Title: Optimal Control Problems with Symmetry Breaking Cost Functions,
Abstract: We investigate symmetry reduction of optimal control problems for
left-invariant control 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
applying a Legendre transformation to it, we recover the Lie-Poisson equations
obtained by A. D. Borum [Master's Thesis, University of Illinois at
Urbana-Champaign, 2015] 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.