On the stabilization problem for nonholonomic distributions

Let M be a smooth connected and complete manifold of dimension n, and be a smooth nonholonomic distribution of rank m n on M. We prove that, if there exists a smooth Riemannian metric on for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of on M. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the cor- responding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and establish fine properties of optimal trajectories.