On topological obstructions to global stabilization of an inverted pendulum

TL;DR

This paper demonstrates fundamental topological limitations preventing the global stabilization of an inverted pendulum under broad control conditions, highlighting inherent obstructions in achieving perfect control.

Contribution

It establishes that no Lipschitz control law can achieve global stabilization of an inverted pendulum, revealing intrinsic topological obstructions and extending results to related systems.

Findings

Global stabilization impossible with Lipschitz control laws.

Existence of solutions avoiding the vertical position.

Results apply to systems with impacts and additional controls.

Abstract

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