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Control and Optimization

CHAPMAN & HALL MATHEMATICS SERIES Editors: Professor Keith Devlin St Mary's College USA

Professor Derek Goldrei Open University UK

Dr James Montaldi Universite de Lille France

OTHER TITLES IN THE SERIES INCLUDE Functions of TwoVariables S. Dineen Network Optimization V. K. Balakrishnan Sets, Functions and Logic A foundation coursein mathematics Second edition K. Devlin

Dynamical Systems Differential equations, mapsand chaotic behaviour D. K. Arrowsmith and C. M. Place Elements of Linear Algebra P.M.Cohn

Algebraic Numbers and Algebraic Functions P.M.Cohn Fullinformation on the complete range ofChapman & Hall mathematicsbooks is availablefrom thepublishers.

Control and Optimization

B. D. Craven Reader in Mathematics University of Melbourne Australia

CHAPMAN & HALL London· Glasgow· Weinheim . New York· Tokyo· Melbourne· Madras

Published by Chapman & Hall, 2-6 Boundary Row, LondonSE18HN, UK Chapman & Hall, 2-6 Boundary Row, London SEI 8HN, UK Blackie Academic & Professional,Wester Cleddens Road, Bishopbriggs, Glasgow G64 2NZ, UK Chapman & Hall GmbH, Pappelallee3, 69469 Weinheim,Germany Chapman & Hall USA, lIS Fifth Avenue, New York, NY 10003, USA Chapman & Hall Japan, ITP-Japan, Kyowa Building, 3F, 2-2-1 Hirakawacho, Chiyoda-ku, Tokyo 102, Japan Chapman & Hall Australia, 102 Dodds Street, South Melbourne, Victoria 3205, Australia Chapman & Hall India, R. Seshadri, 32 Second Main Road, CIT East, Madras 600035, India First edition 1995

e 1995 B. D. Craven Typesetin 10/12 pt Times by Thomson Press (India) Ltd, New Delhi, India Printed in Great Britian by Hartnolls Ltd, Bodmin, Cornwall ISBN 0 412 55890 4 Apart from any fair dealing for the purposes of research or private study, or criticismor review, as permitted under the UK Copyright Designsand Patents Act, 1988, this publication may not be reproduced, stored, or transmitted, in any form or by any means, without the prior permission in writing of the publishers,or in the case of reprographic reproduction only in accordance with the terms of the licences issuedby the Copyright Licensing Agencyin the UK, or in accordance with the terms of licences issuedby the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to the publishers at the London address printed on this page. The publisher makes no representation, expressor implied,with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liabilityfor any errors or omissions that may be made. A catalogue record for this book is available from the British Library Library of CongressCatalog Card Number: 95-68504

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Printed on permanent acid-freetext paper, manufactured in accordance with ANSIINISOZ39.48-1992 and ANSIINISO Z39.48-1984 (Permanence of Paper).

Contents

Preface 1 Optimization - ideas and background 1.1 Introduction 1.2 Simple inventory model 1.3 Mathematical formulation - stage one 1.4 A rocket problem 1.5 Some mathematical background 1.6 References 2 Optimal control models 2.1 Introduction 2.2 An advertising model 2.3 Some other advertising models 2.4 An investment model 2.5 Production and inventory models 2.6 Water management model 2.7 The fish model 2.8 Epidemic models 2.9 Stability? 2.10 Exercises 2.11 References 3 Convexity, linearization and multipliers 3.1 Convexity 3.2 Convex functions 3.3 Convex functions and subdifferentials 3.4 Alternative theorems 3.5 Linearization and Lagrangian conditions 3.6 Invex functions

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1 1

2 3 5 6 21

22

22 23 25 27 28 31 33 34 36 38 39 40 40 47 52 54

60 69

vi Contents

3.7 3.8 3.9 3.10

Conditions for local solvability Duality and quasiduality Nonsmooth optimization References

4 Optimality conditions for control problems 4.1 Formulation 4.2 Lagrange multipliers 4.3 Pontryagin conditions 4.4 Some examples 4.5 Boundary conditions 4.6 Time-optimal control 4.7 Sensitivity and stability 4.8 Exercises 4.9 References

71 72 75

79 81 81 85 92 95

98 101 102 107 109

5 Worked examples of control problems 5.1 Outline of procedure 5.2 Oscillator 5.3 Remark on open-loop and closed-loop control 5.4 Singular arcs 5.5 The 'dodgem car' problem 5.6 Vidale-Wolfe model 5.7 Investment model 5.8 Fish model 5.9 Epidemic models 5.10 Sufficient conditions for a minimum 5.11 Exercises 5.12 References

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6 Algorithms for control problems 6.1 Introduction 6.2 Algorithms for unconstrained minimization 6.3 Constrained minimization 6.4 Computation of optimal control problems 6.5 Sensitivity to parameters 6.6 Examples 6.7 Exercises 6.8 References

