A discrete maximum principle for solving optimal control problems

We develop a discrete maximum principle that yields discrete necessary conditions for optimality. These conditions are in agreement with the usual conditions obtained from the Pontryagin maximum principle and define symplectic algorithms that solve the optimal control problem. We show that our approach allows one to recover most of the classical symplectic algorithms and can be enhanced so that the discrete necessary conditions define symplectic-energy conserving algorithms. Finally we illustrate its use with an example of a sub-Riemannian optimal control problem.