On the global (f, g)-invariance of a class of nonlinear systems

Consider the nonlinear system x = f(x) + ??m i=1 gi (x) ui and the nonsingular involutive distribution ?? Here we discuss the existance of feedback (??, ??) such that [f + g??, ??] ?? ?? and [g??, ??] ?? ??. Single input case is given special attention, and we derive a necessary condition from the view point of characteristic classes of foliations. We give some sufficient conditions in the more general case when the foliation of ?? is such that the space of leaves is a Hausdorff manifold.