JournalsjemsVol. 11 , No. 2DOI 10.4171/jems/148

On the stabilization problem for nonholonomic distributions

  • Ludovic Rifford

    Université de Nice, France
  • Emmanuel Trélat

    Université Pierre et Marie Curie (Paris 6), France
On the stabilization problem for nonholonomic distributions cover

Abstract

Let MM be a smooth connected and complete manifold of dimension nn, and Δ\Delta be a smooth nonholonomic distribution of rank mnm\leq n on MM. We prove that, if there exists a smooth Riemannian metric on Δ\Delta for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of Δ\Delta on MM. 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.