Lagrange-Poincar\'e reduction for optimal control of underactuated mechanical systems

We deal with regular Lagrangian constrained systems which are invariant under the action of a symmetry group. Fixing a connection on the higher-order principal bundle where the Lagrangian and the (independent) constraints are defined, the higher-order Lagrange-Poincar\'e equations of classical mechanical systems with higher-order constraints are obtained from classical Lagrangian reduction. Higher-order Lagrange-Poincar\'e operator is introduced to characterize higher-order Lagrange-Poincar\'e equations. Interesting applications are derived as, for instance, the optimal control of an underactuated Elroy's Beanie and a snakeboard seens as an optimization problem with higher-order constraints.