On the problem of controlling a second-order nonlinear system by means of discrete control under disturbance

The problem of bringing a trajectory to a neighborhood of zero under disturbance is considered in terms of a differential pursuit game. The dynamics are described by a nonlinear autonomous system of second-order differential equations. The set of values of the pursuer's controls is finite, and that of the evader (disturbance) is compact. The goal of the control, that is, the goal of the pursuer, is to bring, within a finite time, the trajectory to any predetermined neighborhood of zero, regardless of the actions of the disturbance. To construct the control, the pursuer knows only the phase coordinates and the value of the velocity at some discrete moments of time and the choice of the disturbance control is unknown. Conditions are obtained for the existence of a set of initial positions, from each point of which a capture occurs in the specified sense. Moreover, this set contains a certain neighborhood of zero. The winning control is constructed constructively and has an additional property specified in the theorem. In addition, an estimate of the time required to bring the speed from one given point to the neighborhood of another given point under disturbance conditions was obtained.