A Topological View on Forced Oscillations and Control of an Inverted Pendulum

We consider a system of a planar inverted pendulum in a gravitational field. First, we assume that the pivot point of the pendulum is moving along a horizontal line with a given law of motion. We prove that, if the law of motion is periodic, then there always exists a periodic solution along which the pendulum never becomes horizontal (never falls). We also consider the case when the pendulum with a moving pivot point is a control system, in which the mass point is constrained to be strictly above the pivot point (the rod cannot fall 'below the horizon'). We show that global stabilization of the vertical upward position of the pendulum cannot be obtained for any smooth control law, provided some natural assumptions.