A Sufficient Condition for Local Controllability

Abstract

A system S of vector fields is locally controllable at point p if, for every positive time t, the set of points reachable from p by an S-trajectory in time $ \leqq t$ contains p in its interior. Let K be the convex hull of the values $X(p)$ of those $X \in S$ for which $X(p) \ne 0$. It is well known that S is l.c. at p if $0 \in \operatorname{interior} (K)$, and that S is not l.c. at p if $0 \notin K$. We prove that these are the only cases in which it is possible to determine if S is l.c. at p by just looking at the values at p of the elements of S. We prove a sufficient condition for local controllability which gives new information for the case when $0 \in K$ but $0 \notin \operatorname{interior}(K)$.