The theory of dynamic equations has many interesting applications in control theory, mathematical economics, mathematical biology, engineering and technology. In some cases, there exists uncertainty, ambiguity, or vague factors in such problems, 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 essential unity achieved in the theory fuzzy dynamic equations, dynamic inclusions and optimal control problems on time scales; and initiate several new extensions to other types of fuzzy dynamic systems and dynamic inclusions. The material is presented in a highly readable, mathematically solid format. Many practical problems are illustrated, displaying a wide variety of solution techniques. The book is primarily intended for senior undergraduate students and beginning graduate students of engineering and science courses. Students in mathematical and physical sciences will find many sections of direct relevance.
Author(s): Svetlin G. Georgiev
Publisher: Springer
Year: 2021
Language: English
Pages: 882
City: Cham
Preface
Contents
1 Calculus of Fuzzy Functions
1.1 First Type Fuzzy Delta Differentiation
1.2 Second Type Fuzzy Delta Differentiation
1.3 Other Properties of First Type and Second Type Fuzzy Delta Differentiation
1.4 First Type Fuzzy Delta Integration
1.5 Second Type Fuzzy Delta Integration
1.6 Shift Operators—Properties
1.7 Complete-Closed Time Scales under Non-translational Shifts
1.8 Shift Almost Periodic Fuzzy Functions
1.9 Advanced Practical Problems
1.10 Notes and References
2 First Order Fuzzy Dynamic Equations
2.1 Linear First Order Fuzzy Dynamic Equations
2.2 Existence and Uniqueness of Solutions
2.3 Continuous Dependence of the Solutions of First Order Fuzzy Dynamic Equations on the Initial Data
2.4 Comparison Results
2.5 Stability Criteria
2.6 Exponential Stability
2.7 Advanced Practical Problems
2.8 Notes and References
3 Second Order Fuzzy Dynamic Equations
3.1 Linear Second Order Fuzzy Dynamic Equations
3.2 Boundary Value Problems for Second Order Fuzzy Dynamic Equations
3.3 Existence and Uniqueness of Solutions of Second Order Fuzzy Dynamic Equations
3.4 Continuous Dependence of the Solutions of Second Order Fuzzy Dynamic Equations on the Initial Data
3.5 Advanced Practical Problems
3.6 Notes and References
4 Functional Fuzzy Dynamic Equations
4.1 Periodic Properties of Time Scales
4.2 The Phase Space
4.3 Periodic Solutions
4.4 Advanced Practical Problems
4.5 Notes and References
5 Impulsive Fuzzy Dynamic Equations
5.1 Linear First Order Impulsive Fuzzy Dynamic Equations
5.2 Existence of Solutions for First Order Nonlinear Impulsive Fuzzy Dynamic Equations-I
5.3 Existence of Solutions for First Order Nonlinear Impulsive Fuzzy Dynamic Equations-II
5.4 Stability of the Solutions of First Order Impulsive Fuzzy Dynamic Equations-I
5.5 Stability of the Solutions of First Order Impulsive Fuzzy Dynamic Equations-II
5.6 Advanced Practical Problems
5.7 Notes and References
6 The Lebesgue Integration. Lp-Spaces. Sobolev Spaces
6.1 The Lebesgue Delta-Integral
6.2 Absolutely Continuous Functions
6.3 Alternative Way for Defining of Lebesgue Type Measure and Integration over T
6.4 The Fundamental Theorem of Calculus
6.5 The Spaces Lp(T)
6.6 Sobolev Type Spaces and Generalized Derivatives
6.7 Weak Solutions of Dynamic Systems
6.8 Euler Solutions for Dynamic Equations
6.9 The Gronwall Type Inequality
6.10 ΔB-Measurable Set-Valued Functions
6.11 Advanced Practical Problems
6.12 Notes and References
