For a given vector field $h$ on a manifold $M$ and an initial point $x \in M$, let $t \mapsto \exp th(x)$ denote the solution to the Cauchy problem $y' = h(y)$, $y(0) = x$. Given two vector fields $f$, $g$, the flows $\exp(tf)$, $\exp(tg)$ in general are not commutative. That is, it may happen that, for some initial point $x$,
$$\exp(-tg) \circ \exp(-tf) \circ \exp(tg) \circ \exp(tf) (x) ≠ x,$$
for small times $t ≠ 0$.
As is well-known, the Lie bracket $[f,g] := Dg \cdot f - Df \cdot g$ measures the local non-commutativity of the flows. Indeed, one has (on any coordinate chart)
$$\exp(-tg) \circ \exp(-tf) \circ \exp(tg) \circ \exp(tf) (x) - x = t^2 [f,g](x) + o(t^2)$$
The non-commutativity of vector fields lies at the basis of many nonlinear issues, like propagation of maxima for solutions of degenerate elliptic PDEs, controllability sufficient conditions in Nonlinear Control Theory, and higher order necessary conditions for optimal controls. The fundamental results concerning commutativity (e.g. Rashevski-Chow's Theorem, also known as Hörmander's full rank condition, or Frobenius Theorem) assume that the vector fields are smooth enough for the involved iterated Lie brackets to be well defined and continuous: for instance, if the bracket $[f,[g,h]]$ is to be used, one posits $g,h \in C^2$ and $f \in C^{1..}$.
We propose a notion of (set-valued) Lie bracket (see [1]-[3]), through which we are able to extend some of the mentioned fundamental results to families of vector fields whose iterated brackets are just measurable and defined almost everywhere.
References.
[1] Rampazzo, F. and Sussmann, H., Set-valued differentials and a nonsmooth version of Chow’s Theorem (2001), Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, 2001 (IEEE Publications, New York), pp. 2613-2618.
[2] Rampazzo F. and Sussmann, H.J., Commutators of flow maps of nonsmooth vector fields (2007), Journal of Differential Equations, 232, pp. 134-175.
[3] Feleqi, E. and Rampazzo, F., Iterated Lie brackets for nonsmooth vector fields (2017), Nonlinear Differential Equations and Applications NoDEA, 24-6.
