Calculus of Variations and Optimal Control Theory - A Concise Introduction.pdf

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Preface xiii
1 Introduction 1
1.1 Optimal control problem 1
1.2 Some background on nite-dimensional optimization 3
1.2.1 Unconstrained optimization . . . . . . . . . . . . . . . 4
1.2.2 Constrained optimization . . . . . . . . . . . . . . . . 11
1.3 Preview of innite-dimensional optimization 17
1.3.1 Function spaces, norms, and local minima . . . . . . . 18
1.3.2 First variation and rst-order necessary condition . . . 19
1.3.3 Second variation and second-order conditions . . . . . 21
1.3.4 Global minima and convex problems . . . . . . . . . . 23
1.4 Notes and references for Chapter 1 24
2 Calculus of Variations 26
2.1 Examples of variational problems 26
2.1.1 Dido's isoperimetric problem . . . . . . . . . . . . . . 26
2.1.2 Light re
ection and refraction . . . . . . . . . . . . . . 27
2.1.3 Catenary . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.4 Brachistochrone . . . . . . . . . . . . . . . . . . . . . 30
2.2 Basic calculus of variations problem 32
2.2.1 Weak and strong extrema . . . . . . . . . . . . . . . . 33
2.3 First-order necessary conditions for weak extrema 34
2.3.1 Euler-Lagrange equation . . . . . . . . . . . . . . . . . 35
2.3.2 Historical remarks . . . . . . . . . . . . . . . . . . . . 39
2.3.3 Technical remarks . . . . . . . . . . . . . . . . . . . . 40
vii
viii
CONTENTS
2.3.4 Two special cases . . . . . . . . . . . . . . . . . . . . . 41
2.3.5 Variable-endpoint problems . . . . . . . . . . . . . . . 42
2.4 Hamiltonian formalism and mechanics 44
2.4.1 Hamilton's canonical equations . . . . . . . . . . . . . 45
2.4.2 Legendre transformation . . . . . . . . . . . . . . . . . 46
2.4.3 Principle of least action and conservation laws . . . . 48
2.5 Variational problems with constraints 51
2.5.1 Integral constraints . . . . . . . . . . . . . . . . . . . . 52
2.5.2 Non-integral constraints . . . . . . . . . . . . . . . . . 55
2.6 Second-order conditions 58
2.6.1 Legendre's necessary condition for a weak minimum . 59
2.6.2 Sucient condition for a weak minimum . . . . . . . . 62
2.7 Notes and references for Chapter 2 68
3 From Calculus of Variations to Optimal Control 71
3.1 Necessary conditions for strong extrema 71
3.1.1 Weierstrass-Erdmann corner conditions . . . . . . . . 71
3.1.2 Weierstrass excess function . . . . . . . . . . . . . . . 76
3.2 Calculus of variations versus optimal control 81
3.3 Optimal control problem formulation and assumptions 83
3.3.1 Control system . . . . . . . . . . . . . . . . . . . . . . 83
3.3.2 Cost functional . . . . . . . . . . . . . . . . . . . . . . 86
3.3.3 Target set . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 Variational approach to the xed-time, free-endpoint problem 89
3.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 89
3.4.2 First variation . . . . . . . . . . . . . . . . . . . . . . 92
3.4.3 Second variation . . . . . . . . . . . . . . . . . . . . . 95
3.4.4 Some comments . . . . . . . . . . . . . . . . . . . . . 96
3.4.5 Critique of the variational approach and preview of
the maximum principle . . . . . . . . . . . . . . . . . 98
3.5 Notes and references for Chapter 3 100
CONTENTS
ix
4 The Maximum Principle 102
4.1 Statement of the maximum principle 102
4.1.1 Basic xed-endpoint control problem . . . . . . . . . . 102
4.1.2 Basic variable-endpoint control problem . . . . . . . . 104
4.2 Proof of the maximum principle 105
4.2.1 From Lagrange to Mayer form . . . . . . . . . . . . . 107
4.2.2 Temporal control perturbation . . . . . . . . . . . . . 109
4.2.3 Spatial control perturbation . . . . . . . . . . . . . . . 110
4.2.4 Variational equation . . . . . . . . . . . . . . . . . . . 112
