Multiple Capture of a Given Number of Evaders in the Problem of Simple Pursuit with Phase Restrictions on Timescales

The problem of pursuing a group of evaders by a group of pursuers with equal opportunities for all participants is considered in the finite-dimensional Euclidean space. It is assumed that the movement of the players is simple in a given timescale. Additionally, we assume that in the process of the game each evader does not move out of a convex set with a nonempty interior. The goal of the pursuers' group is to capture at least q evaders, and each evader must be captured by at least r different pursuers, and the moments of capture may not coincide. Terminal sets are the coordinates origin. Assuming that the evaders use program strategies, and each pursuer catches no more than one evader in terms of initial positions, sufficient conditions for the solvability of the pursuit problem are obtained. In the research, the method of resolving functions is used as the base method, which makes it possible to obtain sufficient conditions for the solvability of the approach problem with one evader in some guaranteed time. To prove the main theorem, Hall's theorem of the system of various representatives is used.