Jurdjevic V. Optimal Control and Geometry: Integrable Systems

Cambridge University Press, UK, 2016. — 435 p. — (Cambridge Studies in Advanced Mathematics 154) — ISBN10: 1107113881.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.The Orbit Theorem and Lie determined systems
Control systems: accessibility and controllability
Lie groups and homogeneous spaces
Symplectic manifolds: Hamiltonian vector fields
Poisson manifolds, Lie algebras, and coadjoint orbits
Hamiltonians and optimality: the Maximum Principle
Hamiltonian view of classic geometry
Symmetric spaces and sub-Riemannian problems
Affine-quadratic problem
Cotangent bundles of homogeneous spaces as coadjoint orbits
Elliptic geodesic problem on the sphere
Rigid body and its generalizations
Isometry groups of space forms and affine systems: Kirchhoff’s elastic problem
Kowalewski–Lyapunov criteria
Kirchhoff–Kowalewski equation
Elastic problems on symmetric spaces: the Delauney–Dubins problem
The non-linear Schroedinger’s equation and Heisenberg’s magnetic equation–solitons