On the stabilization problem for nonholonomic distributions

Rifford, Ludovic, and Trélat, Emmanuel. "On the stabilization problem for nonholonomic distributions." Journal of the European Mathematical Society 011.2 (2009): 223-255. <http://eudml.org/doc/277253>.

@article{Rifford2009,
abstract = {Let $M$ be a smooth connected complete manifold of dimension $n$, and $\Delta $ be a smooth nonholonomic distribution of rank $m\le n$ on $M$. We prove that if there exists a smooth Riemannian metric on1for 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 we establish fine properties of optimal trajectories.},
author = {Rifford, Ludovic, Trélat, Emmanuel},
journal = {Journal of the European Mathematical Society},
keywords = {nonholonomic distributions; stabilization; SRS feedback; minimizing singular path; Martinet case; nonsmooth analysis; nonholonomic distributions; stabilization; SRS feedback; minimizing singular path; Martinet case; nonsmooth analysis},
language = {eng},
number = {2},
pages = {223-255},
publisher = {European Mathematical Society Publishing House},
title = {On the stabilization problem for nonholonomic distributions},
url = {http://eudml.org/doc/277253},
volume = {011},
year = {2009},
}

TY - JOUR
AU - Rifford, Ludovic
AU - Trélat, Emmanuel
TI - On the stabilization problem for nonholonomic distributions
JO - Journal of the European Mathematical Society
PY - 2009
PB - European Mathematical Society Publishing House
VL - 011
IS - 2
SP - 223
EP - 255
AB - Let $M$ be a smooth connected complete manifold of dimension $n$, and $\Delta $ be a smooth nonholonomic distribution of rank $m\le n$ on $M$. We prove that if there exists a smooth Riemannian metric on1for 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 we establish fine properties of optimal trajectories.
LA - eng
KW - nonholonomic distributions; stabilization; SRS feedback; minimizing singular path; Martinet case; nonsmooth analysis; nonholonomic distributions; stabilization; SRS feedback; minimizing singular path; Martinet case; nonsmooth analysis
UR - http://eudml.org/doc/277253
ER -