Asymptotics of accessibility sets along an abnormal trajectory
Université de Bourgogne,
Laboratoire de Topologie, UMR 5584 du CNRS, BP. 47870,
21078 Dijon Cedex, France; email@example.com.
Revised: 28 February 2001
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.
Mathematics Subject Classification: 93B03 / 49K15
Key words: Accessibility set / abnormal trajectory / end-point mapping / single-input affine control system / sub-Riemannian geometry.
© EDP Sciences, SMAI, 2001