ИСТИНА

We consider a mathematical model of a spherical inverted pendulum-cart system. The pendulum consists of the point mass on the end of the rigid massless rod which is attached by a hinge to a moving base (a cart). The cart motion in the horizontal plane is controlled by an external planar force. The control force is assumed to be bounded. We study the minimization problem of the mean square deviation of the pendulum from the upper equilibrium position over an infinite time interval. For the linearized model we find the optimal singular solution and solutions with accumulations of the control switchings (chattering arcs). We prove that there is a one-parameter family of spiral-like optimal trajectories which attain the origin in finite time and undergo an infinite number of rotations. We show that the optimal solutions stabilize the pendulum around its upper equilibrium position. This research was supported in part by the Russian Foundation for Basic Research under grant no. 17-01-00805.