Optimal Control and the Calculus of Variations (Revised)

by Pinch, Enid R

• ISBN13: 9780198514893
• ISBN10: 0198514891
• Edition: 1st
• Format: Paperback
• Copyright: 1995-10-19
• Publisher: Oxford

Summary

Optimal control is a modern development of the calculus of variations and classical optimization theory. For that reason, this introduction to the theory of optimal control starts by considering the problem of minimizing a function of many variables. It moves through an exposition of the calculus of variations, to the optimal control of systems governed by ordinary differential equations. This approach should enable students to see the essential unity of important areas of mathematics, and also allow optimal control and the Pontryagin maximum principle to be placed in a proper context. A good knowledge of analysis, algebra, and methods is assumed. All the theorems are carefully proved, and there are many worked examples and exercises. Although this book is written for the advanced undergraduate mathematician, engineers and scientists who regularly rely on mathematics will also find it a useful text.

Table of Contents

Introduction

The maxima and minima of functions

The calculus of variations

Optimal control

Optimization in

Functions of one variable

Critical points, end-points, and points of discontinuity

Functions of several variables

Minimization with constraints

A geometrical interpretation

Distinguishing maxima from minima

The calculus of variations

Problems in which the end-points are not fixed

Finding minimizing curves

Isoperimetric problems

Sufficiency conditions

Fields of extremals

Hilbert's invariant integral

Semi-fields and the Jacobi condition

Optimal Control I: Theory

Introduction

Control of a simple first-order system

Systems governed by ordinary differential equations

The optimal control problem

The Pontryagin maximum principle

Optimal control to target curves

Optimal Control II: Applications

Time-optimal control of linear systems

Optimal control to target curves

Singular controls

Fuel-optimal controls

Problems where the cost depends on X (t l)

Linear systems with quadratic cost

The steady-state Riccai equation

The calculus of variations revisited

Proof of the Maximum Principle of Pontryagin

Convex sets in

The linearized state equations

Behaviour of H on an optimal path

Sufficiency conditions for optimal control

Table of Contents provided by Publisher. All Rights Reserved.