Qualitative Analysis of Set-Valued Differential Equations PDF
Preview Qualitative Analysis of Set-Valued Differential Equations
Anatoly A. Martynyuk Qualitative Analysis of Set-Valued Differential Equations Anatoly A. Martynyuk Qualitative Analysis of Set-Valued Differential Equations AnatolyA.Martynyuk InstituteofMechanics NationalAcademyofSciencesofUkraine Kiev,Ukraine ISBN978-3-030-07643-6 ISBN978-3-030-07644-3 (eBook) https://doi.org/10.1007/978-3-030-07644-3 LibraryofCongressControlNumber:2018968441 Mathematics Subject Classification (2010): 34A34, 34A37, 34A60, 49J24, 49K24, 34D20, 34D40, 34G99,93B12,93B27,93C41,93D09,93D30 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Simplicityis theonlysoilonwhich wecan erect thebuildingofourgeneralizations. A.Poincaré Preface The construction of the theory of dynamic system trajectories is rooted in the classical works by Poincare and Lyapunov.The notion of phase space introduced byGibbsallowsonetoconsiderthemotionofamechanicalsystemasthemotionof animagepointinsomen-dimensionalspaceofconfigurationsor,whatisthesame, inthephasespacewithgivenmetric.Darbouxtreatedadynamicalsystemasapoint movinginn-dimensionalspace.Thisideawassuccessivelyappliedinthepapersby Hertz,wherethetrajectorieswereconsideredasthegeodesiclines.Inthepapersby Painleve,thetrajectoriesofmechanicalsystemswerestudiedinthe contextofthe ideasofmultidimensionalgeometrywiththeuseofEuclideanmetric. The ideas ofmultidimensionalgeometrywereextensivelyappliedin the inves- tigation of the trajectories of mechanical systems in the works by Ricci and Levi-Civita,Sing,andBelenkiyandmanyothers. The recent studies of complex mechanical and electrophysical systems have required some development of the “tool” for modeling the processes in such systems. As is known, the main requirement to the mathematical model of a processisanadequatedescriptionofthesystemfunctioning.Oneoftheapproaches developed for such systems is based on the consideration of systems of ordinary differentialequationswithmultivaluedright-handpart.Oneofthefactorsgenerating multi-valuednessoftheright-handpartisanuncertaintyindeterminationofsystem parameters. In view of these facts, there arises a necessity to study sets (bundles) of trajectories of system of perturbed motion equations. General requirements to the set of systems of equations are its closedness and self-consistency as well as correctnesswithrespecttotheenteringparameters. The construction of the theory is at the initial stage, and there are many open problemstobeinvestigatedinthisarea.Thesolutionofsomeofthemisbroughtto theattentionofthereaders. Themonographconsistsofeightchapters,alistofreferences,andsubjectindex. In the first chapter, we discuss the general properties of equations with a set oftrajectories.Here,weproposeaprocedureforregularizingthesetofinaccurate equations and establish sufficient conditions for the existence and uniqueness of thesetsolutions.Inaddition,wepresentestimatesofthesolutionsoftheperturbed vii viii Preface motionsystemsin which the changeof the state vectoris subjectto a generalized derivative. In the second chapter, for the set of equations with generalized derivative, sufficient conditions for various types of boundedness of trajectories and for the stabilityofastationarysetoftrajectoriesareestablished.Tothisend,thescalarand vectorLyapunovfunctions,constructedon the basis of an auxiliarymatrix-valued function,areused. Inthethirdchapter,forthesetofdiscrete-timeequations,acomparisonprinciple with matrix Lyapunov function is established and sufficient conditions for the stability of a certain type of stationary solution are obtained. The analysis is carried out on the basis of the matrix Lyapunov function of a special structure. Foressentiallynonlinearmultiplyconnecteddifferencesystemwithswitching,we establish the conditionsthat guaranteethe asymptotic stability of its zero solution underanyswitchinglaw. In the fourth chapter, for the family of impulse equations, a heterogeneous matrixLyapunovfunctionis considered,atheoremoftheprincipleofcomparison isobtained,andstabilityconditionsforasetofstationarysolutionsareestablished. Inthefifthchapter,weconsidersetsofequationswithaftereffectanduncertain values of parameters. As a result of regularization of the family of equations according to the scheme adopted in the book, a set of equations with aftereffect