On topological obstructions to global stabilization of an inverted pendulum

If the configuration space of a controlled system is closed, the feedback control is autonomous and solutions exist, unique, and continuously depend on the initial data, then the system cannot have a globally asymptotically stable equilibrium. It follows from the fact that a closed manifold cannot be contractible. Let us consider a classical controlled system, an inverted pendulum controlled by means of a horizontal motion of its pivot point. We consider the pendulum only in the positions where its mass point is above the pivot point (for instance, we can consider any mechanical model of the rod-plane impact). Although the configuration space in this case is contractible, we prove that, for any mechanical model of the impact, it is impossible to globally stabilize the rod in a given position (we also assume some natural properties of the equilibrium). To be more precise, we prove that there always exists a family of solutions separated from the vertical position and along which the pendulum never becomes horizontal. Similar results can be easily proved for several analogous systems: a pendulum on a cart, a spherical pendulum, and a pendulum with an additional torque control.