On the Problem of Controlling a Nonlinear System by a Discrete Control under Disturbance

We consider the problem of stabilization to zero under disturbance in terms ofa differential pursuit game. The dynamics is described by a nonlinear autonomous system ofdifferential equations. The set of control values of the pursuer is finite, and that of the evader(disturbance) is a compact set. The control objective, i.e., the pursuer’s goal, is to bring thetrajectory to any predetermined neighborhood of zero in finite time regardless of the disturbance.To construct the control, the pursuer knows only the state coordinates at some discrete times, andthe choice of the disturbance’s control is unknown. In the paper, we obtain conditions for theexistence of a neighborhood of zero from each point of which a capture occurs in the indicatedsense. A winning control is constructed constructively and has an additional property specified ina theorem. In addition, an estimate of the capture time sharp in some sense is produced.