变分法和最优控制论 英文PDF|Epub|txt|kindle电子书版本网盘下载

变分法和最优控制论 英文PDF格式电子书版下载

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1 Introduction1

1.1 Optimal control problem1

1.2 Some background on finite-dimensional optimization3

1.2.1 Unconstrained optimization4

1.2.2 Constrained optimization11

1.3 Preview of infinite-dimensional optimization17

1.3.1 Function spaces,norms,and local minima18

1.3.2 First variation and first-order necessary condition19

1.3.3 Second variation and second-order conditions21

1.3.4 Global minima and convex problems23

1.4 Notes and references for Chapter 124

2 Calculus of Variations26

2.1 Examples of variational problems26

2.1.1 Dido's isoperimetric problem26

2.1.2 Light reflection and refraction27

2.1.3 Catenary28

2.1.4 Brachistochrone30

2.2 Basic calculus of variations problem32

2.2.1 Weak and strong extrema33

2.3 First-order necessary conditions for weak extrema34

2.3.1 Euler-Lagrange equation35

2.3.2 Historical remarks39

2.3.3 Technical remarks40

2.3.4 Two special cases41

2.3.5 Variable-endpoint problems42

2.4 Hamiltonian formalism and mechanics44

2.4.1 Hamilton's canonical equations45

2.4.2 Legendre transformation46

2.4.3 Principle of least action and conservation laws48

2.5 Variational problems with constraints51

2.5.1 Integral constraints52

2.5.2 Non-integral constraints55

2.6 Second-order conditions58

2.6.1 Legendre's necessary condition for a weak minimum59

2.6.2 Sufficient condition for a weak minimum62

2.7 Notes and references for Chapter 268

3 From Calculus of Variations to Optimal Control71

3.1 Necessary conditions for strong extrema71

3.1.1 Weierstrass-Erdmann corner conditions71

3.1.2 Weierstrass excess function76

3.2 Calculus of variations versus optimal control81

3.3 Optimal control problem formulation and assumptions83

3.3.1 Control system83

3.3.2 Cost functional86

3.3.3 Target set88

3.4 Variational approach to the fixed-time,free-endpoint problem89

3.4.1 Preliminaries89

3.4.2 First variation92

3.4.3 ？econd variation95

3.4.4 Some comments96

3.4.5 Critique of the variational approach and preview of the maximum principle98

3.5 Notes and references for Chapter 3100

4 The Maximum Principle102

4.1 Statement of the maximum principle102

4.1.1 Basic fixed-endpoint control problem102

4.1.2 Basic variable-endpoint control problem104

4.2 Proof of the maximum principle105

4.2.1 From Lagrange to Mayer form107

4.2.2 Temporal control perturbation109

4.2.3 Spatial control perturbation110

4.2.4 Variational equation112

4.2.5 Terminal cone115

4.2.6 Key topological lemma117

4.2.7 Separating hyperplane120

4.2.8 Adjoint equation121

4.2.9 Properties of the Hamiltonian122

4.2.10 Transversality condition126

4.3 Discussion of the maximum principle128

4.3.1 Changes of variables130

4.4 Time-optimal control problems134

4.4.1 Example:double integrator135

4.4.2 Bang-bang principle for linear systems138

4.4.3 Nonlinear systems,singular controls,and Lie brackets141

4.4.4 Fuller's problem146

4.5 Existence of optimal controls148

4.6 Notes and references for Chapter 4153

5 The Hamilton-Jacobi-Bellman Equation156

5.1 Dynamic programming and the HJB equation156

5.1.1 Motivation:the discrete problem156

5.1.2 Principle of optimality158

5.1.3 HJB equation161

5.1.4 Sufficient condition for optimality165

5.1.5 Historical remarks167

5.2 HJB equation versus the maximum principle168

5.2.1 Example:nondifferentiable value function170

5.3 Viscosity solutions of the HJB equation172

5.3.1 One-sided differentials172

5.3.2 Viscosity solutions of PDEs174

5.3.3 HJB equation and the value function176

5.4 Notes and references for Chapter 5178

6 The Linear Quadratic Regulator180

6.1 Finite-horizon LQR problem180

6.1.1 Candidate optimal feedback law181

6.1.2 Riccati differential equation183

6.1.3 Value function and optimality185

6.1.4 Global existence of solution for the RDE187

6.2 Infinite-horizon LQR problem189

6.2.1 Existence and properties of the limit190

6.2.2 Infinite-horizon problem and its solution193

6.2.3 Closed-loop stability194

6.2.4 Complete result and discussion196

6.3 Notes and references for Chapter 6199

7 Advanced Topics200

7.1 Maximum principle on manifolds200

7.1.1 Differentiable manifolds201

7.1.2 Re-interpreting the maximum principle203

7.1.3 Symplectic geometry and Hamiltonian flows206

7.2 HJB equation,canonical equations,and characteristics207

7.2.1 Method of characteristics208

7.2.2 Canonical equations as characteristics of the HJB equation211

7.3 Riccati equations and inequalities in robust control212

7.3.1 L2 gain213

7.3.2 H∞ control problem216

7.3.3 Riccati inequalities and LMIs219

7.4 Maximum principle for hybrid control systems219

7.4.1 Hybrid optimal controlproblem219

7.4.2 Hybrid maximum principle221

7.4.3 Example: light reflection222

7.5 Notes and references for Chapter 7223

Bibliography225

Index231