Calculus of Variations and Optimal Control Theory: A Concise Introduction (Free preliminary copy)

Preface......Page 9
Optimal control problem......Page 11
Unconstrained optimization......Page 13
Constrained optimization......Page 19
Function spaces, norms, and local minima......Page 24
First variation and first-order necessary condition......Page 26
Second variation and second-order conditions......Page 27
Global minima and convex problems......Page 29
Notes and references for Chapter 1......Page 30
Dido\'s isoperimetric problem......Page 31
Light reflection and refraction......Page 32
Catenary......Page 33
Brachistochrone......Page 34
Weak and strong extrema......Page 36
Euler-Lagrange equation......Page 38
Historical remarks......Page 41
Technical remarks......Page 42
Two special cases......Page 43
Variable-endpoint problems......Page 44
Hamilton\'s canonical equations......Page 46
Legendre transformation......Page 47
Principle of least action and conservation laws......Page 49
Variational problems with constraints......Page 51
Integral constraints......Page 52
Non-integral constraints......Page 54
Second-order conditions......Page 56
Legendre\'s necessary condition for a weak minimum......Page 57
Sufficient condition for a weak minimum......Page 60
Notes and references for Chapter 2......Page 64
Weierstrass-Erdmann corner conditions......Page 67
Weierstrass excess function......Page 71
Calculus of variations versus optimal control......Page 75
Control system......Page 77
Cost functional......Page 79
Target set......Page 80
Preliminaries......Page 81
First variation......Page 83
Second variation......Page 86
Some comments......Page 87
Critique of the variational approach and preview of Metapost......Page 88
Notes and references for Chapter 3......Page 90
Basic fixed-endpoint control problem......Page 91
Basic variable-endpoint control problem......Page 92
Proof of the maximum principle......Page 94
From Lagrange to Mayer form......Page 95
Spatial control perturbation......Page 97
Variational equation......Page 99
Terminal cone......Page 101
Key topological lemma......Page 103
Separating hyperplane......Page 105
Adjoint equation......Page 107
Properties of the Hamiltonian......Page 108
Transversality condition......Page 110
Discussion of Metapost......Page 112
Changes of variables......Page 114
Example: double integrator......Page 117
Bang-bang principle for linear systems......Page 120
Nonlinear systems, singular controls, and Lie brackets......Page 122
Fuller\'s problem......Page 126
Existence of optimal controls......Page 128
Notes and references for Chapter 4......Page 131
Motivation: the discrete problem......Page 135
Principle of optimality......Page 137
HJB equation......Page 139
Sufficient condition for optimality......Page 142
Historical remarks......Page 144
HJB equation versus Metapost......Page 145
Example: nondifferentiable value function......Page 146
One-sided differentials......Page 147
Viscosity solutions of PDEs......Page 150
HJB equation and the value function......Page 151
Notes and references for Chapter 5......Page 153
Finite-horizon LQR problem......Page 155
Candidate optimal feedback law......Page 156
Riccati differential equation......Page 158
Value function and optimality......Page 159
Global existence of solution for the RDE......Page 161
Existence and properties of the limit......Page 163
Infinite-horizon problem and its solution......Page 165
Closed-loop stability......Page 166
Complete result and discussion......Page 167
Notes and references for Chapter 6......Page 170
Maximum principle on manifolds......Page 171
Differentiable manifolds......Page 172
Re-interpreting the maximum principle......Page 173
Symplectic geometry and Hamiltonian flows......Page 176
Method of characteristics......Page 177
Canonical equations as characteristics of the HJB equation......Page 180
L2 gain......Page 181
H control problem......Page 184
Hybrid optimal control problem......Page 186
Hybrid maximum principle......Page 187
Example: light reflection......Page 188
Bibliography......Page 189
Index......Page 191