On the stabilization problem for nonholonomic distributions

Ludovic Rifford (1) , Emmanuel Trélat (2)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

Let $M$ be a smooth connected and complete manifold of dimension $n$, and $\Delta$ be a smooth nonholonomic distribution of rank $m\leq n$ on $M$. We prove that, if there exists a smooth Riemannian metric on $\Delta$ for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of $\Delta$ 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 corresponding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and establish fine properties of optimal trajectories.

Mots clés: stabilization; distributions; sub-Riemannian geometry; singular curve

Domaines: Optimisation et contrôle [math.OC]

Ludovic Rifford, Emmanuel Trélat. On the stabilization problem for nonholonomic distributions. Journal of the European Mathematical Society, 2009, 11, pp.223--255. ⟨hal-00105488v2⟩