Optimal synthesis in the control problem of an $n$-link inverted pendulum with a moving base

In this paper, we consider the problem of stabilization of an $n$-link inverted pendulum on a movable base (cart). A cart is allowed to move along the horizontal axis. A force applied to the cart is considered as a control. The problem is to minimize the mean square deviation of the pendulum from the vertical line. For the linearized model, we show that, for small deviations from the upper unstable equilibrium position, the optimal regime contains trajectories with more and more frequent switchings. Namely, the optimal trajectories with infinite number of switchings are shown to attain, in finite time, the singular surface and then continue these motion with singular control over the singular surface, approaching the origin in an infinite time. It is shown that the costructed solutions are globally optimal.