“Good Lie Brackets” for Control Affine Systems

We consider a smooth system of the form \(\dot{q}=f_0(q)+\sum \limits _{i=1}^ku_if_i(q)\) $\dot{q}=f_0(q)+\sum _{i=1}^k{u}_i{f}_i(q)$ , \(q\in M,\ u_i\in {\mathbb R},\) $q\in M,u_i\in \mathbb{R},$ and study controllability issues on the group \(\textrm{Diff}M\) $\text{Diff}M$ . It is well-known that the system can arbitrarily well approximate the movement in the direction of any Lie bracket polynomial of \(f_1,\ldots ,f_k\) $f_1,\dots ,f_k$ . Any Lie bracket polynomial of \(f_1,\ldots ,f_k\) $f_1,\dots ,f_k$ is good in this sense. Moreover, some combinations of Lie brackets which involve the drift term \(f_0\) $f_0$ are also good but surely not all of them. In this paper, we try to characterize good ones and, in particular, all universal good combinations, which are good for any nilpotent truncation of any system.