# Asymptotics of accessibility sets along an abnormal trajectory

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 $\gamma$ into two sectors, bordered by the first Pontryagin's cone along $\gamma$, called the $\xLinfty$-sector and the $\xLtwo$-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.
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Journal articles

Cited literature [25 references]

https://hal.archives-ouvertes.fr/hal-00086290
Contributor : Emmanuel Trélat <>
Submitted on : Tuesday, July 18, 2006 - 2:05:32 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Tuesday, April 6, 2010 - 12:13:50 AM

### Citation

Emmanuel Trélat. Asymptotics of accessibility sets along an abnormal trajectory. ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2001, 6, pp.387--414. ⟨hal-00086290⟩

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