Asymptotics of accessibility sets along an abnormal trajectory

Trélat, Emmanuel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 387-414.

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 ${\mathrm{L}}^{\infty }$-sector and the ${\mathrm{L}}^2$-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.

Classification : 93B03, 49K15
Mots clés : accessibility set, abnormal trajectory, end-point mapping, single-input affine control system, sub-riemannian geometry

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