The consistency problem in optimal control: The degenerate case

We examine the problem of the consistency of the second-order differential equations associated with optimal control problems. This problem can be treated in a presymplectic framework by means of a constraint algorithm. Two cases may arise: the regular one, already considered in the literature, and the degenerate one. The main contribution of this paper is the proposal of a discrete transition law for the optimal trajectories that reach a singular point. This discrete law respects both the geometry and the dynamical structure of the optimal control problem.