Discrete Clebsch optimal control

In this paper we analyze a class of discrete optimal control problems. These systems are discretizations of a class of optimal control problems defined on invariant submanifolds which we denote embedded optimal control problems. We analyze a particular subset of these called discrete Clebsch optimal control problems where the invariant manifolds are group orbits. The generating Hamiltonian equations for such systems are analyzed. The analysis provides a large class of geometric integrators for mechanical systems. We apply the theory to two example systems: mechanical systems on matrix Lie groups and mechanical systems on the n-sphere.