Conservation of generalized momentum maps in mechanical optimal control problems with symmetry

In this paper, a structure-preserving direct method for the optimal control of mechanical systems is developed. The new method accommodates a large class of one-step integrators for the underlying state equations. The state equations under consideration govern the motion of affine Hamiltonian control systems. If the optimal control problem has symmetry, associated generalized momentum maps are conserved along an optimal path. This is in accordance with an extension of Noether's theorem to the realm of optimal control problems. In the present work, we focus on optimal control problems with rotational symmetries. The newly proposed direct approach is capable of exactly conserving generalized momentum maps associated with rotational symmetries of the optimal control problem. This is true for a variety of one-step integrators used for the discretization of the state equations. Examples are the one-step theta method, a partitioned variant of the theta method, and energy-momentum (EM) consistent integrators. Numerical investigations confirm the theoretical findings. Copyright (c) 2016 John Wiley & Sons, Ltd.