128 128 130 144 152 160 162 166 168

7 Proof of Pontryagin theory and related results 7.1 Introduction 7.2 Fixed-time optimal control problem 7.3 Pontryagin theorem

169 169 169 174

110 111 112 113 113

115 117 119 121 124 126 127

Contents

7.4 7.5 7.6 7.7 7.8 7.9 7.1 0 Index

Sensitivityto perturbations Lipschitz stability Truncation in [2 Approximating a constrained minimum Approximating an optimal control problem Nonsmooth control problems References

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176 179

180 182 183 184 189 191

Preface

Many questions of optimization, and optimal control, arise in management, economics and engineering. An optimum(maximum or minimum) is sought for some function describing the system, subject to constraints on the values ofthe variables. Often the functions are nonlinear, so more specialized methods than linear programming are required. In optimal control, the variables in the problem become functions of time - state and control functions. A state function could be a path to be followed, or describe sales rate or earnings in a business model or water level in a water storage system, and the control function describes what can be controlled, within limits, in order to manage the system. This subject, as wellas contributing to a large diversity ofapplications, has generated a great deal of mathematics and a variety of computational methods. This book presents a systematic theory of optimal control, in relation to a general approach to optimization, applicable in other contexts as well. Chapter I introduces the subject, and summarizes various mathematical tools that are required. Chapter 2 sets up a diversity of optimal control models for the application areas mentioned and various others. Chapter 3 presents the underlying mathematics - some key words are convex, linearization, Lagrangian conditions, duality, nonsmooth optimization - and obtains general conditions describing an optimum. In Chapter 4, these are applied to optimal control problems, in discrete time and in continuous time, to obtain the Pontryagin principle, and also to analyse questions ofsensitivity and stability when a system is perturbed. Chapter 5 is devoted to worked examples of optimal control problems. However, many practical problems in control, or otherwise in optimization, cannot be solved by formulae and require numerical computation. To this end, Chapter 6 describes the principles of some algorithms for control problems. These depend on approximating the control problem by some optimization problem in a finite number of variables so that it can be computed by one of various optimization algorithms. Sensitivity to parameter changes and the treatment of some nonsmooth terms in control problems are also discussed.

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Preface

Chapter 7 presents proofs for Pontryagin's principle and other related results, as well as analysing questions of approximation and sensitivity. Lists ofselected references may be found at the end of chapters. This book includes recent results which may not be found elsewhere, concerning sensitivity and approximation, invex (generalized convex) functions in optimization models and methods for nonsmooth problems (when the functions do not always have derivatives). The book is aimed at mathematics students at senior or graduate level at an American university, at second or third year of an honours course or postgraduate level in the UK and at third or fourth year or graduate level in an Australian university. For an indication of the mathematical background assumed, reference may be made to section 1.5, where the tools needed are summarized, including relevant topics in matrix algebra and nonned spaces. An applied course will emphasize Chapters 2 (models), 5 (worked examples) and portions of 6 (algorithms). Students of pure mathematics will read Chapters 3 and 4, perhaps omitting the proofs in Chapter 7. Exercises are distributed through Chapter 3, and appear at the end of Chapters 2, 4, 5 and 6. It thank Dr Barney Glover for useful suggestions and checking, and my final honours class for finding typing errors (but I am responsible for any remaining errors). B. D. Craven Melbourne November 1994

1

Optimization - ideas and background

1.1 INTRODUCTION Many questions in management and planning lead to mathematical models requiring optimization. Thus, some function of the variables that describe the problem must be maximized (or minimized) by a suitable choice of the variables within some permittedfeasible region. For example, it may be required to calculate the conditions ofoperation ofan industrial process which gives the maximum output or quality, or which gives the minimum cost. Such calculations are always subject to restrictions on the variables - not all combinations of the variables are practicable, there are restrictions on resources available, profit is to be maximized subject to some minimum level of quality being attained, and so on. Such restrictions, or constraints, define the feasible region for the variables of the problem. A mathematical problem in which it is required to calculate the maximum or minimum (the word optimum includes both) of some objective function, usually of a vector variable, subject to constraints, is called a problem of mathematical programming. The numbers of variables and constraints may be finite or infinite - the same underlying theory applies. In the second case, the optimization is with respect to some function, for example a continuous function describing the state of the system being optimized. In a large class of problems, both a statefunction and a controlfunction are involved. Typically, the state function describes the state of the system as a function of time, whereas the control function describes some input to the system, which can be chosen, subject to some constraints, so as to optimize an objective function. Such a problem is called an optimal control problem. It may be noted that it differs from a problem ofthe calculus ofvariations in that the control function is subject to inequality constraints.