Reduction of Sufficient Conditions for Optimal Control Problems with Subgroup Symmetry

In this paper, we apply symmetry reduction techniques from geometric mechanics to sufficient conditions for local optimality in optimal control problems. After reinterpreting some previous results for left-invariant problems on Lie groups, we focus on optimal control problems with subgroup symmetry. For these problems, the necessary conditions for optimality can be simplified by exploiting symmetries so as to reduce the number of variables needed to describe trajectories of the system. We show that sufficient conditions for optimality, based on the non-existence of conjugate points, can be simplified in an analogous way to the necessary conditions. We demonstrate these simplifications by analyzing an optimal control problem that models a spinning top in a gravitational field, and we give particular attention to the example of an axisymmetric sleeping top. The results we derive in this paper allow us to determine which trajectories of a sleeping top are locally optimal solutions of the optimal control problem, which is a new result that has not appeared in previous literature.