Local accessibility, local reachability, and representations of compact groups

In this paper we consider the local controllability problem for nonlinear control systems, or as we prefer to call it, the local reachability problem. We introduce the action of a compact group on the system, and define what it means for the system to be invariant under the group action. Our principal result gives sufficient conditions, in terms of the group action, in order that a locally accessible system is also locally reachable. The technique used is a generalization of one first introduced by P. Brunovsky for “odd” systems invariant under a certainZ2 action. We give both geometric and group representational interpretations of our conditions, and provide examples of our conditions applied to certain “even” systems which do not satisfy Brunovsky's conditions.

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