Optimal Control and Geometry: Integrable Systems

Blending control theory, mechanics, geometry and the calculus of variations, this book is a vital resource for graduates and researchers in engineering, mathematics and physics.

Velimir Jurdjevic (Author)

9781107113886, Cambridge University Press

Hardback, published 4 July 2016

423 pages, 4 tables
23.5 x 15.9 x 2.7 cm, 0.74 kg

'The book is written in a very refreshing style that reflects the author's contagious enthusiasm for the subject. It manages this by focusing on the geometry, using the precise language of differential geometry, while not getting bogged down by analytic intricacies. Throughout, the book pays much attention to historical developments and evolving and contrasting points of view, which is also reflected in the rich bibliography of classical resources.' Matthias Kawski, MathSciNet

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.

Subject Areas: Differential calculus & equations [PBKJ], Mathematics [PB]