Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions

Abstract

The problem of conflict interaction between a group of pursuers and an evader in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players are described by a system of differential equations with fractional derivatives in the form D(α)zi=azi+ui−v,ui,v∈V, where D(α)f is a Caputo derivative of order α of the function f. Additionally, it is assumed that in the process of the game the evader does not move out of a convex polyhedral cone. The set of admissible controls V is a strictly convex compact and a is a real number. The goal of the group of pursuers is to capture of the evader by no less than m different pursuers (the instants of capture may or may not coincide). The target sets are the origin. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evader. The method of resolving functions is used to solve the problem, which is used in differential games of pursuit by a group of pursuers of one evader.