Composition and division theorems and controlled decomposition

An important problem in control theory seeks the existence of an affine feedback law (of as high a degree of regularity, and as global as possible) which will cause the closed loop vector fields of an affine nonlinear system to leave an involutive distribution invariant. We require that a smoothness function be invertible on a dense subset of its domain of existence. This is justified in applications, and will also allow us to obtain the integrability conditions for a set of PDEs for the function. The problem arises in a variety of unrelated situations. In some cases the problem amounts to local decomposition of the control system (via feedback) into subsystems. One can still salvage much of this notion when there are singularities