Optimal Control with Broken Symmetry of Multi-Agent Systems on Lie Groups

In this paper we study reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Hamiltonian and obtain the reduced optimality conditions from a reduced variational principle via Pontryagin Maximum Principle. We apply the results to a collision avoidance problem for multiple unicycles in the presence of an obstacle.