The Geometry of Integrable Systems and Optimal Control

In this talk we discuss a geometric approach to certain optimal control problems and discuss the relationship of the solutions of these problems to some classical integrable dynamical systems and their generalizations. We consider the so-called Clebsch optimal control problem and its relationship to Lie group actions on manifolds. The integrable systems discussed include the rigid body equations, geodesic flows on the ellipsoid, flows on Stiefel manifolds, and the Toda lattice flows. We discuss the Hamiltonian structure of these systems and relate our work to some work of Moser. We also discuss the link to discrete dynamics and symplectic integration.