Fuzzy Dynamic Equations, Dynamic Inclusions, and Optimal Control Problems on Time Scales

882 Pages·2021·5.275 MB·English

by  Svetlin G. Georgiev

Svetlin G. Georgiev
Fuzzy Dynamic
Equations, Dynamic
Inclusions and
Optimal Control
Problems on Time
Scales
Fuzzy Dynamic Equations, Dynamic Inclusions and
Optimal Control Problems on Time Scales
Svetlin G. Georgiev
Fuzzy Dynamic Equations,
Dynamic Inclusions
and Optimal Control
Problems on Time Scales
SvetlinG.Georgiev
FacultyofMathematics&Info
SofiaUniversityStKlimentOhridski
Sofia,Bulgaria
ISBN978-3-030-76131-8 ISBN978-3-030-76132-5 (eBook)
https://doi.org/10.1007/978-3-030-76132-5
MathematicsSubjectClassification:34A60,34N05,34A37,34B37
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Preface
Thetheoryofdynamicequationshasmanyinterestingapplicationsincontroltheory,
mathematical economics, mathematical biology, engineering, and technology. In
somecases,thereexistsuncertainty,ambiguity,orvaguefactorsinsuchproblems,
and fuzzy theory and interval analysis are powerful tools for modeling these
equations on time scales. The aim of this book is to present a systematic account
of recent developments; describe the current state of the useful theory; show the
essentialunityachievedinthetheoryfuzzydynamicequations,dynamicinclusions,
andoptimalcontrolproblemsontimescales;andinitiateseveralnewextensionsto
othertypesoffuzzydynamicsystemsanddynamicinclusions.
The book contains 16 chapters. Chapter 1 deals with fuzzy functions on time
scales. We focus on a new class derivative and a new class integral of such fuzzy
functions. The corresponding fundamental properties of the introduced derivative
and integral are studied and discussed. In this chapter, we introduce the concept
for shifts operators. They are defined shift almost periodic fuzzy functions and
some of their properties are deduced. In Chap.2, certain classes linear fuzzy
dynamic equations are considered, and we give formulas for their solutions. They
are proved some results for existence and uniqueness of the solutions of some
classes nonlinear fuzzy dynamic equations. The continuous dependence of the
solutions of first order fuzzy dynamic equations on the initial data is investigated.
Some comparison results are provided. The Lyapunov functional is introduced,
and criteria for stability and exponential stability of the trivial solution of some
classesnonlinearfuzzydynamicequationsareprovedbyusingit.InChap.3,some
classeslinearsecondorderfuzzydynamicequationsareinvestigatedandformulas
for their solutions are deduced. They are formulateds and proved some existence
and uniqueness results. The continuous dependence of the solutions of second
order fuzzy dynamic equations on the initial data is investigated. Some classes
of BVPs for second order fuzzy dynamic equations are investigated and formulas
for their solutions are given. In Chap.4, some basic results for periodic functions
on the periodic time scales are formulated and proved. A phase space is built
for fuzzy dynamic equations with infinite delays on the periodic time scales. By
usingthemonotonemethods,wesystematicallyconsidertheexistenceofperiodic
v
vi Preface
solutionsforthefuzzydynamicequationswithinfinitedelaysontheperiodictime
scales, which generalize and incorporate, as special cases, some known results
for fuzzy differential equations and fuzzy difference equations. In Chap.5, linear
first order impulsive fuzzy dynamic equations on time scales are considered, and
we deduce formulas for their solutions. In the chapter, some classes nonlinear
first order impulsive fuzzy dynamic equations are investigated, and existence and
uniqueness of their solutions are proved. We introduce the conception for stable
trivial solution of some classes nonlinear first order impulsive fuzzy dynamic
equations and some stability criteria are deduced. In Chap.6, the concept for Δ-
measurable functions ontimescalesandΔ-Lebesgue integration ontimescalesis
introduced.Inthechapter,someofthepropertiesoftheΔ-measurablefunctionsand
the Δ-Lebesgue integral are deduced. The spaces L1(T) and the Sobolev spaces
are introduced. The concept for absolutely continuous functions on time scales is
establishedandacharacterizationofsuchfunctionsisprovided.Inthischapter,we
introduceweaksolutionsandtheEulersolutionsforfirstorderdynamicequations
on time scales and an analogue of the Gronwall inequality is formulated and
proved. In Chap.7, some IVPs for first order dynamic inclusions are investigated.
