Geodesic
On the stabilization problem for nonholonomic distributions
Journal of the European Mathematical Society, Tome 11 (2009) no. 2, pp. 223-255 Cet article a éte moissonné depuis la source EMS Press

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Let M be a smooth connected and complete manifold of dimension n, and Δ be a smooth nonholonomic distribution of rank m≤n on M. We prove that, if there exists a smooth Riemannian metric on Δ for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of Δ 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.
DOI : 10.4171/jems/148
Classification : 93-XX, 34-XX, 00-XX
Keywords: Nonholonomic distributions, stabilization, SRS feedback, minimizing singular path, Martinet case, nonsmooth analysis 
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     author = {Ludovic Rifford and Emmanuel Tr\'elat},
     title = {On the stabilization problem for nonholonomic distributions},
     journal = {Journal of the European Mathematical Society},
     pages = {223--255},
     year = {2009},
     volume = {11},
     number = {2},
     doi = {10.4171/jems/148},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/148/}
}
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Ludovic Rifford; Emmanuel Trélat. On the stabilization problem for nonholonomic distributions. Journal of the European Mathematical Society, Tome 11 (2009) no. 2, pp. 223-255. doi: 10.4171/jems/148

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