Functional equations with causal operators

185 Pages·2002·0.855 MB·English

by  C. Corduneanu

Functional Equations with
Causal Operators
StabilityandControl:Theory,MethodsandApplications
Aseriesofbooksandmonographsonthetheoryofstabilityandcontrol
EditedbyA.A.Martynyuk
InstituteofMechanics,Kiev,Ukraine
V.Lakshmikantham
FloridaInstituteofTechnology,USA
Volume1
TheoryofIntegro-DifferentialEquations
V.LakshmikanthamandM.RamaMohanaRao
Volume2
StabilityAnalysis:NonlinearMechanicsEquations
A.A.Martynyuk
Volume3
StabilityofMotionofNonautonomousSystems(MethodofLimitingEquations)
J.Kato,A.A.MartynyukandA.A.Shestakov
Volume4
ControlTheoryanditsApplications
E.O.Roxin
Volume5
AdvancesinNonlinearDynamics
EditedbyS.SivasundaramandA.A.Martynyuk
Volume6
SolvingDifferentialProblemsbyMultistepInitialandBoundaryValueMethods
L.BrugnanoandD.Trigiante
Volume7
DynamicsofMachineswithVariableMass
L.Cveticanin
Volumes8
OptimizationofLinearControlSystems:AnalyticalMethodsandComputationalAlgorithms
F.A.AlievandV.B.Larin
Volume9
DynamicsandControl
EditedbyG.Leitmann,F.E.UdwadiaandA.V.Kryazhimskii
Volume10
VolterraEquationsandApplications
EditedbyC.CorduneanuandI.W.Sandberg
Volume11
NonlinearProblemsinAviationandAerospace
EditedbyS.Sivasundaram
Volume12
StabilizationofProgrammedMotion
E.Ya.Smirnov
PleaseseethebackofthisbookforothertitlesintheStabilityandControl: Theory,
MethodsandApplicationsseries.
Functional Equations with
Causal Operators
C. Corduneanu
London and New York
Firstpublished2002
byTaylor&Francis
11NewFetterLane,LondonEC4P4EE
SimultaneouslypublishedintheUSAandCanada
byTaylor&FrancisInc,
29West35thStreet,NewYork,NY10001
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Contents
IntroductiontotheSeries vii
Preface ix
1 Introduction 1
1.1 Classesoffunctionalequations 1
1.2 Equationswithcausaloperators 4
1.3 Causaloperatorsinapplications 7
2 Auxiliaryconcepts 10
2.1 Abstractspaces 10
2.2 Functionspaces 16
2.3 Operatorsonfunctionspaces 22
2.4 Fixedpointsandotherauxiliaryresults 28
3 Existencetheoryforfunctionalequationswith
causaloperators 31
3.1 Theequationx(t)=(Vx)(t)inthespaceofcontinuousfunctions 31
3.2 Theequationx(t)=(Vx)(t)inspacesofmeasurablefunctions 37
3.3 Existenceandapproximationofsolutionsbymeansofthesingular
perturbationtechnique 43
3.4 Globalexistenceresultsforfunctionalequationswithcausal
operators 46
3.5 Someglobalresultsofexistenceanduniqueness 56
3.6 Functionalinequalities 65
3.7 Initialvalueproblemsofthesecondkindforfunctional
differentialequations 71
4 Linearandquasilinearequationswithcausaloperators 79
4.1 Globalexistenceanduniquenessforlinearfunctional
differentialequations 79
4.2 Globalexistenceanduniquenessforlinearfunctionalequations 81
4.3 Integralrepresentationofsolutionsoflinearfunctional
differentialequations 84
vi Contents
4.4 Initialvalueproblemswithfunctionaldataforlinearfunctional
differentialequations 89
4.5 Quasilinearfunctionalequations 92
4.6 Two-pointboundaryvalueproblems 98
4.7 Commentsandreferences 100
5 Stabilitytheory 101
5.1 Definitionandgeneralities 101
5.2 Stabilityoflinearsystems 103
5.3 Stabilityofquasilinearsystems 110
5.4 Comparisonmethodinstability 114
5.5 Admissibilityconcepts 119
6 Neutralfunctionalequations 123
6.1 Theconceptofneutralfunctionalequationwithcausaloperators 123
6.2 Existenceresultsinthecontinuouscase 124
6.3 Existenceresultsinspacesofmeasurablefunctions 129
6.4 Moreresultsonneutralfunctionaldifferentialequations 131
6.5 Thelinearandquasilinearcases 134
7 Miscellanea(applicationsandgeneralizations) 137
