Optimal Control and Geometry: Integrable Systems

The synthesis of symplectic geometry, the calculus of variations and control theory offered in this book provides a crucial foundation for the understanding of many problems in applied mathematics. Focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the Maximum Principle of optimality, shed new light on the theory of integrable systems. These Hamiltonians provide an original and unified account of the existing theory of integrable systems. The book particularly explains much of the mystery surrounding the Kepler problem, the Jacobi problem and the Kovalevskaya Top. It also reveals the ubiquitous presence of elastic curves in integrable systems up to the soliton solutions of the non-linear Schroedinger's equation. Containing a useful blend of theory and applications, this is an indispensable guide for graduates and researchers in many fields, from mathematical physics to space control.1. The orbit theorem and Lie determined systems; 2. Control systems. Accessibility and controllability; 3. Lie groups and homogeneous spaces; 4. Symplectic manifolds. Hamiltonian vector fields; 5. Poisson manifolds, Lie algebras and coadjoint orbits; 6. Hamiltonians and optimality: the Maximum Principle; 7. Hamiltonian view of classic geometry; 8. Symmetric spaces and sub-Riemannian problems; 9. Affine problems on symmetric spaces; 10. Cotangent bundles as coadjoint orbits; 11. Elliptic geodesic problem on the sphere; 12. Rigid body and its generalizations; 13. Affine Hamiltonians on space forms; 14. Kowalewski-Lyapunov criteria; 15. Kirchhoff-Kowalewski equation; 16. Elastic problems on symmetric spaces: Delauney-Dubins problem; 17. Non-linear Schroedinger's equation and Heisenberg's magnetic equation. Solitons.