Optimal control and geodesic flows

In this paper we analyze and generalize, from the point of view of the maximum principle, a class of nonlinear optimal control problems originally introduced in Brockett (1994). The optimal control problems discussed here lie on the adjoint orbits of compact Lie groups or on Grassmann manifolds and the structure of the equations that arise depends on the metric used - the so-called normal metric. We consider some special cases of the general problem and their meaning from the point of view of Riemannian geometry and Hamiltonian mechanics.