Optimal Control and Geometry: Integrable Systems (Cambridge Studies in Advanced Mathematics)
Cambridge University Press, 7/4/2016
EAN 9781107113886, ISBN10: 1107113881
Hardcover, 423 pages, 22.8 x 15.2 x 2.7 cm
Language: English
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.