利伯逊著的《变分法和最优控制论》是一部讲述变量微积分和优化控制理论的严谨、详尽、自成体系的工程类、应用数学、及相关科目的研究生教程。书中的内容从变量微积分开始，为优化控制做准备，特别适合一学期教程。然后给出了最大原理的完整证明和囊括了动态规划的Hamilton－Jacobi－Bellman理论和线性二次优化控制。目次：导论；变量微积分；从变量微积分到优化控制；最大值原理；Hamilton－Jacobi－Bellman方程；线性二次型调节器；高等论题。读者对象：数学、工程和相关专业的科研人员。

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，andlocalminima

1.3.2 First variation and first-order necessary condition

1.3.3 Second variation and second-order conditions

1.3.4 Globalminima 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 reflection and refraction

2.1.3 Catenary

2.1.4 BruehistOChrone

 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 remarkss

2.3.4 TWO special cases

2.3.5 Variable-endpoint problems

 2.4 Hamiltonian 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 conditionfor aweakminimum

 2.7 Notes and references for Chapter 2

3 From 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 assumptions

3.3.1 Control system

3.3.2 Cost functional

3.3.3 Target set

 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

 4.1 Statement of the maximum principle

4.1.1 Basic fixed-endpoint control problem

4.1.2 Basic variable-endpoint control problem

 4.2 Proof of the maximum principle

4.2.1 From Lagrange to Mayer form

4.2.2 Temporal control perturbation

4.2.3 Spatial control perturbation

4.2.4 Variational equation.

4.2.5 Terminal cone.

4.2.6 Key topological lemma

4.2.7 Separating hyperplane

4.2.8 Adjoint equation

4.2.9 Properties ofthe Hamiltonian

4.2.10 Transversality condition

 4.3 Discussion of the maximum principle

4.3.1 Changes ofvariables

 4.4 Time-optimal control problems

4.4.1 Example：double integrator

4.4.2 Bang-bang principle for linear systems

4.4.3 Nonlinear systems，singular controls，and Lie brackets

4.4.4 Fuller'S problem

 4.5 Existence of optimal controls

 4.6 Notes and references for Chapter 4

5 The Hamilton-Jacobi.Bellman Equation

 5.1 Dynamic programming and the HJB equation

5.1 Motivation：the discrete problem

5.1.2 Principle of optimality

5.1.3 HJB equation

5.1.4 Sufficient condition for optimality.

5.1.5 Historical remarks

 5.2 HJB equation versus the maximum principle

5.2.1 Example：nondifferentiable value function

 5.3 Viscosity solutions of the HJB equation

5.3.1 One-sided difierentials

5.3.2 Viscosity solutions ofPDEs

5.3.3 HJB equation andthe valuefunction

 5.4 Notes and references for Chapter 5

6 The Linear Quadratic Regulator

 6.1 Finite-horizon LQR problem

6.1.1 Candidate optimalfeedbacklaw.

6.1.2 Riccati differential equatio

6.1.3 Value function and optimality

6.1.4 Global existence of solutionfortheRDE.

 6.2 Infinite-horizon LQR problem

6.2.1 Existence and properties ofthelimit

6.2.2 Infinite-horizon problem and its solution

6.2.3 Closed-loop stability

6.2.4 Complete result and discussion

 6.3 Notesand referencesforChapter 6

7 Advanced Topics

 7.1 Maximum principle on manifolds

7.1.1 Differentiable manifolds

7.1.2 Re-interpreting the maximum principle

7.1.3 ymplectic geometry and Hamiltonian flows

 7.2 HJB equation，canonical equations，and characteristics

7.2.1 Method of characteristics

7.2.2 Canonical equations as characteristics of the HJB

equation

 7.3 Riccati equations and inequalities in robust control

7.3.1 L2 gain

7.3.2 H∞ control problem

7.3.3 Riccati inequalities and LMIs

 7.4 Maximum principle for hybrid control systems

7.4.1 Hybrid optimal control problem

7.4.2 Hybrid maximum principle

7.4.3 Example：light reflection

 7.5 Notes and references for Chapter 7

Bibliography

Index