isobtained,forwhichtheconditionsfortheexistenceofsolutionsareestablished, theestimateofthedistancebetweentheextremesetsofsolutionsisgiven,andthe stability conditions for the set of stationary solutions on a finite-time interval are foundaswellasthedampingconditionsforthesetoftrajectories. Inthesixthchapter,thefamiliesofequationswithaftereffectanduncertainvalues ofparametersareconsidered.Asaresultofregularizationofthefamilyofequations, a new family of equations with aftereffect is obtained, for which conditions for the existence of solutionsare established, an estimate of the distance between the extremesetsofsolutionsisgiven,andthestabilityconditionsforthesetofstationary solutionson a finite-time intervalare foundas wellas the dampingconditionsfor thesetoftrajectories. The seventh chapter presents the results of dynamic analysis of the family equationswithacausalrobustoperator.Theconditionsoflocalandglobalexistence of solutions of the regularized equation are found, an estimate of the funnel for theset oftrajectoriesis given,andthe stability conditionsfortheset ofstationary solutionsare found.The generalizeddirectLyapunovmethodand the comparison principlewiththeLyapunovmatrixfunctionareapplied. In Chap.8, for standard form nonlinear equations with generalized derivative, estimates of deviation of the set of exact solutions from the averaged ones are established and the deviation of the set of trajectoriesof averagedequationsfrom theequilibriumstateisspecifiedintermsofpseudo-linearintegralinequalities.Sets ofaffinesystemsandproblemsofapproximateintegrationandstabilityoverfinite intervalareconsideredasapplications. Preface ix Inthisbook,forthefirsttime,wepresent: 1. A procedureforregularizinga familyofequationswithrespecttoan uncertain parameter 2. A techniquefor estimatingthe “funnel”solutionsof S.A.Chaplyginfor the set oftrajectoriesofthefamiliesofequations 3. AgeneralizationofthedirectLyapunovmethodonthebasisofthematrix-valued functionforadynamicalanalysisoffamiliesofequations 4. Theideaofapseudo-linearrepresentationofnonlinearintegralinequalitieswith respecttotheproblemsofdeviationofthesetoftrajectoriesfromtheequilibrium state 5. The stability conditions for the set of trajectories of the family of difference equationsforanyswitchinglaw Thisbookisdesignedfortheexpertsworkinginthefieldofqualitativeanalysis ofdifferentialandothertypesofequations.Theapplicationofgeneralresultsgiven inthebookisassociatedwiththeneedtoconstructsuitableLyapunovfunctionson compact convex spaces or the use of classical results of the theory of differential andintegralinequalities. Acknowledgments Acquainted with some chapters of this book are Professor N.A.Izobov, an academician of the National Academy of Sciences of Belarus, and Professors A.Yu.Aleksandrov (Russia), T.A.Burton (USA), I.M.Stamova (USA), and A.S.Vatsala (USA). Their comments and discussion of specific issues have contributedtoimprovingthepresentationoftheresults. Employeesof the Processes Stability Departmentof the S.P.TimoshenkoInsti- tuteofMechanicsNASofUkraine,L.N.ChernetskayaandS.N.Rasshyvalova,have doneagreatworkonthepreparationofthismanuscriptforprinting. Myheartfeltgratitudetoallthescientistsandspecialistsmentionedforthework done. Finally,wearegratefultotheeditorsandproductionstaffofBirkhäuserfortheir assistance,goodideas,andpatienceinthepublicationofthisbook. Kiev,Ukraine AnatolyA.Martynyuk 2018 Contents 1 GeneralPropertiesofSet-ValuedEquations.............................. 1 1.1 Introduction............................................................ 1 1.2 ElementsofMultivaluedAnalysis.................................... 2 1.3 RegularizationofaFamilyofUncertainEquations.................. 6 1.4 EstimatesfortheFunnelofFamilyEquations ....................... 8 1.5 ExistenceConditionsforaSetofTrajectories ....................... 13 1.6 MonotoneIterativeTechnique ........................................ 17 1.7 IterativeTechniqueforaFamilyofEquations ....................... 20 1.8 ConditionsforGlobalExistenceofSolutions........................ 26 1.9 ApproximateSolutionoftheFamilyEquations...................... 28 1.10 EulerSolutionsfortheFamilyofEquations(1.4) ................... 29 1.11 InvarianceoftheSetofEulerSolutions.............................. 32 1.12 DeviationofTrajectoriesfromtheEquilibriumState................ 36 1.13 NotesandReferences.................................................. 