  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.