7 First Order Dynamic Inclusions
7.1 Existence and Approximations of Solutions of First Order Dynamic Inclusions
7.2 Existence Results for First Order Dynamic Inclusions with Nonlocal Initial Conditions
7.3 Existence of Solutions of First Order Dynamic Inclusions with General Boundary Conditions
7.4 Existence of Solutions of First Order Dynamic Inclusions with Periodic Boundary Conditions
7.5 The Dual Time Scales
7.6 Existence of Solutions of First Order Dynamic Inclusions via Duality
7.7 Advanced Practical Problems
7.8 Notes and References
8 Second Order Dynamic Inclusions
8.1 Fixed Point Results
8.2 Existence Results for Second Order Dynamic Inclusions
8.3 Existence Results for Second Order Dynamic Inclusions with m-Point Boundary Value Conditions
8.4 Advanced Practical Problems
8.5 Notes and References
9 Boundary Value Problems for First Order Impulsive Dynamic Inclusions
9.1 Lower and Upper Solutions for First Order Impulsive Dynamic Inclusions
9.2 Periodic Boundary Value Problems for First Order Linear Dynamic Inclusions with Impulses
9.3 Periodic Boundary Value Problems for First Order Nonlinear Dynamic Inclusions with Impulses-I
9.4 Periodic Boundary Value Problems for First Order Nonlinear Dynamic Inclusions with Impulses-II
9.5 Extremal Solutions of Periodic Boundary Value Problems for First Order Impulsive Integro-Dynamic Inclusions of Mixed Type
9.6 Multiple Positive Solutions for First Order Impulsive Integral Boundary Value Problems
9.7 Advanced Practical Problems
9.8 Notes and References
10 Controllability, Bang–Bang Principle
10.1 Basic Definitions
10.2 Controllability of Linear Equations
10.3 Observability
10.4 Bang–Bang Principle
10.5 Advanced Practical Problems
10.6 Notes and References
11 Linear Time-Optimal Control
11.1 Existence of Time-Optimal Controls
11.2 The Maximum Principle for Linear Time-Optimal Control
11.3 Advanced Practical Problems
11.4 Notes and References
12 The Pontryagin Maximum Principle
12.1 Adjoint Linear Dynamics
12.2 Variations of the Control
12.3 Simple Control Variations
12.4 The Free Endpoint Problem
12.5 The Pontryagin Maximum Principle
12.6 Advanced Practical Problems
12.7 Notes and References
13 Dynamic Programming
13.1 The Hamilton–Jacobi–Bellman Partial Dynamic Equations
13.2 The Dynamic Programming Method
13.3 The Linear Dynamics
13.4 Dynamic Games: Definitions
13.5 The Isaacs Equations
13.6 Advanced Practical Problems
13.7 Notes and References
14 Weak Solutions and Optimal Control Problems for Some Classes of Linear First Order Dynamic Systems
14.1 Optimal Control Governed by a Class of First Order Linear Dynamic Systems-I
14.2 Optimal Control Governed by a Class of First Order Linear Dynamic Systems-II
14.3 Advanced Practical Problems
14.4 Notes and References
15 Nonlinear Dynamic Equations and Optimal Control Problems
15.1 Existence of Optimal Controls
15.2 Necessary Conditions of Optimality
15.3 Advanced Practical Problems
15.4 Notes and References
16 Nonlinear Integro-Dynamic Equations and Optimal Control Problems
16.1 Existence of Optimal Controls
16.2 Necessary Conditions of Optimality
16.3 Advanced Practical Problems
16.4 Notes and References
A Fuzzy Sets
A.1 Basic Definitions
A.2 Basic Operations with Fuzzy Sets
A.3 Fuzzy Numbers
A.4 Multiplication of Compact Intervals
A.5 Notes and References
B Set-Valued Maps
B.1 Limits of Sets
B.2 Definition of Set-Valued Maps
B.3 Continuity Concepts
C Alaoglu's Theorem. Krein–Milman Theorem
D Mazur's Theorem
References
Index
