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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Calculus of Fuzzy Functions

Abstract
In this chapter are introduced new classes of delta derivatives called first type fuzzy delta derivative and second type fuzzy delta derivative. Their existence and uniqueness are proved. Some of their basic properties such as fuzzy delta differentiation of a sum of two fuzzy functions and fuzzy delta differentiation of a multiplication of a fuzzy function with a constant are deducted. α-levels of fuzzy functions are defined, and formulae for their fuzzy delta derivatives are given. In the chapter are defined six types of multiplication of fuzzy functions, and for each type a formula for its fuzzy delta derivative is deducted. First type fuzzy delta integration and second type fuzzy delta integration of fuzzy functions are defined, and some of their basic properties are deducted. Shift operators are investigated, and complete-closed time scales under non-translational shifts are introduced. Shift almost periodic fuzzy functions are defined, and some of their properties are given.
Svetlin G. Georgiev

### Chapter 2. First Order Fuzzy Dynamic Equations

Abstract
This chapter is devoted to a qualitative analysis of first order fuzzy dynamic equations. First, we deducted formulae for the solutions of linear first order fuzzy dynamic equations and then investigated the Cauchy problem for first order fuzzy dynamic equations for existence and uniqueness. For this aim we introduced Lipschitz fuzzy functions and determined some of their properties. In this chapter we investigated the solutions of first order fuzzy dynamic equations for continuous dependence on the initial data. Some criteria as to when the trivial solution of first order fuzzy dynamic equations is equi-stable, uniformly stable, uniformly asymptotically stable, equi-asymptotically stable, exponentially stable, uniformly exponentially stable, and uniformly asymptotically stable are discussed.
Svetlin G. Georgiev

### Chapter 3. Second Order Fuzzy Dynamic Equations

Abstract
This chapter is devoted on a qualitative analysis of second order fuzzy dynamic equations. Firstly, the α-levels of the solutions of linear second order fuzzy dynamic equations are investigated. Then, it is introduced a class of BVPs for linear second order fuzzy dynamic equations and for it is given the corresponding Green function. In the chapter, it is investigated the Cauchy problem for a class of second order fuzzy dynamic equations for existence and uniqueness of the solutions and for the continuous dependence of the solutions on the initial data. For this aim, it is defined a suitable Banach space.
Svetlin G. Georgiev

### Chapter 4. Functional Fuzzy Dynamic Equations

Abstract
This chapter is devoted to a qualitative analysis of functional fuzzy dynamic equations. First, we deducted chain rules for fuzzy functions. These chain rules can be considered as analogues of the well known chain rules in the dynamic calculus on time scales. Then we investigated the periodic properties of time scales. In this chapter we deducted the periodic solutions of linear first order functional fuzzy dynamic equations. We considered the lower solutions and upper solutions for first order functional fuzzy dynamic equations.
Svetlin G. Georgiev

### Chapter 5. Impulsive Fuzzy Dynamic Equations

Abstract
This chapter is devoted to a qualitative analysis for impulsive fuzzy dynamic equations. First, we deducted the solutions of linear first order impulsive dynamic equations. Then we investigated some problems for first order impulsive fuzzy dynamic equations for existence and uniqueness of the solutions, using the Krasnosel’skii fixed point theorem and Sadovskii fixed point theorem. In this chapter we give criteria as to when the trivial solution of first order impulsive fuzzy dynamic equations is stable.
Svetlin G. Georgiev

### Chapter 6. The Lebesgue Integration. L p-Spaces. Sobolev Spaces

Abstract
In this chapter we define the Lebesgue measure and Lebesgue integral on time scales. We show the difference between the classical Lebesgue integral and the time scale Lebesgue integral. We introduce the absolutely continuous functions and deduct some of their properties. In this chapter we define functions of bounded variation and give some of their properties. We introduce L p spaces, Sobolev spaces, and generalized derivatives. As their applications, we investigate the weak solutions and Euler solutions of dynamic systems. We prove an analogue of the Gronwall type inequality. We define $$\varDelta \times \mathcal {B}$$-measurable set-valued functions and deduct some of their basic properties.
Svetlin G. Georgiev