7.1 Alinear–quadraticoptimalcontrolproblemwithcausaloperators 137
7.2 Amaximumprincipleapproach 142
7.3 Asymptoticbehaviorinsecond-ordersystems 144
7.4 Globalexistencefortheequationx(t)=(Lx)(t)+(Nx)(t) 148
7.5 Reviewoffurtherresultsandtopics 150
Appendix 155
References 160
Index 167
Introduction to the Series
Theproblemsofmodernsocietyarebothcomplexandinterdisciplinary.Despitetheapparent
diversityofproblems,toolsdevelopedinonecontextareoftenadaptabletoanentirelydif-
ferent situation. For example, consider the Lyapunov’s well known second method. This
interestingandfruitfultechniquehasgainedincreasingsignificanceandhasgivenadecisive
impetusformoderndevelopmentofthestabilitytheoryofdifferentialequations.Amanifest
advantageofthismethodisthatitdoesnotdemandtheknowledgeofsolutionsandtherefore
hasgreatpowerinapplication.ItisnowwellrecognizedthattheconceptofLyapunov-like
functionsandthetheoryofdifferentialandintegralinequalitiescanbeutilizedtoinvestigate
qualitativeandquantitativepropertiesofnonlineardynamicsystems. Lyapunov-likefunc-
tionsserveasvehiclestotransformthegivencomplicateddynamicsystemsintoarelatively
simplersystemandthereforeitissufficienttostudythepropertiesofthissimplerdynamic
system.Itisalsobeingrealizedthatthesameversatiletoolscanbeadaptedtodiscussentirely
differentnonlinearsystems,andthatothertools,suchasthevariationofparametersandthe
methodofupperandlowersolutionsprovideequallyeffectivemethodstodealwithproblems
ofasimilarnature.Moreover,interestingnewideashavebeenintroducedwhichwouldseem
toholdgreatpotential.
Control theory, on the other hand, is that branch of application-oriented mathematics
thatdealswiththebasicprinciplesunderlyingtheanalysisanddesignofcontrolsystems.
To control an object implies the influence of its behavior so as to accomplish a desired
goal.Inordertoimplementthisinfluence,practitionersbuilddevicesthatincorporatevar-
ious mathematical techniques. The study of these devices and their interaction with the
object being controlled is the subject of control theory. There have been, roughly speak-
ing, twomainlinesofworkincontroltheorywhicharecomplementary. Oneisbasedon
the idea that a good model of the object to be controlled is available and that we wish
tooptimizeitsbehavior, andtheotherisbasedontheconstraintsimposedbyuncertainty
about the model in which the object operates. The control tool in the latter is the use of
feedbackinordertocorrectfordeviationsfromthedesiredbehavior.Mathematically,sta-
bilitytheory,dynamicsystemsandfunctionalanalysishavehadastronginfluenceonthis
approach.
Volume 1, Theory of Integro-Differential Equations, is a joint contribution by
V.Lakshmikantham(USA)andM.RamaMohanaRao(India).
Volume 2, Stability Analysis: Nonlinear Mechanics Equations, is by A.A. Martynyuk
(Ukraine).
viii IntroductiontotheSeries
Volume 3, Stability of Motion of Nonautonomous Systems: The Method of Limiting
Equations,isacollaborativeworkbyJ.Kato(Japan),A.A.Martynyuk(Ukraine)and
A.A.Shestakov(Russia).
Volume4,ControlTheoryanditsApplications,isbyE.O.Roxin(USA).
Volume 5, Advances in Nonlinear Dynamics, is edited by S. Sivasundaram (USA) and
A.A.Martynyuk(Ukraine)andisamultiauthorvolumededicatedtoProfessorS.Leela.
Volume6,SolvingDifferentialProblemsbyMultistepInitialandBoundaryValueMethods,
isajointcontributionbyL.Brugnano(Italy)andD.Trigiante(Italy).
Volume7,DynamicsofMachineswithVariableMass,isbyL.Cveticanin(Yugoslavia).
Volume8,OptimizationofLinearControlSystems:AnalyticalMethodsandComputational
Algorithms,isajointworkbyF.A.Aliev(Azerbaijan)andV.B.Larin(Ukraine).
