# On the stabilization problem for nonholonomic distributions

Abstract : 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.
Keywords :
Type de document :
Article dans une revue
Journal of the European Mathematical Society, European Mathematical Society, 2009, 11, pp.223--255

Littérature citée [47 références]

https://hal.archives-ouvertes.fr/hal-00105488
Contributeur : Emmanuel Trélat <>
Soumis le : dimanche 24 août 2008 - 19:43:59
Dernière modification le : jeudi 3 mai 2018 - 15:32:06
Document(s) archivé(s) le : mardi 21 septembre 2010 - 17:40:07

### Fichiers

RTstab.pdf
Fichiers produits par l'(les) auteur(s)

### Citation

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

### Métriques

Consultations de la notice

## 370

Téléchargements de fichiers