```Contents
1 Introduction to Optimal Control  . . . . . . . . . . . . . . . . . . . 1
1.1 Optimal Control and Maximum Principle  . . . . . . . . . . . . . . . 2
1.1.1 Preliminaries  . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 The Weak Maximum Principle . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Affine Control Systems and Connection with General
Control Systems  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.5 Computation of Singular Controls . . . . . . . . . . . . . . . . . 7
1.1.6 Singular Trajectories and Feedback Classifcation . . . . . . . . . 8
1.1.7 Maximum Principle with Fixed Time  . . . . . . . . . . . . . . . . 9
1.1.8 Maximum Principle, the General Case . . . . . . . . . . . . . . . 11
1.1.9 Examples  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.10 The Shooting Equation  . . . . . . . . . . . . . . . . . . . . . 13
1.2 Second Order Necessary and Sufficient Conditions in the
Generic Case  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 Second Order Conditions in the Classical Calculus of
Variations  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 Symplectic Geometry and Second Order Optimality
Conditions under Generic Assumptions  . . . . . . . . . . . . . . . . . 19
1.2.3 Second Order Optimality Conditions in the Affine Case . . . . . . 31
1.2.4 Existence Theorems in Optimal Control . . . . . . . . . . . . . . 47
2 Riemannian Geometry and Geometric Control Theory  . . . . . . . . . . 49
2.1 Generalities about SR-Geometry  . . . . . . . . . . . . . . . . . . 50
2.1.1 Optimal Control Theory Formulation  . . . . . . . . . . . . . . . 51
2.1.2 Computation of the Extremals and Exponential
Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2 A Property of the Distance Function . . . . . . . . . . . . . . . . 54
2.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Classi¯cation of SR Problems. . . . . . . . . . . . . . . . . . . . 55
2.4 Two Cases Studies . . . . . . . . . . . . . . . . . . . . . . . . . 55
XII Contents
2.4.1 The Heisenberg Case . . . . . . . . . . . . . . . . . . . . . . . 55
2.4.2 The Martinet Flat Case. . . . . . . . . . . . . . . . . . . . . . 58
2.4.3 The Generalizations . . . . . . . . . . . . . . . . . . . . . . . 61
2.4.4 A Conclusion about SR Spheres . . . . . . . . . . . . . . . . . . 63
2.5 The Riemannian Case . . . . . . . . . . . . . . . . . . . . . . . . 63
2.5.1 A Brief Review of Riemannian Geometry . . . . . . . . . . . . . . 63
2.5.2 Clairaut-Liouville Metrics. . . . . . . . . . . . . . . . . . . . 66
2.5.3 The Optimality Problem. . . . . . . . . . . . . . . . . . . . . . 68
2.5.4 Conjugate and Cut Loci on Two-Spheres of Revolution . . . . . . . 68
2.6 An Example of Almost Riemannian Structure: The Grushin Model  . . . 73
2.6.1 The Grushin Model on R2 . . . . . . . . . . . . . . . . . . . . . 74
2.6.2 The Grushin Model on S2 . . . . . . . . . . . . . . . . . . . . . 75
2.6.3 Generalization of the Grushin Case  . . . . . . . . . . . . . . . 77
2.6.4 Conjugate and Cut Loci for Metrics on the
Two-Sphere with Singularities . . . . . . . . . . . . . . . . . . . . . 78
2.6.5 Homotopy on Clairaut-Liouville Metrics and
Continuation Technique. . . . . . . . . . . . . . . . . . . . . . . . . 79
2.7 Extension of SR Geometry to Systems with Drift  . . . . . . . . . . 79
2.7.1 Examples  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.8 Generic Extremals Analysis  . . . . . . . . . . . . . . . . . . . . 82
2.8.1 An Application to SR Problems with Drift in Dimension 4 . . . . . 84
3 Orbital Transfer Problem  . . . . . . . . . . . . . . . . . . . . . . 87
