Group pursuit of two evaders in a linear game with a simple matrix

Abstract

We consider a linear nonstationary problem of pursuit of a group of two evaders by a group of pursuers with equal dynamic capabilities of all participants. The laws of motion of the participants have the form $\dot z(t)+a(t)z=u(t).$ For $t=t_0$ initial conditions are given. Phase state restrictions are imposed on the evaders state under the assumption that they use the same control. Geometric constraints on the controls are strictly convex compact set with a smooth boundary, terminal sets are the origin of coordinates. It is assumed that in the process of the game the evaders do not leave the limits of the halfspace $D=\{y\colon y\in \mathbb{R}^k, \langle p_1, y\rangle \leqslant 0\},$ where $p_1$ is a unit vector. The aim of the pursuers is to capture both evaders, the aim of the group of evaders is the opposite. It is also assumed that all evaders use tightly coordinated management, which is determined at each point in time, taking into account the positions of other players in the game. The capture occurs if there are times $\tau_1,$ $\tau_2$ such that the coordinates of the pursuers and evaders coincide, and the capture times may not coincide. For a nonstationary simple pursuit problem in terms of initial positions and game parameters, sufficient conditions for catching two evaders are obtained. An example is given that illustrates the results obtained.