Volume9,DynamicsandControl,iteditedbyG.Leitmann(USA),F.E.Udwadia(USA)
andA.V.Kryazhimskii(Russian)andisamultiauthorvolume.
Volume10,VolterraEquationsandApplications,iseditedbyC.Corduneanu(USA)and
I.W.Sandberg(USA)andisamultiauthorvolume.
Volume11,NonlinearProblemsinAviationandAerospace,iseditedbyS.Sivasundaram
(USA)andisamultiauthorvolume.
Volume12,StabilizationofProgrammedMotion,isbyE.Ya.Smirnov(Russia).
Volume 13, Advances in Stability Theory at the end of the 20th Century, is edited by
A.A.Martynyuk.
Volume 14, Dichotomies and Stability in Nonautonomous Linear Systems, is by Yu.A.
Mitropolskii,A.M.SamoilenkoandV.L.Kulik.
Volume15, AlmostPeriodicSolutionsofDifferentialEquationsinBanachSpaces, isby
YoshiyukiHino,ToshikiNaito,NguyenVanMinhandJongSonShin
Volume16,FunctionalEquationswithCausalOperators,isbyC.Corduneanu.
Due to the increased interdependency and cooperation among the mathematical sciences
across the traditional boundaries, and the accomplishments thus far achieved in the areas
ofstabilityandcontrol,thereiseveryreasontobelievethatmanybreakthroughsawaitus,
offeringexistingprospectsfortheseversatiletechniquestoadvancefurther.Itisinthisspirit
that we see the importance of the “Stability and Control” series, and we are immensely
thankfultoTaylorandFrancisfortheirinterestandcooperationinpublishingthisseries.
Preface
Thisbookisdedicatedtotheinvestigationoffunctionalorfunctionaldifferentialequations
involving causal operators. These operators are also called nonanticipative, or abstract
Volterraoperators.Theterm“causal”isprevalentintheengineeringliterature.
Thedefinitionofcausaloperatorsisverysimple:anoperatorV,actingonagivenfunction
spaceE([0,T],Rn),iscalledcausal,ifforanypairoffunctionx,y ofE,suchthatx and
y coincideonaninterval[0,t],t ≤ T,Vx andVy alsocoincideonthatinterval.Inother
words,thevaluesofVxuptoagivenpointtaredeterminedonlybythevaluestakenbythe
functionxontheinterval[0,t].
TheideaofconsideringsuchoperatorsappearsimplicitlyinVolterra’swork,butasharp
definitionandfurtherconsiderationappearinthepaperofL.Tonelli[1].Inthispaper,the
functionalequation
x(t)=f(t)+A(t,xt(s))
0
isconsidered,wherethesecondterminparanthesesmeanstherestrictionofthefunctionx
totheinterval[0,t]. ThenotationisobviouslyinspiredbyVolterra’swork, andtheabove
equationremindsusinstantlyoftheVolterraintegralequation
(cid:1)
t
x(t)=f(t)+ k(t,s,x(s))ds.
0
Tonelli’spaperwasdedicatedtoprovingtheexistenceanduniquenessofthesolutionof
thefunctionalequationhehasdevisedbymeansofcausaloperators.Theequationhasbeen
investigatedinthespaceofcontinuousfunctionsandthehypothesesareformulatedinsuch
awaythatthecompactnessoftheoperatorAisassured.ThisresultofTonelliis,verylikely,
thefirstexistenceresultforequationswithgeneralcausaloperators.
Thenextsignificantstepindevelopingthetheoryoffunctionalequationsinvolvingcauasal
operatorswasmadein1938byA.N.Tychonoff[1]. ThedefinitiongivenbyTychonoffto
causaloperatorsisasformulatedabove,andbesidestheexistenceofsolutionstheimportance
ofthesetypesofoperatorsorequationsforotherfieldsisemphasized.
Inretrospect, itmayappearsomewhatstrangethattheconceptofacausal(or, asboth
TonelliandTychonoffcallit, Volterra)operatordidnotattracttheimmediateattentionof
researchers. A possible explanation may be that at the time this concept was advanced,
therelativelynewmethodsoffunctionalanalysisdidnotconstitutethemaintoolofmany
investigators.
Gradually,thetheoryoffunctionalequationswithcausaloperatorshascaughttheattention
ofresearchers.Fromthe1960swementionthepapersbyR.Driver[1],C.Corduneanu[2],