3.1 The Model for the Controlled Kepler Equation  . . . . . . . . . . . 87
3.1.1 First Integrals of Kepler Equation and Orbit Elements . . . . . . 88
3.1.2 Connection with a Linear Oscillator . . . . . . . . . . . . . . . 88
3.1.3 Orbit Elements for Elliptic Orbits  . . . . . . . . . . . . . . . 89
3.2 A Review of Geometric Controllability Techniques and Results  . . . 92
3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.2.2 Basic Controllability Results . . . . . . . . . . . . . . . . . . 93
3.2.3 Controllability and Enlargement Technique . . . . . . . . . . . . 94
3.3 Lie Bracket Computations and Controllability in Orbital Transfer  . 96
3.3.1 Lie Bracket Computations  . . . . . . . . . . . . . . . . . . . . 96
3.3.2 Controllability Results . . . . . . . . . . . . . . . . . . . . . 98
3.4 Constructing a Feedback Control Using Stabilization Techniques  . . 98
3.4.1 Stability Results . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4.2 Stabilization of Nonlinear Systems . . . . . . . . . . . . . . . 100
3.4.3 Application to the Orbital Transfer  . . . . . . . . . . . . . . 101
3.5 Optimal Control Problems in Orbital Transfer . . . . . . . . . . . 102
3.5.1 Physical Problems  . . . . . . . . . . . . . . . . . . . . . . . 102
3.5.2 Extremal Trajectories  . . . . . . . . . . . . . . . . . . . . . 103
Contents XIII
3.6 Preliminary Results on the Time-Minimal Control Problem  . . . . . 107
3.6.1 Homotopy on the Maximal Thrust . . . . . . . . . . . . . . . . . 107
3.6.2 Conjugate Points . . . . . . . . . . . . . . . . . . . . . . . . 107
3.7 Extremals for Single-Input Time-Minimal Control  . . . . . . . . . 107
3.7.1 Singular Extremals . . . . . . . . . . . . . . . . . . . . . . . 108
3.7.2 Classi¯cation of Regular Extremals . . . . . . . . . . . . . . . 109
3.7.3 The Fuller Phenomenon  . . . . . . . . . . . . . . . . . . . . . 111
3.8 Application to Time Minimal Transfer with Cone Constraints . . . . 112
3.9 Averaged System in the Energy Minimization Problem . . . . . . . . 113
3.9.1 Averaging Techniques for Ordinary Differential
Equations and Extensions to Control Systems  . . . . . . . . . . . . . 113
3.9.2 Controllability Property and Averaging Techniques  . . . . . . . 114
3.9.3 Riemannian Metric of the Averaged Controlled Kepler Equation . . 115
3.9.4 Computation of the Averaged System in Coplanar Orbital Transfer  118
3.10 The Analysis of the Averaged System . . . . . . . . . . . . . . . 120
3.10.1 Analysis of ¹g1 . . . . . . . . . . . . . . . . . . . . . . . . 121
3.10.2 Integrability of the Extremal Flow  . . . . . . . . . . . . . . 122
3.10.3 Geometric Properties of ¹g2 . . . . . . . . . . . . . . . . . . 124
3.10.4 A Global Optimality Result with Application to Orbital Transfer 125
3.10.5 Riemann Curvature and Injectivity Radius in Orbital Transfer  . 127
3.10.6 Cut Locus on S2 and Global Optimality Results in
Orbital Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.11 The Averaged System in the Tangential Case  . . . . . . . . . . . 129
3.11.1 Construction of the Normal Form . . . . . . . . . . . . . . . . 129
3.11.2 The Metric g1 . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.11.3 The Metric g2 . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.11.4 The Integration of the Extremal Flow  . . . . . . . . . . . . . 131
3.11.5 A Continuation Result . . . . . . . . . . . . . . . . . . . . . 131
3.12 Conclusion in Both Cases  . . . . . . . . . . . . . . . . . . . . 131
3.13 The Averaged System in the Orthoradial Case . . . . . . . . . . . 133
3.14 Averaged System for Non-Coplanar Transfer . . . . . . . . . . . . 133
3.15 The Energy Minimization Problem in the Earth-Moon Space Mission . 134
