# Topics in Optimization

## 1st Edition

**Editor:**George Leitmann

**eBook ISBN:**9781483266817

**Imprint:**Academic Press

**Published Date:**1st January 1967

**Page Count:**486

## Description

Mathematics in Science and Engineering, Volume 31: Topics in Optimization compiles contributions to the field of optimization of dynamical systems.

This book is organized into two parts. Part 1 covers reported investigations that are based on variational techniques and constitute essentially extensions of the classical calculus of variations. The contributions to optimal control theory and its applications, where the arguments are primarily geometric in nature, are discussed in Part 2. This volume specifically discusses the inequalities in a variational problem, singular extremals, mathematical foundations of system optimization, and synthesis of optimal controls.

This publication is recommended for both theoreticians and practitioners.

## Table of Contents

List of Contributors

Preface

Part 1 A Variational Approach

Chapter 1. Inequalities in a Variational Problem

1.0 Introduction

1.1 Condition

1.2 Conditions II and III

Part A—Case (a)

1.3 Preliminary Considerations

1.4 Singularities in Case (a)

1.5 The Extremaloid Index

1.6 The Imbedding Construction

1.7 Condition IV

1.8 Proof of Sufficiency

1.9 Numerical Example

Part B—Case (b)

1.10 Preliminary Considerations

1.11 Singularities in Case (b)

1.12 The Imbedding Construction

1.13 Numerical Example

1.14 Discussion of the Results

References

Chapter 2. Discontinuities in a Variational Problem

2.0 Introduction

2.1 Conditions Ic and Id

Part A—Case (a)

2.2 Conditions Ia, Ib, II, and III

2.3 Preliminary Considerations

2.4 The Function h(y')

2.5 Zermelo Diagram

2.6 The Imbedding Construction

2.7 Condition IV'

2.8 The Hilbert Integral

2.9 Proof of Sufficiency

2.10 Numerical Example

2.11 Discussion of the Results

Part B—Case (b)

2.12 Conditions Ia, Ib, II, and III

2.13 Preliminary Considerations

2.14 Zermelo Diagram

2.15 Corner Manifolds

2.16 Conditions II' and IIN'

2.17 Free Corners

2.18 A Special Case

2.19 The Imbedding Construction

2.20 Proof of Sufficiency

2.21 Numerical Example

2.22 Discussion of the Results

References

Chapter 3. Singular Extremals

3.0 Introduction

3.1 Second Variation Test for Singular Extremals

3.2 A Transformation Approach to the Analysis of Singular Subarcs

3.3 Examples

References

Chapter 4. Thrust Programming in a Central Gravitational Field

4.1 General Equations Governing the Motion of a Boosting Vehicle in a Central Gravitational Field

4.2 Integrals of the Basic System of Equations

4.3 Boundary Conditions: Various Types of Motion

4.4 Orbits on a Spherical Surface

4.5 Boosting Devices of Limited Propulsive Power

4.6 Singular Control Regimes

References

Chapter 5. The Mayer-Bolza Problem for Multiple Integrals: Some Optimum Problems for Elliptic Differential Equations Arising in Magnetohydrodynamics

5.0 Introduction

5.1 Optimum Problems for Partial Differential Equations: Necessary Conditions of Optimality

5.2 Optimum Problems in the Theory of Magnetohydrodynamical Channel Flow

5.3 Application to the Theory of MHD Power Generation: Minimization of End Effects in an MHD Channel

Appendix

References

Part 2 A Geometric Approach

Chapter 6. Mathematical Foundations of System Optimization

6.0 Introduction

6.1 Dynamical Polysystem

6.2 Optimization Problem

6.3 The Principle of Optimal Evolution

6.4 Statement of the Maximum Principle

6.5 Proof of the Maximum Principle for an Elementary Dynamical Polysystem

6.6 Proof of the Maximum Principle for a Linear Dynamical Polysystem

6.7 Proof of the Maximum Principle for a General Dynamical Polysystem

6.8 Uniformly Continuous Dependence of Trajectories with Respect to Variations of the Control Functions

6.9 Some Uniform Estimates for the Approximation z(t;U) of the Variational Trajectory y(t;U)

6.10 Convexity of the Range of a Vector Integral Over the Class A of Subsets of [0,1]

6.11 Proof of the Fundamental Lemma

6.12 An Intuitive Approach to the Maximum Principle

Appendix A. Some Results from the Theory of Ordinary Differential Equations

Appendix B. The Geometry of Convex Sets

References

Chapter 7. On the Geometry of Optimal Processes

7.0 Introduction

7.1 Dynamical System

7.2 Augmented State Space and Trajectories

7.3 Limiting Surfaces and Optimal Isocost Surfaces

7.4 Some Properties of Optimal Isocost Surfaces

7.5 Some Global Properties of Limiting Surfaces

7.6 Some Local Properties of Limiting Surfaces

7.7 Some Properties of Local Cones

7.8 Tangent Cone CΣ(Χ)

7.9 A Nice Limiting Surface

7.10 A Set of Admissible Rules

7.11 Velocity Vectors in Augmented State Space

7.12 Separability of Local Cones

7.13 Regular and Nonregular Points of a Limiting Surface

7.14 Some Properties of a Linear Transformation

7.15 Properties of Separable Local Cones

7.16 Attractive and Repulsive Subsets of a Limiting Surface

7.17 Regular Subset of a Limiting Surface

7.18 Antiregular Subset of a Limiting Surface

7.19 Symmetrical Subset of Local Cone ℓ(x)

7.20 A Maximum Principle

7.21 Boundary Points of ℰ*

7.22 Boundary and Interior Points of ℰ*

7.23 Degenerated Case

7.24 Some Illustrative Examples

Appendix

Bibliography

Chapter 8. The Pontryagin Maximum Principle

8.0 Introduction

8.1 The Extended Problem

8.2 The Control Lemma

8.3 The Controllability Theorem

8.4 The Maximum Principle (Part I)

8.5 The Maximum Principle (Part II)

8.6 The Bang-Bang Principle

Reference

Chapter 9. Synthesis of Optimal Controls

9.0 Introduction

9.1 Neustadt's Synthesis Method

9.2 Computational Considerations

9.3 Final Remarks

References

Chapter 10. The Calculus of Variations, Functional Analysis, and Optimal Control Problems

10.0 Introduction

10.1 The Problem of Mayer

10.2 Optimal Control Problems

10.3 Abstract Analysis, Basic Concepts

10.4 The Multiplier Rule in Abstract Analysis

10.5 Necessary Conditions for Optimal Controls

10.6 Variation of Endpoints and Initial Conditions

10.7 Examples of Optimal Control Problems

10.8 A Convergent Gradient Procedure

10.9 Computations Using the Convergent Gradient Procedure

References

Author Index

Subject Index

## Details

- No. of pages:
- 486

- Language:
- English

- Copyright:
- © Academic Press 1967

- Published:
- 1st January 1967

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483266817

## About the Editor

### George Leitmann

### Affiliations and Expertise

DIVISION OF APPLIED MECHANICS UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA