Optimal Control and higher-order mechanics for systems with symmetries

In this paper we will develop and design numerical methods for optimal control problems for a class of underactuated Lagrangian mechanical systems where the configuration manifold is a trivial principal bundle . We will construct these geometric integrators using discrete variational calculus, deriving a discrete version of the higher-order Euler-Lagrange equations on trivial principal bundles. The analysis applies to systems subject to higher-order constraints (that is, depending of higher-order derivatives as, for example, the acceleration). Interesting applications as, for instance, a discrete derivation of the Euler-Lagrange equations for higher-order Lagrangians and higher-order reduced Lagrangians, respectively, are shown. We find interesting applications both in the optimal control of an underactuated vehicle and the well-known plate ball problem seen as an optimization problem with nonholonomic constraints .