A constructive Lyapunov function for controlling the inverted pendulum

A constructive control strategy is presented for the stabilization of the inverted pendulum system. Under the assumption that the pendulum is initialized above the upper half plane. To carried out, a suitable procedure is presented in order to find a candidate Lyapunov function, for the whole system. Then the control is proposed directly from the obtained Lyapunov, in such a way, that its time derivative be semi-definite negative with respect to a convenient manifold. Finally, the asymptotically stability of the closed-loop system is concluded by LaSalle theorem.