# Controllability Results for Non-Linear Neutral Functional Differential Equations

ABSTRACT

In this work, necessary and sufficient conditions are investigated and proved for the controllability of nonlinear functional neutral differential equations. The existence, form, and uniqueness of the optimal control of the linear systems are also derived. Global uniform asymptotic stability for nonlinear infinite neutral differential systems are investigated and proved and ultimately, the Shaefers’ fixed point theorem is used to forge a new and far- reaching result for the existence of mild solutions of nonlinear neutral differential equations in Banach Spaces.

KEY WORDS:

Relative Controllability, Volterra Integro-Differential Equation, Optimal Control, Complete State, Unsymmetric Fubini Theorem, Neutral Systems, Linearization, Exponential Estimate, , Stability in the Large, mild solution .

CHAPTER 1: GENERAL INTRODUCTION

1. 1.1  Background

2. 1.2  Statement of the Problem/Objective

3. 1.3  Scope of Study.

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

Reference. 69

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APA

Orakie, P (2021). Controllability Results for Non-Linear Neutral Functional Differential Equations. Afribary. Retrieved from https://afribary.com/works/controllability-results-for-non-linear-neutral-functional-differential-equations

MLA 8th

Orakie, Paul "Controllability Results for Non-Linear Neutral Functional Differential Equations" Afribary. Afribary, 22 Feb. 2021, https://afribary.com/works/controllability-results-for-non-linear-neutral-functional-differential-equations. Accessed 11 Sep. 2024.

MLA7

Orakie, Paul . "Controllability Results for Non-Linear Neutral Functional Differential Equations". Afribary, Afribary, 22 Feb. 2021. Web. 11 Sep. 2024. < https://afribary.com/works/controllability-results-for-non-linear-neutral-functional-differential-equations >.

Chicago

Orakie, Paul . "Controllability Results for Non-Linear Neutral Functional Differential Equations" Afribary (2021). Accessed September 11, 2024. https://afribary.com/works/controllability-results-for-non-linear-neutral-functional-differential-equations

##### Document Details
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