Optimal Control with Applications in Space and Quantum Dynamics

Cover......Page 1
Optimal Control with Applications in Space and Quantum Dynamics......Page 2
ISBN-10: 1601330138 ISBN-13: 9781601330130......Page 4
Preface......Page 6
Contents......Page 12
1 Introduction to Optimal Control......Page 18
1.1.1 Preliminaries......Page 19
1.1.2 The Weak Maximum Principle......Page 20
1.1.5 Computation of Singular Controls......Page 23
1.1.6 Singular Trajectories and Feedback Classification......Page 25
1.1.7 Maximum Principle with Fixed Time......Page 26
1.1.8 Maximum Principle, the General Case......Page 28
1.1.9 Examples Smooth Calculus of Variations......Page 29
1.1.10 The Shooting Equation......Page 30
1.2.1 Second order conditions in the Classical Calculus of Variations Preliminaries......Page 31
1.2.2 Symplectic Geometry and Second Order Optimality Conditions under Generic Assumptions Symplectic Geometry and Lagrangian Manifolds......Page 35
1.2.3 Second Order Optimality Conditions in the Affine Case General Properties......Page 48
1.2.4 Existence Theorems in Optimal Control......Page 63
2 Riemannian Geometry and Extension Arising in Geometric Control Theory......Page 66
2.1 Generalities About SR-Geometry......Page 67
2.1.1 Optimal Control Theory Formulation......Page 68
2.1.2 Computation of the Extremals and Exponential Mapping......Page 69
2.2.1 Preliminaries......Page 71
2.4.1 The Heisenberg Case......Page 72
2.4.2 The Martinet Flat Case......Page 75
2.4.3 The Generalizations......Page 77
2.5.1 A Brief Review of Riemannian Geometry......Page 80
2.5.2 Clairaut-Liouville Metrics......Page 83
2.5.4 Conjugate and Cut Loci on Two-Spheres of Revolution......Page 85
2.6 An Example of Almost Riemannian Structure: the Grushin Model......Page 90
2.6.1 The Grushin Model on R2......Page 91
2.6.2 The Grushin Model on S2......Page 92
2.6.3 Generalization of the Grushin case......Page 94
2.6.4 Conjugate and cut loci for metrics on the two-sphere with singularities......Page 95
2.7.1 Examples......Page 96
2.8 Generic Extremals Analysis......Page 99
2.8.1 An Application to SR Problems with Drift in Dimension 4......Page 101
3.1 The Model for the Controlled Kepler Equation......Page 104
3.1.2 Connection with a Linear Oscillator......Page 105
3.1.3 Orbit Elements for Elliptic Orbits......Page 106
3.2.1 Preliminaries......Page 109
3.2.2 Basic Controllability Results......Page 110
3.2.3 Controllability and Enlargement Technique......Page 111
3.3.1 Lie Bracket Computations......Page 113
3.3.2 Controllability Results......Page 114
3.4.1 Stability Results......Page 115
3.4.2 Stabilization of Nonlinear Systems via La Salle Theorem......Page 117
3.4.3 Application to the Orbital Transfer......Page 118
3.5.1 Physical Problems......Page 119
3.5.2 Extremal Trajectories......Page 120
3.6 Preliminary results on the time-minimal control problem......Page 123
3.7.1 Singular Extremals......Page 124
3.7.2 Classification of Regular Extremals......Page 125
3.7.3 The Fuller Phenomenon......Page 128
3.8 Application to Time Minimal Transfer with Cone Constraints......Page 129
3.9.1 Averaging Techniques for Ordinary Di®erential Equations and Extensions to Control Systems......Page 130
3.9.2 Controllability Property and Averaging Techniques......Page 131
3.9.3 Riemannian Metric of the Averaged Controlled Kepler Equation......Page 132
3.9.4 Computation of the Averaged System in Coplanar Orbital Transfer......Page 135
3.10 The Analysis of the Averaged System......Page 136
3.10.1 Analysis of g¹1......Page 137
3.10.2 Integrability of the Extremal Flow......Page 138
3.10.3 Geometric Properties of g¹2......Page 140
3.10.4 A Global Optimality Result with Application to Orbital Transfer......Page 141
3.10.5 Riemann Curvature and Injectivity Radius in Orbital Transfer......Page 143
3.10.6 Cut Locus on S2 and Global Optimality Results in Orbital Transfer......Page 144
3.11.1 Construction of the Normal Form......Page 145
3.11.3 The Metric g2......Page 146
3.12 Conclusion in Both Cases......Page 147
3.13 The Averaged System in the Orthoradial Case......Page 148
3.14 Averaged System for Non-Coplanar Transfer......Page 149
3.15.1 Mathematical model and presentation of the problem.......Page 150
3.15.2 The circular restricted 3-body problem in Jacobi coordinates......Page 151
3.15.4 Equilibrium points......Page 152
3.15.5 The continuation method in the Earth-Moon transfer......Page 153
4.1 Introduction......Page 164
4.2.1 Quantum Mechanics of Open Systems......Page 166
4.2.2 The Kossakowski-Lindblad equation......Page 173
4.2.3 Construction of the Model......Page 175
4.3.1 Preliminaries......Page 177
4.3.2 The case of SL(2; R)......Page 179
4.3.3 Controllability on Sp(n; R)......Page 188
4.4 Geometric analysis of the time minimal control of the Kossakowski-Lindblad equation......Page 189
4.4.1 Symmetry of revolution......Page 190
4.4.2 Spherical coordinates......Page 191
4.4.3 Lie Brackets Computations......Page 193
4.4.4 Singular trajectories......Page 195
4.4.5 The Time-Optimal Control Problem......Page 196
4.5.1 Introduction......Page 197
4.5.2 Methodology......Page 198
4.5.3 Four Different Illustrative Examples......Page 202
4.5.5 Complete classification......Page 206
4.6.1 The integrable case......Page 211
4.6.2 Numerical determination of the conjugate locus......Page 215
4.6.3 Geometric Interpretation of the Integrable Case......Page 217
4.6.4 The Generic Case gamma_= 0:......Page 219
4.6.5 Regularity Analysis......Page 221
4.6.6 Abnormal Analysis......Page 224
4.6.7 Singular value decomposition The normal case......Page 225
4.6.8 Continuation method......Page 228
4.7.1 Geometric analysis of the extremal curves Maximum principle......Page 233
4.7.2 The optimality problem......Page 252
4.7.3 Numerical simulations......Page 271
4.8 Application to Nuclear Magnetic Resonance......Page 274
4.9 The contrast imaging problem in NMR......Page 279
4.9.1 The model system......Page 280
4.9.2 The geometricTgfjqj jj··......Page 282
4.9.3 Second-order necessary and sufficient optimality conditions......Page 284
4.9.4 An example of the contrast problem......Page 285
References......Page 288
Index......Page 296