Stability of dynamical systems: continuous, discontinuous, and discrete systems

515 Pages·2008·2.786 MB·English

by  Anthony N. Michel, Ling Hou, Derong Liu

Systems & Control: Foundations & Applications
SeriesEditor
TamerBas¸ar,UniversityofIllinoisatUrbana-Champaign
EditorialBoard
KarlJohanA˚stro¨m,LundUniversityofTechnology,Lund,Sweden
Han-FuChen,AcademiaSinica,Beijing
WilliamHelton,UniversityofCalifornia,SanDiego
AlbertoIsidori,UniversityofRome(Italy)and
WashingtonUniversity,St.Louis
PetarV.Kokotovic´,UniversityofCalifornia,SantaBarbara
AlexanderKurzhanski,RussianAcademyofSciences,Moscow
andUniversityofCalifornia,Berkeley
H.VincentPoor,PrincetonUniversity
MeteSoner,Koc¸ University,Istanbul
Anthony N. Michel
Ling Hou
Derong Liu
Stability of
Dynamical Systems
Continuous, Discontinuous,
and Discrete Systems
Birkha¨user
Boston • Basel • Berlin
AnthonyN.Michel LingHou
DepartmentofElectricalEngineering DepartmentofElectricaland
UniversityofNotreDame ComputerEngineering
NotreDame,IN46556 St.CloudStateUniversity
U.S.A. St.Cloud,MN56301
U.S.A.
DerongLiu
DepartmentofElectricaland
ComputerEngineering
UniversityofIllinoisatChicago
Chicago,IL60607
U.S.A.
MathematicsSubjectClassification:15-XX,15A03,15A04,15A06,15A09,15A15,15A18,15A21,
15A42, 15A60, 15A63, 26-XX, 26Axx, 26A06, 26A15, 26A16, 26A24, 26A42, 26A45, 26A46,
26A48,26Bxx,26B05,26B10,26B12,26B20,26B30,26E05,26E10,26E25,34-XX,34-01,34Axx,
34A12,34A30,34A34,34A35,34A36,34A37,34A40,34A60,34Cxx,34C25,34C60,34Dxx,34D05,
34D10,34D20,34D23,34D35,34D40,34Gxx,34G10,34G20,34H05,34Kxx,34K05,34K06,34K20,
34K30,34K40,35-XX,35Axx,35A05,35Bxx,35B35,35Exx,35E15,35F10,35F15,35F25,35F30,
35Gxx,35G10,35G15,35G25,35G30,35Kxx,35K05,35K25,35K30,35K35,35Lxx,35L05,35L25,
35L30,35L35,37-XX,37-01,37C75,37Jxx,37J25,37N35,39-XX,39Axx,39A11,45-XX,45A05,
45D05,45J05,45Mxx,45M10,46-XX,46Bxx,46B25,46Cxx,46E35,46N20,47-XX,47Axx,47A10,
47B44, 47Dxx, 47D03, 47D06, 47D60, 47E05, 47F05, 47Gxx, 47G20, 47H06, 47H10, 47H20,
54-XX, 54E35, 54E45, 54E50, 70-XX, 70Exx, 70E50, 70Hxx, 70H14, 70Kxx, 70K05, 70K20,
93-XX,93B18,93C10,93C15,93C20,93C23,93C62,93C65,93C73,93Dxx,93D05,93D10,93D20,
93D30
LibraryofCongressControlNumber:2007933709
ISBN-13:978-0-8176-4486-4 e-ISBN-13:978-0-8176-4649-3
Printedonacid-freepaper.
