Variational integrators for mechanical control systems with symmetries

Optimal control problems for underactuatedmechanical systems can be seen as a higher-order variational problemsubject to higher-order constraints (that is, when the Lagrangianfunction and the constraints depend on higher-order derivatives suchas the acceleration, jerk or jounces). In this paper we discuss thevariational formalism for the class of underactuated mechanicalcontrol systems when the configuration space is a trivial principalbundle and the construction of variational integrators for suchmechanical control systems. An interesting family of geometric integrators can be defined usingdiscretizations of the Hamilton's principle of critical action. Thisfamily of geometric integrators is called variational integrators,being one of their main properties the preservation of geometricfeatures as the symplecticity, momentum preservation and goodbehavior of the energy. We construct variationalintegrators for higher-order mechanical systems on trivial principalbundles and their extension for higher-order constrained systems, paying particular attention to the case of underactuatedmechanical systems.