Invariant distributions, global decompositions of nonlinear systems, and applications

Several necessary and sufficient conditions for a given (possibly singular) distribution to be invariant for a control system (or for a single vector field) are given. These conditions provide a modest first step in the search for computable criteria for the existence of invariant distributions. It is shown how the presence of an invariant distribution leads to a tractable notion of a global decomposition of a nonlinear system. These ideas are illustrated by examining some controllability considerations for a certain class of control semilinear systems