3.15.1 Mathematical Model and Presentation of the Problem  . . . . . . 134
3.15.2 The Circular Restricted 3-Body Problem in Jacobi Coordinates  . 135
3.15.3 Jacobi Integral and Hill Regions  . . . . . . . . . . . . . . . 136
3.15.4 Equilibrium Points  . . . . . . . . . . . . . . . . . . . . . . 137
3.15.5 The Continuation Method in the Earth-Moon Transfer  . . . . . . 137
XIV Contents
4 Optimal Control of Quantum Systems . . . . . . . . . . . . . . . . . 149
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.2 Control of Dissipative Quantum Systems . . . . . . . . . . . . . . 151
4.2.1 Quantum Mechanics of Open Systems  . . . . . . . . . . . . . . . 151
4.2.2 The Kossakowski-Lindblad Equation  . . . . . . . . . . . . . . . 158
4.2.3 Construction of the Model  . . . . . . . . . . . . . . . . . . . 160
4.3 Controllability of Right-Invariant Systems on Lie Groups . . . . . 162
4.3.1 Preliminaries  . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.3.2 The Case of SL(2; R) . . . . . . . . . . . . . . . . . . . . . . 164
4.3.3 Controllability on Sp(n; R)  . . . . . . . . . . . . . . . . . . 173
4.4 Time Minimal Control of the Lindblad Equation  . . . . . . . . . . 175
4.4.1 Symmetry of Revolution . . . . . . . . . . . . . . . . . . . . . 175
4.4.2 Spherical Coordinates  . . . . . . . . . . . . . . . . . . . . . 176
4.4.3 Lie Brackets Computations  . . . . . . . . . . . . . . . . . . . 178
4.4.4 Singular Trajectories  . . . . . . . . . . . . . . . . . . . . . 180
4.4.5 The Time-Optimal Control Problem . . . . . . . . . . . . . . . . 181
4.5 Single-Input Time-Optimal Control Problem  . . . . . . . . . . . . 182
4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.5.2 Methodology  . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.5.3 Four Di®erent Illustrative Examples  . . . . . . . . . . . . . . 187
4.5.4 Physical Interpretation  . . . . . . . . . . . . . . . . . . . . 191
4.5.5 Complete Classi¯cation . . . . . . . . . . . . . . . . . . . . . 191
4.6 The Two-Input Time-Optimal Case  . . . . . . . . . . . . . . . . . 196
4.6.1 The Integrable Case  . . . . . . . . . . . . . . . . . . . . . . 196
4.6.2 Numerical Determination of the Conjugate Locus . . . . . . . . . 201
4.6.3 Geometric Interpretation of the Integrable Case  . . . . . . . . 203
4.6.4 The Generic Case °¡ = 0 6 :  . . . . . . . . . . . . . . . . . . 204
4.6.5 Regularity Analysis  . . . . . . . . . . . . . . . . . . . . . . 205
4.6.6 Abnormal Analysis  . . . . . . . . . . . . . . . . . . . . . . . 209
4.6.7 Singular Value Decomposition . . . . . . . . . . . . . . . . . . 211
4.6.8 Continuation Method  . . . . . . . . . . . . . . . . . . . . . . 215
4.7 The Energy Minimization Problem  . . . . . . . . . . . . . . . . . 218
4.7.1 Geometric Analysis of the Extremal Curves  . . . . . . . . . . . 219
4.7.2 The Optimality Problem . . . . . . . . . . . . . . . . . . . . . 238
4.7.3 Numerical Simulations  . . . . . . . . . . . . . . . . . . . . . 256
4.8 Application to Nuclear Magnetic Resonance  . . . . . . . . . . . . 261
4.9 The Contrast Imaging Problem in NMR  . . . . . . . . . . . . . . . 265
4.9.1 The Model System . . . . . . . . . . . . . . . . . . . . . . . . 267
4.9.2 The Geometric Necessary Optimality Conditions and the Dual
Problem of Extremizing the Transfer Time to a Given Manifold . . . . . 268
4.9.3 Second-Order Necessary and Sufficient Optimality Conditions  . . 270
4.9.4 An Example of the Contrast Problem . . . . . . . . . . . . . . . 270
Contents XV
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Index  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281```