Distributions and controllability problems (II)

In [C. Giannotti, A. Spiro, M. Zoppello, {\it Distributions and controllability problems (I)}, preprint posted on ArXiv (2024)], we introduced a new approach to the real analytic non-linear control systems of the form $\dot q^i = f^i(t, q, w)$, with controls $w = (w^\alpha)$ running in a connected open set $\mathcal{K}$ of $\mathbb{R}^m$ and states represented by points $q = (q^i)$ in a configuration space $\mathcal{Q} := \mathbb{R}^n$. The new approach consists of a differential-geometric study of (a) the oriented piecewise regular curves in the {\it extended space-time} $\mathcal{M} = \mathbb{R} \times \mathcal{Q} \times \mathcal{K}$, which are the (completed) graphs of the piecewise real analytic solutions $t \mapsto (q(t), w(t))$ of the control system, and (b) the local structure of the sets of points of $\mathcal{M}$ that are reachable from an initial point $x_o = (t_o, q_o, w_o) \in \mathcal{M}$ through such (completed) graphs. The main results of that paper are two new criterions which can be used to establish the small time local controllability near stable points of real analytic non-linear systems. The goal of this paper is to offer a friendly user's guide to those criterions, illustrating them by several examples. In particular, we analyse certain non-linear control systems, for which the new criterions show that they are small time locally controllable at their stable points, while, at the best of our knowledge, all other previous criterions are either inconclusive or not applicable.