One Game Problem for Oscillatory Systems

Abstract

The paper concerns the linear differential game of approaching a cylindrical terminal set. We study the case when classic Pontryagin’s condition does not hold. Instead, the modified considerably weaker condition, dealing with the function of time stretching, is used. The latter allows expanding the range of problems susceptible to analytical solution by the way of passing to the game with delayed information. Investigation is carried out in the frames of Pontryagin’s First Direct method that provides hitting the terminal set by a trajectory of the conflict-controlled process at finite instant of time. In so doing, the pursuer’s control, realizing the game goal, is constructed on the basis of the Filippov-Castaing theorem on measurable choice. The outlined scheme is applied to solving the problem of pursuit for two different second-order systems, describing damped oscillations. For this game, we constructed the function of time stretching and deduced conditions on the game parameters, ensuring termination of the game at a finite instant of time, starting from arbitrary initial states and under all admissible controls of the evader. Keywords: differential game, time-variable information delay, Pontryagin’s condition, Aumann’s integral, principle of time stretching, Minkowski’ difference, damped oscillations.