On extremal shift strategies in delayed systems

Abstract

We consider a differential game in which the motion of a conflict-controlled dynamical system is described by an equation with delay, the initial condition is determined by a piecewise continuous function, and the performance index assesses the history of the motion realized by the terminal time and involves an integral estimate for the realizations of the players' controls. The optimality of the players' positional strategies constructed by the method of extremal shift to an accompanying point is proved. The main result of the paper states that the accompanying point is chosen from a finite-dimensional neighborhood of the current state of the system. © 2021 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.