### Chapter 7. First Order Dynamic Inclusions

Abstract
This chapter is devoted to a qualitative analysis of some classes of first order dynamic inclusions. They are investigated for existence of solutions of first order dynamic inclusions with local and nonlocal initial conditions, general boundary conditions, and periodic boundary conditions. Dual time scales are introduced and some of their basic properties are deducted. As their applications, the existence of solutions for some IVPs for first order dynamic inclusions is proved.
Svetlin G. Georgiev

### Chapter 8. Second Order Dynamic Inclusions

Abstract
This chapter is devoted to a qualitative analysis of second order dynamic inclusions. Firstly, some fixed point results are proved. As their applications, the existence of solutions for second order dynamic inclusions with periodic boundary conditions and the existence of solutions of some multi-point BVPs for second order dynamic inclusions are proved.
Svetlin G. Georgiev

### Chapter 9. Boundary Value Problems for First Order Impulsive Dynamic Inclusions

Abstract
This chapter is devoted to a qualitative analysis of some classes of second order impulsive dynamic inclusions. They are investigated for existence of solutions of some periodic BVPs for second order impulsive dynamic inclusions. Some criteria for the existence of lower solutions and upper solutions for some classes of BVPs for second order impulsive dynamic inclusions are given. Next, extremal solutions for BVPs for second order impulsive integro-dynamic inclusions are considered. In the chapter, some BVPs for second order dynamic inclusions for existence of multiple positive solutions are investigated.
Svetlin G. Georgiev

### Chapter 10. Controllability, Bang–Bang Principle

Abstract
In this chapter we formulate the basic form of a controlled dynamic equation and define its control. We introduce the payoff functional, running payoff, and terminal payoff. We give some criteria as to when the control is optimal and make a characterization of an optimal control. We study linear dynamic equations for observability and controllability and deduct the bang–bang principle.
Svetlin G. Georgiev

### Chapter 11. Linear Time-Optimal Control

Abstract
This chapter is devoted to a qualitative analysis of time-optimal controls. We prove their existence and deduct the maximum principle for linear time-optimal controls.
Svetlin G. Georgiev

### Chapter 12. The Pontryagin Maximum Principle

Abstract
This chapter is devoted to a qualitative analysis of some adjoint linear dynamics. We investigate the free endpoint control problem. In this chapter, we define the simple variation of a control. We study the variation of the terminal payoff. The Pontryagin maximum principle is deducted.
Svetlin G. Georgiev

### Chapter 13. Dynamic Programming

Abstract
In this chapter we introduce the theory of dynamic programming on time scales. We deduct the Hamilton–Jacobi–Bellman equations and give a method for their solving. Using the solutions of the Hamilton–Jacobi–Bellman equations, a representation of the value function is found. In this chapter we introduce dynamic games on time scales and define the payoff function of a game. The Isaacs equations are deducted.
Svetlin G. Georgiev

### Chapter 14. Weak Solutions and Optimal Control Problems for Some Classes of Linear First Order Dynamic Systems

Abstract
This chapter is devoted to some optimal control problems governed by a class of linear first order dynamic systems. We define a Hamiltonian and prove existence of optimal controls. We give necessary conditions for existence of admissible controls.
Svetlin G. Georgiev

### Chapter 15. Nonlinear Dynamic Equations and Optimal Control Problems

Abstract
This chapter is devoted to some optimal control problems governed by a class of nonlinear first order dynamic systems. We prove a necessary condition for existence of a minimum of a functional. We give necessary conditions for existence of optimal controls.
Svetlin G. Georgiev

### Chapter 16. Nonlinear Integro-Dynamic Equations and Optimal Control Problems

Abstract
This chapter is devoted to some optimal control problems governed by a class of nonlinear integro-dynamic systems. We give necessary conditions for existence of optimal controls.
Svetlin G. Georgiev

### Backmatter

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