Calculus of Variations and Optimal Control Theory: A Concise Introduction

5.3.1 One-sided differentials......Page 1
Title Page......Page 4
Table of Contents......Page 8
Preface......Page 14
1.1 Optimal control problem......Page 20
1.2 Some background on finite-dimensional optimization......Page 22
1.2.1 Unconstrained optimization......Page 23
1.2.2 Constrained optimization......Page 30
1.3 Preview of infinite-dimensional optimization......Page 36
1.3.1 Function spaces, norms, and local minima......Page 37
1.3.2 First variation and first-order necessary condition......Page 38
1.3.3 Second variation and second-order conditions......Page 40
1.3.4 Global minima and convex problems......Page 42
1.4 Notes and references for Chapter 1......Page 43
2.1.1 Dido\'s isoperimetric problem......Page 45
2.1.2 Light reflection and refraction......Page 46
2.1.3 Catenary......Page 47
2.1.4 Brachistochrone......Page 49
2.2 Basic calculus of variations problem......Page 51
2.2.1 Weak and strong extrema......Page 52
2.3 First-order necessary conditions for weak extrema......Page 53
2.3.1 Euler-Lagrange equation......Page 54
2.3.2 Historical remarks......Page 58
2.3.3 Technical remarks......Page 59
2.3.4 Two special cases......Page 60
2.3.5 Variable-endpoint problems......Page 61
2.4 Hamiltonian formalism and mechanics......Page 63
2.4.1 Hamilton\'s canonical equations......Page 64
2.4.2 Legendre transformation......Page 65
2.4.3 Principle of least action and conservation laws......Page 67
2.5 Variational problems with constraints......Page 70
2.5.1 Integral constraints......Page 71
2.5.2 Non-integral constraints......Page 74
2.6 Second-order conditions......Page 77
2.6.1 Legendre\'s necessary condition for a weak minimum......Page 78
2.6.2 Sufficient condition for a weak minimum......Page 81
2.7 Notes and references for Chapter 2......Page 87
3.1.1 Weierstrass-Erdmann corner conditions......Page 90
3.1.2 Weierstrass excess function......Page 95
3.2 Calculus of variations versus optimal control......Page 100
3.3.1 Control system......Page 102
3.3.2 Cost functional......Page 105
3.3.3 Target set......Page 107
3.4.1 Preliminaries......Page 108
3.4.2 First variation......Page 111
3.4.3 Second variation......Page 114
3.4.4 Some comments......Page 115
3.4.5 Critique of the variational approach and preview of the maximum principle......Page 117
3.5 Notes and references for Chapter 3......Page 119
4.1.1 Basic fixed-endpoint control problem......Page 121
4.1.2 Basic variable-endpoint control problem......Page 123
4.2 Proof of the maximum principle......Page 124
4.2.1 From Lagrange to Mayer form......Page 126
4.2.2 Temporal control perturbation......Page 128
4.2.3 Spatial control perturbation......Page 129
4.2.4 Variational equation......Page 131
4.2.5 Terminal cone......Page 134
4.2.6 Key topological lemma......Page 136
4.2.7 Separating hyperplane......Page 139
4.2.8 Adjoint equation......Page 140
4.2.9 Properties of the Hamiltonian......Page 141
4.2.10 Transversality condition......Page 145
4.3 Discussion of the maximum principle......Page 147
4.3.1 Changes of variables......Page 149
4.4 Time-optimal control problems......Page 153
4.4.1 Example: double integrator......Page 154
4.4.2 Bang-bang principle for linear systems......Page 157
4.4.3 Nonlinear systems, singular controls, and Lie brackets......Page 160
4.4.4 Fuller\'s problem......Page 165
4.5 Existence of optimal controls......Page 167
4.6 Notes and references for Chapter 4......Page 172
5.1.1 Motivation: the discrete problem......Page 175
5.1.2 Principle of optimality......Page 177
5.1.3 HJB equation......Page 180
5.1.4 Sufficient condition for optimality......Page 184
5.1.5 Historical remarks......Page 186
5.2 HJB equation versus the maximum principle......Page 187
5.2.1 Example: nondifferentiable value function......Page 189
5.3.2 Viscosity solutions of PDEs......Page 193
5.3.3 HJB equation and the value function......Page 195
5.4 Notes and references for Chapter 5......Page 197
6.1 Finite-horizon LQR problem......Page 199
6.1.1 Candidate optimal feedback law......Page 200
6.1.2 Riccati differential equation......Page 202
6.1.3 Value function and optimality......Page 204
6.1.4 Global existence of solution for the RDE......Page 206
6.2 Infinite-horizon LQR problem......Page 208
6.2.1 Existence and properties of the limit......Page 209
6.2.2 Infinite-horizon problem and its solution......Page 212
6.2.3 Closed-loop stability......Page 213
6.2.4 Complete result and discussion......Page 215
6.3 Notes and references for Chapter 6......Page 218
7.1 Maximum principle on manifolds......Page 219
7.1.1 Differentiable manifolds......Page 220
7.1.2 Re-interpreting the maximum principle......Page 222
7.1.3 Symplectic geometry and Hamiltonian ows......Page 225
7.2 HJB equation, canonical equations, and characteristics......Page 226
7.2.1 Method of characteristics......Page 227
7.2.2 Canonical equations as characteristics of the HJB equation......Page 230
7.3 Riccati equations and inequalities in robust control......Page 231
7.3.1 L2 gain......Page 232
7.3.2 H** control problem......Page 235
7.4.1 Hybrid optimal control problem......Page 238
7.4.2 Hybrid maximum principle......Page 240
7.4.3 Example: light reflection......Page 241
7.5 Notes and references for Chapter 7......Page 242
Bibliography......Page 244
Index......Page 250