Some classes first order dynamic inclusions with nonlocal initial conditions are
investigated for existence of solutions. In the chapter, BVPs for some classes first
orderdynamicinclusionswithgeneralboundaryconditionsandperiodicboundary
conditions are studied. Also, the dual time scales are introduced and a method
for investigations for existence of solutions of first order dynamic inclusions via
dualityisproposed.InChap.8,aperiodicboundaryvalueproblemforsecondorder
dynamicinclusionsforexistenceofsolutionsisinvestigated.Inthechapter,m-point
boundary value problem for second order dynamic inclusions is considered and
some existence results are deduced. In Chap.9, some classes first order impulsive
dynamicinclusionsareinvestigated.Theyareintroducedloweranduppersolutions
for first order impulsive dynamic inclusions. They are deducted some existence
results for some classes boundary value problems for different classes first order
impulsivedynamicinclusions.InChap.10,thecontrollabilityandobservabilityof
linearequationsandthebang-bangprincipleareinvestigated.Chapter11isdevoted
on linear time optimal control. The existence of time-optimal controls and the
maximum principle for linear time-optimal control are proven. Chapter 12 deals
withthePontryaginmaximumprinciple.Adjointlineardynamics,variationsofthe
control, simple control variations, and the free endpoint problem are investigated.
Dynamic programming is introduced in Chap.13. The Hamilton-Jacobi-Bellman
partial dynamic equations are deduced and the dynamic programming method is
deduced.ThedynamicgamesareintroducedandtheIsaacsequationsarededuced.
Chapter 14 is devoted on weak solutions and optimal control problems for some
classeslinearfirstorderdynamicsystems.Nonlineardynamicequationsandoptimal
control problems are investigated in Chap.15 and some necessary conditions for
optimality are provided. In Chap.16, nonlinear integro-dynamic equations are
investigatedandsomenecessaryconditionsforoptimalityarededuced.
Preface vii
The text material of this book is presented in highly readable, mathematically
solid format. Many practical problems are illustrated, displaying a wide variety of
solutiontechniques.
Theauthorwelcomessuggestionsfortheimprovementofthetext.
Paris,France SvetlinG.Georgiev
March2021
Contents
1 CalculusofFuzzyFunctions .............................................. 1
1.1 FirstTypeFuzzyDeltaDifferentiation.............................. 1
1.2 SecondTypeFuzzyDeltaDifferentiation........................... 63
1.3 OtherPropertiesofFirstTypeandSecondTypeFuzzy
DeltaDifferentiation ................................................. 85
1.4 FirstTypeFuzzyDeltaIntegration.................................. 93
1.5 SecondTypeFuzzyDeltaIntegration............................... 108
1.6 ShiftOperators—Properties ......................................... 112
1.7 Complete-ClosedTimeScalesunderNon-translationalShifts..... 120
1.8 ShiftAlmostPeriodicFuzzyFunctions............................. 124
1.9 AdvancedPracticalProblems........................................ 131
1.10 NotesandReferences ................................................ 135
2 FirstOrderFuzzyDynamicEquations .................................. 137
2.1 LinearFirstOrderFuzzyDynamicEquations...................... 137
2.2 ExistenceandUniquenessofSolutions............................. 167
2.3 ContinuousDependenceoftheSolutionsofFirstOrder
FuzzyDynamicEquationsontheInitialData ...................... 175
2.4 ComparisonResults.................................................. 176
2.5 StabilityCriteria...................................................... 184
2.6 ExponentialStability................................................. 192
2.7 AdvancedPracticalProblems........................................ 201
2.8 NotesandReferences ................................................ 203
3 SecondOrderFuzzyDynamicEquations................................ 205
3.1 LinearSecondOrderFuzzyDynamicEquations................... 205
3.2 Boundary Value Problems for Second Order Fuzzy
DynamicEquations .................................................. 226
3.3 ExistenceandUniquenessofSolutionsofSecondOrder
FuzzyDynamicEquations........................................... 239
3.4 Continuous Dependence of the Solutions of Second
OrderFuzzyDynamicEquationsontheInitialData............... 248
ix
x Contents
3.5 AdvancedPracticalProblems........................................ 252
3.6 NotesandReferences ................................................ 255
4 FunctionalFuzzyDynamicEquations ................................... 257
4.1 PeriodicPropertiesofTimeScales.................................. 257
4.2 ThePhaseSpace...................................................... 274
4.3 PeriodicSolutions.................................................... 290
4.4 AdvancedPracticalProblems........................................ 306
4.5 NotesandReferences ................................................ 308
5 ImpulsiveFuzzyDynamicEquations..................................... 309
