Asymptotics of accessibility sets along an abnormal trajectory

Université de Bourgogne, Laboratoire de Topologie, UMR 5584 du CNRS, BP. 47870, 21078 Dijon Cedex, France; trelat@topolog.u-bourgogne.fr.

Received: 8 November 2000
Revised: 28 February 2001

Abstract

We describe precisely, under generic conditions, the contact of the accessibility set at time T with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer γ into two sectors, bordered by the first Pontryagin's cone along γ, called the L^∞-sector and the L²-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.