Bloch, Anthony M.Colombo, Leonardo J.Gupta, RohitOhsawa, Tomoki2019-06-282019-06-282017-10-302470-6566https://hdl.handle.net/10735.1/6648We 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-Poincare 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.en©2017 Society for Industrial and Applied MathematicsEuler characteristicLagrange equationsLie groupsarticleBloch, Anthony M., Leonardo J. Colombo, Rohit Gupta, and Tomoki Ohsawa. 2017. "Optimal control problems with symmetry breaking cost functions." SIAM Journal on Applied Algebra and Geometry 1(1): 626-646, doi:10.1137/16M109165411