Optimal Control and the Calculus of Variations / Edition 1

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More About This Textbook


This book provides the core material for undergraduate courses on optimal control, the modern development that has grown out of the calculus of variations and classical optimization theory.
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Product Details

  • ISBN-13: 9780198514893
  • Publisher: Oxford University Press, USA
  • Publication date: 9/28/1995
  • Edition description: Reprint
  • Edition number: 1
  • Pages: 248
  • Product dimensions: 6.13 (w) x 9.19 (h) x 0.61 (d)

Table of Contents

1 Introduction 1
The maxima and minima of functions 1
The calculus of variations 4
Optimal control 9
2 Optimization In R[superscript n] 13
Functions of one variable 13
Critical points, end-points, and points of discontinuity 16
Functions of several variables 18
Minimization with constraints 22
A geometrical interpretation 25
Distinguishing maxima from minima 28
3 The calculus of variations 33
The fixed end-point problem 33
Problems in which the end-points are not fixed 41
Finding minimizing curves 46
Isoperimetric problems 54
Sufficiency conditions 58
Fields of extremals 59
Hilbert's invariant integral 62
Semi-fields and the Jacobi condition 66
4 Optimal control I: Theory 70
Introduction 70
Control of a simple first-order system 70
Systems governed by ordinary differential equations 72
The optimal control problem 74
The Pontryagin maximum principle 80
Optimal control to target curves 100
5 Optimal control II: Applications 103
Introduction 103
Time-optimal control of linear systems 103
Optimal control to target curves 139
Singular controls 149
Fuel-optimal control 151
Problems where the cost depends on x(t[subscript 1]) 159
Linear systems with quadratic cost 163
The steady-state Riccati equation 168
The calculus of variations revisited 170
6 Proof of the maximum principle of Pontryagin 175
Convex sets in R[superscript n] 176
The linearized state equations 182
The behaviour of H on an optimal path 184
Sufficiency conditions for optimal control 206
Appendix: Answers and hints for the exercises 208
Bibliography 231
Index 233
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