Singular reduction for nonlinear control systems

We discuss smooth nonlinear control systems with symmetry. For a free and proper action of the symmetry group, the reduction of symmetry gives rise to a reduced smooth nonlinear control system. If the action of the symmetry group is only proper, the reduced nonlinear control system need not be smooth. Using the smooth calculus on nonsmooth spaces, provided by the theory of differential spaces of Sikorski, we prove a generalization of Sussmann’s theorem on orbits of families of smooth vector fields.