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Lagrange-Poincaré 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-Poincaré equations of classical mechanical systems with higher-order constraints are obtained from classical Lagrangian reduction. Higher-order Lagrange-Poincaré operator is introduced to characterize higher-order Lagrange-Poincaré 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.