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...

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...