4.2.5 Terminal cone . . . . . . . . . . . . . . . . . . . . . . . 115
4.2.6 Key topological lemma . . . . . . . . . . . . . . . . . . 117
4.2.7 Separating hyperplane . . . . . . . . . . . . . . . . . . 120
4.2.8 Adjoint equation . . . . . . . . . . . . . . . . . . . . . 121
4.2.9 Properties of the Hamiltonian . . . . . . . . . . . . . . 122
4.2.10 Transversality condition . . . . . . . . . . . . . . . . . 126
4.3 Discussion of the maximum principle 128
4.3.1 Changes of variables . . . . . . . . . . . . . . . . . . . 130
4.4 Time-optimal control problems 134
4.4.1 Example: double integrator . . . . . . . . . . . . . . . 135
4.4.2 Bang-bang principle for linear systems . . . . . . . . . 138
4.4.3 Nonlinear systems, singular controls, and Lie brackets 141
4.4.4 Fuller's problem . . . . . . . . . . . . . . . . . . . . . 146
4.5 Existence of optimal controls 148
4.6 Notes and references for Chapter 4 153
5 The Hamilton-Jacobi-Bellman Equation 156
5.1 Dynamic programming and the HJB equation 156
5.1.1 Motivation: the discrete problem . . . . . . . . . . . . 156
5.1.2 Principle of optimality . . . . . . . . . . . . . . . . . . 158
5.1.3 HJB equation . . . . . . . . . . . . . . . . . . . . . . . 161
5.1.4 Sucient condition for optimality . . . . . . . . . . . 165
5.1.5 Historical remarks . . . . . . . . . . . . . . . . . . . . 167
5.2 HJB equation versus the maximum principle 168
x
CONTENTS
5.2.1 Example: nondierentiable value function . . . . . . . 170
5.3 Viscosity solutions of the HJB equation 172
5.3.1 One-sided dierentials . . . . . . . . . . . . . . . . . . 172
5.3.2 Viscosity solutions of PDEs . . . . . . . . . . . . . . . 174
5.3.3 HJB equation and the value function . . . . . . . . . . 176
5.4 Notes and references for Chapter 5 178
6 The Linear Quadratic Regulator 180
6.1 Finite-horizon LQR problem 180
6.1.1 Candidate optimal feedback law . . . . . . . . . . . . 181
6.1.2 Riccati dierential equation . . . . . . . . . . . . . . . 183
6.1.3 Value function and optimality . . . . . . . . . . . . . . 185
6.1.4 Global existence of solution for the RDE . . . . . . . . 187
6.2 Innite-horizon LQR problem 189
6.2.1 Existence and properties of the limit . . . . . . . . . . 190
6.2.2 Innite-horizon problem and its solution . . . . . . . . 193
6.2.3 Closed-loop stability . . . . . . . . . . . . . . . . . . . 194
6.2.4 Complete result and discussion . . . . . . . . . . . . . 196
6.3 Notes and references for Chapter 6 199
7 Advanced Topics 200
7.1 Maximum principle on manifolds 200
7.1.1 Dierentiable manifolds . . . . . . . . . . . . . . . . . 201
7.1.2 Re-interpreting the maximum principle . . . . . . . . 203
7.1.3 Symplectic geometry and Hamiltonian
ows . . . . . . 206
7.2 HJB equation, canonical equations, and characteristics 207
7.2.1 Method of characteristics . . . . . . . . . . . . . . . . 208
7.2.2 Canonical equations as characteristics of the HJB
equation . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.3 Riccati equations and inequalities in robust control 212
7.3.1 L2 gain . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.3.2 H1 control problem . . . . . . . . . . . . . . . . . . . 216
7.3.3 Riccati inequalities and LMIs . . . . . . . . . . . . . . 219
7.4 Maximum principle for hybrid control systems 219
CONTENTS
xi
7.4.1 Hybrid optimal control problem . . . . . . . . . . . . . 219
7.4.2 Hybrid maximum principle . . . . . . . . . . . . . . . 221
7.4.3 Example: light re
ection . . . . . . . . . . . . . . . . . 222
7.5 Notes and references for Chapter 7 223
Bibliography 225
Index 231

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关键词：introduction Variations troduction variation Calculus background necessary variation control problem

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