Distributions and controllability problems (I)

We consider a non-linear real analytic control system of first order $\dot q^i = f^i(t, q, w)$, with controls $w = (w^\alpha)$ in a connected open set $\mathcal{K} \subset \mathbb{R}^m$ and configurations $q = (q^i)$ in $\mathcal{Q} := \mathbb{R}^n$. The set of points in the extended space-time $\mathcal{M} = \mathbb{R} \times \mathcal{Q} \times \mathcal{K}$, which can be reached from a triple $x_o = (t_o , q_o, w_o) \in \mathcal{M}$ through a continuous graph completion $\gamma(s) = \big(t(s), q(t(s)), w(t(s))\big)$ of the graph of a solution $t \to (q(t), w(t))$, $t \in [t_o ,t_o + T]$, with piecewise real analytic controls, is called the {\it $\mathcal{M}$-attainable set of $x_o$ in time $T$}. We prove that if $y_o$ is an $\mathcal{M}$-attainable point of $x_o$, a large set of other nearby $\mathcal{M}$-attainable points of $x_o$ can be determined starting directly from $y_o$ and applying an appropriate ordered composition of flows of vector fields in a distinguished distribution $\mathcal{D}^{II} \subset T \mathcal{M}$, canonically associated with the control system. We then determine sufficient conditions for such neighbouring points to constitute an orbit of the pseudogroup of local diffeomorphisms generated by the vector fields in $\mathcal{D}^{II}$. If such conditions are satisfied and if the tangent spaces of these orbits have maximal rank projections onto $\mathcal{Q}$, the control system is locally accessible and has the small time local controllability property near the state points of equilibrium. These results lead to new proofs of classical local controllability criterions and yield new methods to establish the accessibility and the small time local controllability of non-linear control systems.