44 2 AnalysisofContinuousEquations ......................................... 47 2.1 Introduction............................................................ 47 2.2 IdeaswithManyApplications ........................................ 48 2.3 Lyapunov-LikeFunctionsandTheirApplications................... 53 2.4 StabilityoftheSetofStationarySolutions........................... 56 2.5 TheoremsonStability................................................. 58 2.6 ApplicationofStrengthenedLyapunovFunction.................... 65 2.7 TheoremsonBoundedness............................................ 70 2.8 DifferentialEquationsonProductofConvexSpaces................ 73 2.9 Hyers–Ulam–RassiasStabilityoftheSetofEquations.............. 79 2.10 NotesandReferences.................................................. 83 3 Discrete-TimeSystemswithSwitching.................................... 85 3.1 Introduction............................................................ 85 3.2 Preliminaries........................................................... 86 3.3 StatementoftheProblem.............................................. 87 3.4 StructureofAuxiliaryMatrixFunction............................... 88 xi xii Contents 3.5 StabilityofaStationarySolution ..................................... 89 3.6 Multi-ConnectedSwitchedDifferenceSystem ...................... 91 3.7 ConstructionofaComparisonSystem ............................... 92 3.8 ConstructionofaCommonLyapunovFunction ..................... 94 3.9 Example................................................................ 98 3.10 NotesandReferences.................................................. 100 4 QualitativeAnalysisofImpulsiveEquations ............................. 103 4.1 Introduction............................................................ 103 4.2 Preliminaries........................................................... 104 4.3 ComparisonPrinciple ................................................. 106 4.4 StabilityAnalysis...................................................... 108 4.5 MonotoneIterativeTechnique ........................................ 113 4.6 NotesandReferences.................................................. 121 5 StabilityofSystemswithAftereffect ...................................... 123 5.1 Introduction............................................................ 123 5.2 StatementoftheProblemandDesignations.......................... 124 5.3 ExistenceoftheSetofSolutions...................................... 125 5.4 FunnelfortheSetofTrajectories..................................... 126 5.5 Finite-TimeStability................................................... 128 5.6 DampingTimefortheSetTrajectories............................... 132 5.7 NotesandReferences.................................................. 134 6 ImpulsiveSystemswithAftereffect........................................ 135 6.1 Introduction............................................................ 135 6.2 Preliminaries........................................................... 136 6.3 ComparisonPrinciple ................................................. 137 6.4 StabilityoftheSetofStationarySolutions........................... 141 6.5 FunnelfortheSetofUncertainEquations ........................... 143 6.6 RegularizationoftheSetofEquationswithAftereffect............. 145 6.7 NotesandReferences.................................................. 149 7 DynamicsofSystemswithCausalOperator.............................. 151 7.1 Introduction............................................................ 151 7.2 Preliminaries........................................................... 152 7.3 RegularizationoftheSetofCausalEquations....................... 152 7.4 FunnelfortheSetofSolutionsofCausalEquations ................ 153 7.5 GlobalExistenceofSolutions......................................... 156 7.6 ComparisonPrinciple ................................................. 161 7.7 StabilityAnalysis...................................................... 162 7.8 Hyers–Ulam–RassiasStabilityoftheSetofCausalEquations...... 167 7.9 NotesandReferences.................................................. 170 8 Finite-TimeStabilityofStandardSystemsSets .......................... 173 8.1 Introduction............................................................ 173 8.2 StatementoftheProblem.............................................. 174
The list of books you might like
