Controllability Results for Non-Linear Neutral Functional Differential Equations

CHAPTER 1: GENERAL INTRODUCTION

1. 1.1  Background

2. 1.2  Statement of the Problem/Objective

3. 1.3  Scope of Study.

TABLE OF CONTENTS

1. 1.4  Definitions and Preliminaries

2. 1.5  Functional Differential Equations

3. 1.6  121.6 Difference between Retarded and Neutral Functional Differential Equations. 12

1.10 Existence and Uniqueness of Optimal Control for linear neutral

General Form of Functional Differential Equations. 12

General Solution Format or Variation of Parameters. 13

Volterra Integral Equation 13

Volerra Integro-Differential system. 14

1. 1.11  Global Uniform Asymptotic Stability for nonlinear Infinite Neutral differential systems 16

2. 1.12  Existence of Mild Solution of Nonlinear Neutral Differential

   Equations in Banach Spaces 19

CHAPTER 2: REVIEW OF THE LITERATURE 22

2.1 Introduction

1. 2.2  Controllability of Ordinary Differential Systems. 25

2. 2.3  Computable Criterion for Controllability. 26

3. 2.4  Controllability of Nonlinear Systems. 27

4. 2.5  Relative Controllability of Nonlinear Neutral Systems. 27

5. 2.6  Null Controllability of Functional Differential Systems 28

CHAPTER 3: METHODOLOGY AND CONTROLLABILITY 31

3.1 Introduction 31

3.2 Description of System. 32

3.3 Variation of constant formula 32

3.4. The following Methods/Techniques were used to obtain Results 34

3.4.1 Relative Controllability Technique 34

3.4.2 The Energy Method of Alexander Mikhailovick Lyapunov(1829) 34

1. 3.5  Basic set functions and properties 35

2. 3.6  Controllability Conditions or Controllability Standard 35

3.6.1 Optimality Control 35

3.7 Bang-Bang Principle 36

1. 3.8  Control theorem on nonlinear systems 36

2. 3.9  Lyapunov Stability Results (theorem) 36

3. 3.10  Existence Theorems 37

CHAPTER4: MAIN RESULTS 38

4.1. Relative Controllability Results 38

4.2. Optimality Conditions For The Linear Neutral Volterra Intego-Differential Systems. 40

4.3. Existence of an Optimal Control. 41

4.4 The Form of Optimal Control. 43

4.5. Uniqueness of Optimal Control. 45

4.6. Global Uniform Asymptotic Stability for Nonlinear Infinite Neutral Differential Systems. 46

4.6.1 Results on Stability on Neutral Systems. 48

4.6.2 Results on the Exponential Asymptotic Stability in the Large for

Nonlinear Neutral Systems. 48

4.7. Existence of Mild Solution of Nonlinear Neutral Differential Equations in

Banach Spaces. 55

4.9.1. Existence of Solutions by Fixed Point Technique. 56

CHAPTER5: SUMMARY, CONCLUSION, RECOMMENDATION AND CONTRIBUTION TO KNOWLEDGE 61

5.1. Summary 61

5.2. Conclusion 67

5.3. Recommendation 68

5.4. Contribution to Knowledge 68