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Table of Contents
Intro; Preface; Acknowledgements; Contents; About the Authors; 1 Historical Part-Calculus of Variations; 1.1 Statement of the Problem in the Holonomic Case; 1.2 Hamiltonian Equations; 1.3 Hamilton-Jacobi-Bellman Equation; 1.4 Second Order Conditions; 1.5 The Accessory Problem and the Jacobi Equation; 1.6 Conjugate Point and Local Morse Theory; 1.7 From Calculus of Variations to Optimal Control Theory and Hamiltonian Dynamics; 2 Weak Maximum Principle and Application to Swimming at Low Reynolds Number; 2.1 Pre-requisite of Differential and Symplectic Geometry; 2.2 Controllability Results
2.2.1 Sussmann-Nagano Theorem2.2.2 Chow-Rashevskii Theorem; 2.3 Weak Maximum Principle; 2.4 Second Order Conditions and Conjugate Points; 2.4.1 Lagrangian Manifold and Jacobi Equation; 2.4.2 Numerical Computation of the Conjugate Loci Along a Reference Trajectory; 2.5 Sub-riemannian Geometry; 2.5.1 Sub-riemannian Manifold; 2.5.2 Controllability; 2.5.3 Distance; 2.5.4 Geodesics Equations; 2.5.5 Evaluation of the Sub-riemannian Ball; 2.5.6 Nilpotent Approximation; 2.5.7 Conjugate and Cut Loci in SR-Geometry; 2.5.8 Conjugate Locus Computation; 2.5.9 Integrable Case
2.5.10 Nilpotent Models in Relation with the Swimming Problem2.6 Swimming Problems at Low Reynolds Number; 2.6.1 Purcell's 3-Link Swimmer; 2.6.2 Copepod Swimmer; 2.6.3 Some Geometric Remarks; 2.6.4 Purcell Swimmer; 2.7 Numerical Results; 2.7.1 Nilpotent Approximation; 2.7.2 True Mechanical System; 2.7.3 Copepod Swimmer; 2.8 Conclusion and Bibliographic Remarks; 3 Maximum Principle and Application to Nuclear Magnetic Resonance and Magnetic Resonance Imaging; 3.1 Maximum Principle; 3.2 Special Cases; 3.3 Application to NMR and MRI; 3.3.1 Model; 3.3.2 The Problems
2.2.1 Sussmann-Nagano Theorem2.2.2 Chow-Rashevskii Theorem; 2.3 Weak Maximum Principle; 2.4 Second Order Conditions and Conjugate Points; 2.4.1 Lagrangian Manifold and Jacobi Equation; 2.4.2 Numerical Computation of the Conjugate Loci Along a Reference Trajectory; 2.5 Sub-riemannian Geometry; 2.5.1 Sub-riemannian Manifold; 2.5.2 Controllability; 2.5.3 Distance; 2.5.4 Geodesics Equations; 2.5.5 Evaluation of the Sub-riemannian Ball; 2.5.6 Nilpotent Approximation; 2.5.7 Conjugate and Cut Loci in SR-Geometry; 2.5.8 Conjugate Locus Computation; 2.5.9 Integrable Case
2.5.10 Nilpotent Models in Relation with the Swimming Problem2.6 Swimming Problems at Low Reynolds Number; 2.6.1 Purcell's 3-Link Swimmer; 2.6.2 Copepod Swimmer; 2.6.3 Some Geometric Remarks; 2.6.4 Purcell Swimmer; 2.7 Numerical Results; 2.7.1 Nilpotent Approximation; 2.7.2 True Mechanical System; 2.7.3 Copepod Swimmer; 2.8 Conclusion and Bibliographic Remarks; 3 Maximum Principle and Application to Nuclear Magnetic Resonance and Magnetic Resonance Imaging; 3.1 Maximum Principle; 3.2 Special Cases; 3.3 Application to NMR and MRI; 3.3.1 Model; 3.3.2 The Problems