| Abstract: |
| {\it Small time local controllability} for a control system of ODEs consists in the fact that, for every sufficiently small $t>0$, the points reached at a time $s\leq t$ by the trajectories of the system starting from a point
$x_*$ form a full neighborhood of $x_*$. In the case of a dynamics which is (nonlinear but) linear in the controls, the celebrated Rashevskii-Chow`s theorem states that the so-called {\it full rank-condition} --namely the fact that iterated Lie brackets of the involved vector fields generate the whole tangent space-- is sufficient for small time local controllability.
Rashevskii-Chow`s theorem is classically obtained under hypotheses of $C^\infty$ regularity. In this talk I will illustrate how this result (and other similar ones)
can be extended by means of a notion of {\it set-valued Lie bracket} up to the point that tha highest order brackets involved in the full rank condition are just almost everywhere defined, bounded, maps. There are several applications of such controllability results, e.g. in the study of subelliptic pde`s and of Carnot-Carath\`eodory spaces. In particular, the presented result might represent a basis for a highly non-smooth subRiemannian geometry. |
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