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Contents
Preface xi
1 Introduction 1
1.1 DynamicalSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ABriefPerspectiveontheDevelopmentofStabilityTheory . . . . 4
1.3 ScopeandContentsoftheBook . . . . . . . . . . . . . . . . . . . 6
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 DynamicalSystems 17
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 DynamicalSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 OrdinaryDifferentialEquations . . . . . . . . . . . . . . . . . . . 20
2.4 OrdinaryDifferentialInequalities. . . . . . . . . . . . . . . . . . . 26
2.5 DifferenceEquationsandInequalities . . . . . . . . . . . . . . . . 26
2.6 DifferentialEquationsandInclusionsDefinedonBanachSpaces . . 28
2.7 FunctionalDifferentialEquations. . . . . . . . . . . . . . . . . . . 31
2.8 VolterraIntegrodifferentialEquations . . . . . . . . . . . . . . . . 34
2.9 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.10 PartialDifferentialEquations . . . . . . . . . . . . . . . . . . . . . 46
2.11 CompositeDynamicalSystems . . . . . . . . . . . . . . . . . . . . 51
2.12 DiscontinuousDynamicalSystems . . . . . . . . . . . . . . . . . . 52
2.13 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 FundamentalTheory: ThePrincipalStabilityandBoundedness
ResultsonMetricSpaces 71
3.1 SomeQualitativeCharacterizationsofDynamicalSystems . . . . . 73
3.2 ThePrincipalLyapunovandLagrangeStabilityResultsfor
DiscontinuousDynamicalSystems . . . . . . . . . . . . . . . . . . 82
vii
viii Contents
3.3 ThePrincipalLyapunovandLagrangeStabilityResultsfor
ContinuousDynamicalSystems . . . . . . . . . . . . . . . . . . . 92
3.4 ThePrincipalLyapunovandLagrangeStabilityResultsfor
Discrete-TimeDynamicalSystems . . . . . . . . . . . . . . . . . . 103
3.5 ConverseTheoremsforDiscontinuousDynamicalSystems . . . . . 112
3.6 ConverseTheoremsforContinuousDynamicalSystems . . . . . . 125
3.7 ConverseTheoremsforDiscrete-TimeDynamicalSystems . . . . . 133
3.8 Appendix: SomeBackgroundMaterialonDifferentialEquations . . 137
3.9 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4 FundamentalTheory: SpecializedStabilityandBoundedness
ResultsonMetricSpaces 149
4.1 AutonomousDynamicalSystems . . . . . . . . . . . . . . . . . . . 149
4.2 InvarianceTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.3 ComparisonTheory . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.4 UniquenessofMotions . . . . . . . . . . . . . . . . . . . . . . . . 165
4.5 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5 ApplicationstoaClassofDiscrete-EventSystems 173
5.1 AClassofDiscrete-EventSystems . . . . . . . . . . . . . . . . . . 173
5.2 StabilityAnalysisofDiscrete-EventSystems . . . . . . . . . . . . 175
5.3 AnalysisofaManufacturingSystem . . . . . . . . . . . . . . . . . 176
5.4 LoadBalancinginaComputerNetwork . . . . . . . . . . . . . . . 179
5.5 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6 Finite-DimensionalDynamicalSystems 185
6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.2 ThePrincipalStabilityandBoundednessResultsforOrdinary
DifferentialEquations . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.3 ThePrincipalStabilityandBoundednessResultsforOrdinary
DifferenceEquations . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.4 ThePrincipalStabilityandBoundednessResultsforDiscontinuous
DynamicalSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.5 ConverseTheoremsforOrdinaryDifferentialEquations. . . . . . . 232
6.6 ConverseTheoremsforOrdinaryDifferenceEquations . . . . . . . 241
Contents ix
6.7 ConverseTheoremsforFinite-DimensionalDDS . . . . . . . . . . 243
6.8 Appendix: SomeBackgroundMaterialonDifferentialEquations . . 245
6.9 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7 Finite-DimensionalDynamicalSystems: SpecializedResults 255
7.1 AutonomousandPeriodicSystems . . . . . . . . . . . . . . . . . . 256
7.2 InvarianceTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
7.3 DomainofAttraction . . . . . . . . . . . . . . . . . . . . . . . . . 263
7.4 LinearContinuous-TimeSystems . . . . . . . . . . . . . . . . . . 266
7.5 LinearDiscrete-TimeSystems . . . . . . . . . . . . . . . . . . . . 285
7.6 PerturbedLinearSystems . . . . . . . . . . . . . . . . . . . . . . . 295
7.7 ComparisonTheory . . . . . . . . . . . . . . . . . . . . . . . . . . 316
7.8 Appendix: BackgroundMaterialonDifferentialEquationsand
