# Optimality of singular trajectories and asymptotics of accessibility sets under generic assumptions

Abstract : We investigate minimization problems along a singular trajectory of a single-input affine control system with constraint on the control, and then as an application of a sub-Riemannian system of rank 2. Under generic assumptions we get necessary and sufficient conditions for optimality of such a singular trajectory. Moreover we describe precisely the contact of the accessibility sets at time $T$ with the singular direction. 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 $L^\infty$-sector and the $L^2$-sector.
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Conference papers
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https://hal.archives-ouvertes.fr/hal-00086429
Contributor : Emmanuel Trélat <>
Submitted on : Tuesday, July 18, 2006 - 6:48:40 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Monday, April 5, 2010 - 10:28:20 PM

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• HAL Id : hal-00086429, version 1

### Citation

Emmanuel Trélat. Optimality of singular trajectories and asymptotics of accessibility sets under generic assumptions. Contemporary trends in nonlinear geometric control theory and its applications (Mexico City, 2000), 2002, Mexico, Mexico. pp.441--458. ⟨hal-00086429⟩

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