Optimal Control Problems with Symmetry Breaking Cost Functions

We investigate symmetry reduction of optimal control problems for left-invariant control affine systems on Lie groups, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles and considers a discrete-time setting as well as the standard continuous-time formulation. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the Euler--Poincaré equations from a variational principle. By using a Legendre transformation, we recover the Lie--Poisson equations obtained by Borum and Bretl [ IEEE Trans. Automat. Control, 62 (2017), pp. 3209--3224] in the same context. We also discretize the variational principle in time and obtain the discrete-time Lie--Poisson equations. We illustrate the theory with some practical examples including a motion planning problem in the presence of an obstacle.