DifferenceEquations . . . . . . . . . . . . . . . . . . . . . . . . . 320
7.9 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . 328
7.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
8 ApplicationstoFinite-DimensionalDynamicalSystems 337
8.1 AbsoluteStabilityofRegulatorSystems . . . . . . . . . . . . . . . 338
8.2 HopfieldNeuralNetworks . . . . . . . . . . . . . . . . . . . . . . 344
8.3 DigitalControlSystems. . . . . . . . . . . . . . . . . . . . . . . . 353
8.4 Pulse-Width-ModulatedFeedbackControlSystems . . . . . . . . . 364
8.5 DigitalFilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
8.6 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . 387
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
9 Infinite-DimensionalDynamicalSystems 395
9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
9.2 ThePrincipalLyapunovStabilityandBoundednessResultsfor
DifferentialEquationsinBanachSpaces . . . . . . . . . . . . . . . 398
9.3 ConverseTheoremsforDifferentialEquationsinBanachSpaces . . 408
9.4 InvarianceTheoryforDifferentialEquationsinBanachSpaces . . . 409
9.5 ComparisonTheoryforDifferentialEquationsinBanachSpaces . . 413
9.6 CompositeSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 415
9.7 AnalysisofaPointKineticsModelofaMulticoreNuclearReactor . 420
9.8 ResultsforRetardedFunctionalDifferentialEquations . . . . . . . 423
9.9 ApplicationstoaClassofArtificialNeuralNetworkswithTime
Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
x Contents
9.10 DiscontinuousDynamicalSystemsDeterminedbyDifferential
EquationsinBanachSpaces . . . . . . . . . . . . . . . . . . . . . 449
9.11 DiscontinuousDynamicalSystemsDeterminedbySemigroups . . . 463
9.12 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . 479
9.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
Index 489
Preface
Intheanalysisandsynthesisofcontemporarysystems, engineersandscientistsare
frequently confronted with increasingly complex models that may simultaneously
includecomponentswhosestatesevolvealongcontinuoustime(continuousdynam-
ics)anddiscreteinstants(discretedynamics); componentswhosedescriptionsmay
exhibit hysteresis nonlinearities, time lags or transportation delays, lumped param-
eters, spatiallydistributedparameters, uncertaintiesintheparameters, andthelike;
andcomponentsthatcannotbedescribedbytheusualclassicalequations(ordinary
differentialequations,differenceequations,functionaldifferentialequations,partial
differential equations, and Volterra integrodifferential equations), as in the case of
discrete-eventsystems,logiccommands,Petrinets,andthelike. Thequalitativeanal-
ysisofsystemsofthistypemayrequireresultsforfinite-dimensionalsystemsaswell
asinfinite-dimensionalsystems;continuous-timesystemsaswellasdiscrete-timesys-
tems;continuouscontinuous-timesystemsaswellasdiscontinuouscontinuous-time
systems(DDS);andhybridsystemsinvolvingamixtureofcontinuousanddiscrete
dynamics.
Presently,therearenobooksonstabilitytheorythataresuitabletoserveasasingle
sourcefortheanalysisofsystemmodelsofthetypedescribedabove. Mostexisting
engineeringtextsonstabilitytheoryaddressfinite-dimensionalsystemsdescribedby
ordinary differential equations, and discrete-time systems are frequently treated as
analogousafterthoughts,orarerelegatedtobooksonsampled-datacontrolsystems.
On the other hand, books on the stability theory of infinite-dimensional dynamical
systemsusuallyfocusonspecificclassesofsystems(determined,e.g.,byfunctional
differentialequations,partialdifferentialequations,andsoforth). Finally,theliter-
atureonthestabilitytheoryofdiscontinuousdynamicalsystems(DDS)ispresently
scatteredthroughoutjournalsandconferenceproceedings. Consequently,tobecome
reasonablyproficientinthestabilityanalysisofcontemporarydynamicalsystemsof
thetypedescribedabovemayrequireconsiderableinvestmentoftime. Thepresent
bookaimstofillthisvoid. Toaccomplishthis, thebookaddressesfourgeneralar-
eas: therepresentationandmodelingofavarietyofdynamicalsystemsofthetype
describedabove;thepresentationoftheLyapunovandLagrangestabilitytheoryfor
dynamical systems defined on general metric spaces; the specialization of this sta-
bilitytheorytofinite-dimensionaldynamicalsystems;andthespecializationofthis
stabilitytheorytoinfinite-dimensionaldynamicalsystems. Throughoutthebook,the
applicabilityofthedevelopedtheoryisdemonstratedbymeansofnumerousspecific
examplesandapplicationstoimportantclassesofsystems.
xi