5.1 LinearFirstOrderImpulsiveFuzzyDynamicEquations .......... 310
5.2 Existence of Solutions for First Order Nonlinear
ImpulsiveFuzzyDynamicEquations-I ............................. 324
5.3 Existence of Solutions for First Order Nonlinear
ImpulsiveFuzzyDynamicEquations-II............................. 337
5.4 StabilityoftheSolutionsofFirstOrderImpulsiveFuzzy
DynamicEquations-I................................................. 350
5.5 StabilityoftheSolutionsofFirstOrderImpulsiveFuzzy
DynamicEquations-II................................................ 353
5.6 AdvancedPracticalProblems........................................ 357
5.7 NotesandReferences ................................................ 359
6 TheLebesgueIntegration.Lp-Spaces.SobolevSpaces ................ 361
6.1 TheLebesgueDelta-Integral......................................... 361
6.2 AbsolutelyContinuousFunctions................................... 377
6.3 AlternativeWayforDefiningofLebesgueTypeMeasure
andIntegrationoverT ............................................... 383
6.4 TheFundamentalTheoremofCalculus............................. 386
6.5 TheSpacesLp(T) ................................................... 390
6.6 SobolevTypeSpacesandGeneralizedDerivatives................. 393
6.7 WeakSolutionsofDynamicSystems ............................... 408
6.8 EulerSolutionsforDynamicEquations............................. 419
6.9 TheGronwallTypeInequality....................................... 423
6.10 Δ×B-MeasurableSet-ValuedFunctions.......................... 427
6.11 AdvancedPracticalProblems........................................ 433
6.12 NotesandReferences ................................................ 434
7 FirstOrderDynamicInclusions .......................................... 435
7.1 Existence and Approximations of Solutions of First
OrderDynamicInclusions........................................... 435
7.2 ExistenceResultsforFirstOrderDynamicInclusions
withNonlocalInitialConditions .................................... 455
7.3 ExistenceofSolutionsofFirstOrderDynamicInclusions
withGeneralBoundaryConditions.................................. 471
7.4 ExistenceofSolutionsofFirstOrderDynamicInclusions
withPeriodicBoundaryConditions................................. 489
Contents xi
7.5 TheDualTimeScales................................................ 500
7.6 ExistenceofSolutionsofFirstOrderDynamicInclusions
viaDuality............................................................ 510
7.7 AdvancedPracticalProblems........................................ 511
7.8 NotesandReferences ................................................ 514
8 SecondOrderDynamicInclusions ....................................... 515
8.1 FixedPointResults................................................... 515
8.2 ExistenceResultsforSecondOrderDynamicInclusions.......... 522
8.3 ExistenceResultsforSecondOrderDynamicInclusions
withm-PointBoundaryValueConditions .......................... 537
8.4 AdvancedPracticalProblems........................................ 548
8.5 NotesandReferences ................................................ 550
9 Boundary Value Problems for First Order Impulsive
DynamicInclusions......................................................... 551
9.1 LowerandUpperSolutionsforFirstOrderImpulsive
DynamicInclusions.................................................. 551
9.2 PeriodicBoundaryValueProblemsforFirstOrderLinear
DynamicInclusionswithImpulses.................................. 563
9.3 Periodic Boundary Value Problems for First Order
NonlinearDynamicInclusionswithImpulses-I .................... 574
9.4 Periodic Boundary Value Problems for First Order
NonlinearDynamicInclusionswithImpulses-II ................... 601
9.5 ExtremalSolutionsofPeriodicBoundaryValueProblems
forFirstOrderImpulsiveIntegro-DynamicInclusionsof
MixedType........................................................... 609
9.6 MultiplePositiveSolutionsforFirstOrderImpulsive
IntegralBoundaryValueProblems.................................. 631
9.7 AdvancedPracticalProblems........................................ 658
9.8 NotesandReferences ................................................ 660
10 Controllability,Bang–BangPrinciple.................................... 661
10.1 BasicDefinitions ..................................................... 661
10.2 ControllabilityofLinearEquations ................................. 668
10.3 Observability ......................................................... 684
10.4 Bang–BangPrinciple................................................. 695
10.5 AdvancedPracticalProblems........................................ 698
10.6 NotesandReferences ................................................ 699
11 LinearTime-OptimalControl............................................. 701
11.1 ExistenceofTime-OptimalControls................................ 701
11.2 TheMaximumPrincipleforLinearTime-OptimalControl........ 702
11.3 AdvancedPracticalProblems........................................ 714
11.4 NotesandReferences ................................................ 715