Application of the Euler-Lagrange-Method for solving optimal control problems

Table of Contents

1.0 Preamble

1.1 Optimization

1.2 Classification of Optimization Problems

1.2.1 Classification Based on the Existence of Constraints

1.2.2 Classification Based on the Nature of Equations Involved

1.2.3 Classification Based on the Permissible Values of the Design Variable

1.2.4 Classification Based on the Deterministic Nature of the Variable Involved

1.2.5 Classification Based on the Number of Objective Functions

1.3 Optimal Control Problems

1.3.1 Continuous Optimal Control Problems

1.3.2 Discrete Optimal Control Problems

1.4 Regulator Problems

1.5 Application of Optimization

1.6 General Procedures for Solving Optimization Problems

1.7 Aim and Objectives of the Research

1.8 Motivation

1.9 Scope and Limitation of the Research

2.1 Introduction

2.2 Highlight of Some Numerical Optimization Methods

2.2.1 Introduction

2.2.2 Parametric Optimization: Control Parameterization

2.2.3 Riccati Equations

2.2.4 Shooting Methods

2.2.5 Newton’s Method

2.2.6 Sequential Quadratic Programming

2.2.7 Constraint Handling Techniques

2.2.8 Gradient Methods

2.2.9 The Extended Conjugate Gradient Method Algorithm

2.2.10 The Continuous Case of the Extended Conjugate Gradient Method Algorithm

2.2.11 The Discrete Case of the Extended Conjugate Gradient Method Algorithm

3.0 Introduction

3.1 Derivation of Euler’s Method

3.2 Necessary Condition for an Optimal Control Problem

3.3 Necessary Condition for a General Optimal Control Problem with n Equality Constraints

3.4 Necessary Condition for a General Optimal Control Problem with Mixed Constraints

4.0 Introduction

4.1 Mathematical Computation of Euler’s Method

4.2 Algorithm for Euler’s Method Approach for Solving Optimal Control Problems

4.3 Computational Results

4.4 Discussion of the Results

4.5 Generalization of Euler Lagrange Method for Solving General Form of Continuous Time Linear Regulator Problems

5.1 Conclusion

5.2 Recommendations

5.3 Contribution to Knowledge

Research Objectives and Topics

The primary research objective is to develop and investigate the Euler-Lagrange method as an alternative approach for solving optimal control problems, particularly those involving one-dimensional and generalized forms with constraints. The study aims to provide an algorithmic framework that circumvents the difficulties often associated with constructing control operators in existing methods like the Conjugate Gradient Method.

• Investigation of the Extended Conjugate Gradient Method (ECGM) for solving optimal control problems.
• Development of the Euler-Lagrange method as a modification of Calculus of Variation techniques.
• Comparison of numerical results obtained via the Euler-Lagrange approach with existing computational methods.
• Generalization of the method to continuous-time linear regulator problems.
• Implementation of the proposed algorithms and validation through test problems.

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Summary of Chapters

CHAPTER ONE: INTRODUCTION: This chapter introduces the core concepts of optimization, classifying different types of problems and defining optimal control as a vital tool for decision-making in engineering and social sciences.

CHAPTER TWO: REVIEW OF RELATED LITERATURES: This section examines existing numerical optimization methods, including gradient-based techniques and the genesis of the Conjugate Gradient Method, highlighting the transition toward the Extended Conjugate Gradient Method (ECGM).

CHAPTER THREE: METHODOLOGY: This chapter details the derivation of the Euler-Lagrange method, converting optimal control problems into functional problems through Lagrange Multipliers to establish necessary conditions for optimality.

CHAPTER FOUR: RESULTS AND DISCUSSION: This chapter provides numerical computations of the Euler-Lagrange algorithm applied to various test problems, comparing the resulting objective function values against previously established benchmarks.

CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS: The final chapter summarizes the advantages of the Euler-Lagrange approach in circumventing traditional control operator construction challenges and offers recommendations for future research in partial differential equations.

Keywords

Optimization, Optimal Control, Euler-Lagrange Method, Calculus of Variation, Conjugate Gradient Method, Regulator Problems, Objective Function, Constraints, Numerical Methods, Algorithms, Performance Index, Linear Regulator, Differential Equations, Functional Optimization, Computational Results.

Frequently Asked Questions

What is the core focus of this dissertation?

This work focuses on applying the Euler-Lagrange method to solve optimal control problems, specifically addressing both one-dimensional and generalized forms with constraints, aiming to improve upon existing methods.

What are the central thematic fields addressed?

The research sits at the intersection of applied mathematics, engineering optimization, and numerical analysis, focusing on optimal control theory and the development of computational algorithms.

What is the primary research goal?

The primary goal is to provide an alternative, more efficient approach for solving optimal control problems that simplifies the construction of control operators, which are otherwise complex in methods like the Conjugate Gradient Method.

Which scientific methods are utilized?

The study utilizes variational techniques, the Euler-Lagrange approach, the Extended Conjugate Gradient Method (ECGM), and numerical finite-difference approximations to solve constrained optimal control problems.

What is covered in the main body of the work?

The main body covers the theoretical background of optimization, a comprehensive literature review of numerical methods, the derivation of the Euler-Lagrange optimality conditions, and the implementation of these algorithms on ten distinct numerical test problems.

Which keywords characterize this research?

Key terms include Optimization, Optimal Control, Euler-Lagrange Method, Calculus of Variation, and Conjugate Gradient Method, among others related to numerical algorithms.

How does this method handle constraints?

The proposed Euler-Lagrange approach incorporates penalty functions to convert constrained problems into unconstrained ones, effectively managing state and control constraints within the dynamical system.

What distinguishes this method from standard Conjugate Gradient methods?

Unlike standard Conjugate Gradient methods which often require the complex construction of specific control operators, the Euler-Lagrange method offers a direct derivation of optimality conditions that streamlines the solution process for a broader class of problems.