthestabilityofneuralnetworksinChapter10. Chapter7describestheiterationmethodthatusestheconvergenceofiterationfor stabilityanalysesandavoidsthedifficultyencounteredinconstructingLyapunov functions.Particularly,wediscussiterationmethodsofPicardandGauss–Seidel types and their applications in examining the extreme stability and the station- aryoscillation,inimprovingthefreezingcoefficientmethod,andininvestigating the robust stability of interval systems. Dynamical systems with temporal delay are often modeled by differential-difference equations (DDE). The stability of suchsystemsisdiscussedinChapter8.WefirstpresenttheLyapunovfunctional method and the Lyapunov function method with the Razumikhin technique for thestabilityanalysesoftime-delayingnonlineardifferentialequationsandDDE withseparablevariables.WethenapplytheeigenvaluemethodandM matrixthe- ory to develop an algebraic method for modeling the stability of linear systems withconstantcoefficientsandconstanttime-delays.Finally,weusetheiteration methodin Chapter7 toexaminethestabilityofneutralDDEsystems withtem- poraldelays.Chapter9coverstheabsolutestabilityofLuriecontrolsystems.The topicscontainsomealgebraicsufficientconditionsforstability,andthenecessary Preface vii andsufficientconditionsfortheabsolutestabilityofdirect,indirect,criticaland time-delayingLuriecontrolsystemsandfortheabsolutestabilityofLuriecontrol systemswithmultiplenonlinearcontrolsorwithfeedbackloops.Alsodiscussed inthischapteristheapplicationofLurietheoryinchaossynchronization.Chap- ter10focusesonstabilities(Lyapunov,globallyasymptotic,globallyexponential) andexponentialperiodicityofvariousneuralnetworks(Hopfieldwithandwith- outtime-delay,Roskobidirectionalassociativememory,cellular,generalized).In Chapter11,wepresentthecomputationalmethodsofnormalformandlimitcy- cle,thecontrolofHopfbifurcationsandtheirengineeringapplications. WeacknowledgewithgratitudethesupportreceivedfromtheHuazhongUni- versity of Science and Technology (Department of Control Science and Engi- neering), the University of Hong Kong (Department of Mechanical Engineer- ing) and the University of Western Ontario (Department of Applied Mathemat- ics). The support of our research program by the Natural Science Foundation of China (NSFC 60274007 and 60474011), the Research Grant Council of the Hong Kong Special Administration Region of China (RGC HKU7086/00E and HKU7049/06P) and the Natural Sciences and Engineering Research Council of Canada (NSERC R2686A02) is also greatly appreciated. We are very grateful to Dr. Zhen Chen and Mr. Fei Xu who typed part of the manuscript. And, of course,weowespecialthankstoourrespectivefamiliesfortheirtoleranceofthe obsession and the late nights that seemed necessary to bring this monograph to completion.Lookingahead,wewillappreciateitverymuchifuserswillwriteto callourattentiontotheimperfectionsthatmayhaveslippedintothefinalversion. XiaoxinLiao,LiqiuWang,PeiYu London,Canada December2006 This page intentionally left blank Contents Preface v Chapter1. FundamentalConceptsandMathematicalTools 1 1.1. Fundamentaltheoremsofordinarydifferentialequations 1 1.2. Lyapunovfunction 4 1.3. K-classfunction 7 1.4. Diniderivative 10 1.5. Differentialandintegralinequalities 13 1.6. Aunifiedsimpleconditionforstablematrix,p.d.matrixandM matrix 16 1.7. DefinitionofLyapunovstability 21 1.8. Someexamplesofstabilityrelation 24 Chapter2. LinearSystemswithConstantCoefficients 35 2.1. NASCsforstabilityandasymptoticstability 35 2.2. SufficientconditionsofHurwitzmatrix 43 2.3. AnewmethodforsolvingLyapunovmatrixequation:BA+ATB = C 53 2.4. AsimplegeometricalNASCforHurwitzmatrix 61 2.5. Thegeometrymethodforthestabilityoflinearcontrolsystems 69 Chapter3. Time-VaryingLinearSystems 77 3.1. Stabilitiesbetweenhomogeneousandnonhomogeneoussystems 77 3.2. Equivalentconditionforthestabilityoflinearsystems 80 3.3. Robuststabilityoflinearsystems 84 3.4. TheexpressionofCauchymatrixsolution 90 3.5. Linearsystemswithperiodiccoefficients 95 3.6. Spectralestimationforlinearsystems 100 3.7. Partialvariablestabilityoflinearsystems 104 ix