Second order conditions for the controllability of nonlinear systems with drift

In this paper we study controllability of control systems in $\mathbb R^n$ of the form $\dot x=f(x)+\sum_{i=1}^m$ $u_ig_i(x)$ with $u\in\mathcal U$ compact convex subset of $\mathbb R^n$ with a rather general target. The symmetric (driftless) case, i.e. $f=0$, is a very classical topic, and in this case the results on controllability and Hölder continuity of the minimal time function $T$ are related to certain properties of the Lie algebra generated by the $g_i$'s. Here, we want to extend some results on controllability and Hölder continuity of $T$ to some cases where $f\ne 0$.