变分法和最优控制论

Preface

1 Introduction

1.1 Optimal control problem

1.2 Some background on finite-dimensional optimization

1.2.1 Unconstrained optimization

1.2.2 Constrained optimization

1.3 Preview of infinite-dimensional optimization

1.3.1 Function spaces, norms, and local minima

1.3.2 First variation and first-order necessary condition

1.3.3 Second variation and second-order conditions

1.3.4 Global minima and convex problems

1.4 Notes and references for Chapter 1

2 Calculus of Variations

2.1 Examples of variational problems

2.1.1 Dido's isoperimetric problem

2.1.2 Light refiection and refraction

2.1.3 Catenary

2.1.4 Brachistochrone

2.2 Basic calculus of variations problem

2.2.1 Weak and strong extrema

2.3 First-order necessary conditions for weak extrema

2.3.1 Euler-Lagrange equation

2.3.2 Historical remarks

2.3.3 Technical remarks

2.3.4 Two specialcases

2.3.5 Variable-endpoint problems

2.4 Hanultonian formalism and mechanics

2.4.1 Hamilton's canonical equations

2.4.2 Legendre transformation

2.4.3 Principle of least action and conservation laws

2.5 Variational problems with constraints

2.5.1 Integral constraints

2.5.2 Non-integral constraints

2.6 Second-order conditions

2.6.1 Legendre's necessary condition for a weak minimum

2.6.2 Sufficient condition for a weak minimum

2.7 Notes and references for Chapter 2

3 Wom Calculus of Variations to Optimal Control

3.1 Necessary conditions for strong extrema

3.1.1 Weierstrass-Erdmann corner conditions

3.1.2 Weierstrass excess function

3.2 Calculus of variations versus optimal control

3.3 Optimal control problem formulation and as8umptions

3.3.1 Controlsystem

3.3.2 Cost functional

3.3.3 Targetset

3.4 Variational approach to the fixed-time, free-endpoint problem

3.4.1 Preliminaries

3.4.2 First variation

3.4.3 Second variation

3.4.4 Some comments

3.4.5 Critique of the variational approach and preview of the maximum principle

3.5 Notes and references for Chapter 3

……

4 The Maximum Principle

5 The Hamilton-Jacobi-Bellman Equation

6 The Linear Quadratic Regulator

7 Advanced Topics

Bibliography

Index

This book grew out of my lecture notes for a graduate course on optimal control theory which I taught at the University of Illinois at Urbana-Champaign during the period from 2005 to 2010. While preparing the lectures， I have accumulated an entire shelf of textbooks on calculus of variations and optimal control systems.