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