Complexity of the minimum-time damping of a physical pendulum

We study the minimum-time damping of a physical pendulum by means of a bounded control. In the similar problem for a linear oscillator each optimal trajectory possesses a finite number of control switchings from the maximal to the minimal value. If one considers simultaneously all optimal trajectories with any initial state, the number of switchings can be arbitrary large. We show that for the nonlinear pendulum there is a uniform bound for the switching number for all optimal trajectories. We find asymptotics for this bound as the control amplitude goes to zero.