Persecution of rigidly coordinated evaders in a linear problem with fractional derivatives and a simple matrix

Abstract

In the finite-dimensional Euclidean space, the problem of pursuit of a group of evaders by a group of pursuers is considered, which is described by a system of the form $$D^{(\alpha)} z_{ij} = a z_{ij} + u_i - v,$$ where $D^{(\alpha)} f$ is the Caputo derivative of the order $\alpha \in (0,1)$ of the function $f$. It is assumed that all evaders use the same control. The goal of the pursuers is to catch at least one of the evaders. The evaders use piecewise-program strategies, and the pursuers use piecewise-program counterstrategies. Every pursuer catches not more than one evader. The set of admissible controls is a ball of unit radius with the center at the origin, the target sets are the origin. In terms of initial positions and game parameters, a sufficient conditions for the capture are obtained.