  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.
