On topological obstructions to global stabilization of an inverted\n pendulum

We consider a classical problem of control of an inverted pendulum by means\nof a horizontal motion of its pivot point. We suppose that the control law can\nbe non-autonomous and non-periodic w.r.t. the position of the pendulum. It is\nshown that global stabilization of the vertical upward position of the pendulum\ncannot be obtained for any Lipschitz control law, provided some natural\nassumptions. Moreover, we show that there always exists a solution separated\nfrom the vertical position and along which the pendulum never becomes\nhorizontal. Hence, we also prove that global stabilization cannot be obtained\nin the system where the pendulum can impact the horizontal plane (for any\nmechanical model of impact). Similar results are presented for several\nanalogous systems: a pendulum on a cart, a spherical pendulum, and a pendulum\nwith an additional torque control.\n