EVALUATION OF THE PURSUIT TIME IN DIFFERENTIAL GAMES OF MANY PLAYERS ON A CONVEX COMPACT

Abstract

The paper is devoted to the study of the problem of constructing a pursuit strategy in simple differential games of many persons with phase constraints in the state of the players, in the sense of getting into a certain neighborhood of the evader. The game takes place in -dimensional Euclidean space on a convex compact set. The pursuit problem is considered when the number of pursuing players is , that is, less than , in the sense of — captures. A structure for constructing pursuit controls is proposed, which will ensure the completion of the game in a finite time. An upper bound is obtained for the game time for the completion of the pursuit. An auxiliary problem of simple pursuit on a unit cube in the first orthant is considered, and strategies of pursuing players are constructed to complete the game with special initial positions. The results obtained are used to solve differential games with arbitrary initial positions. For this task, a structure for constructing a pursuit strategy is proposed that will ensure the completion of the game in a finite time. The generalization of the problem in the sense of complicating the obstacle is also considered. A more general problem of simple pursuit on a cube of arbitrary size in the first orthant is considered. With the help of the proposed strategies, the possibilities of completing the pursuit are proved and an estimate of the time is obtained. As a consequence of this result, lower and upper bounds are obtained for the pursuit time in a game with ball-type obstacles. Estimates are obtained for the pursuit time when the compact set is an arbitrarily convex set. The concept of a convex set in a direction relative to a section, which is not necessarily convex, is defined. And in it the problem of simple pursuit in a differential game of many players is studied and the possibilities of completing the pursuit using the proposed strategy are shown. The time of completion of the pursuit of the given game is estimated from above.