Capture of two coordinated evaders in a problem with fractional derivatives, phase restrictions and a simple matrix

N. N. Petrov ; A. I. Machtakova

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 56 (2020), pp. 50-62

Résumé

In the finite-dimensional Euclidean space, a task of pursuing two evaders by a group of pursuers is considered, described by a system of the form $$D^{(\alpha)} z_{ij} = a z_{ij} + u_i - v,$$ where $D^{(\alpha)}f$ is the Caputo fractional derivative of order $\alpha \in (0, 1)$ of the function $f$, and $a$ is a real number. It is assumed that all evaders use the same control and that the evaders do not leave a convex cone with vertex at the origin. The aim of the group of pursuers is to capture two evaders. The pursuers use program counterstrategies based on information about the initial positions and the control history of the evaders. The set of admissible controls is a unit ball centered at zero, the target sets are the origins. In terms of initial positions and game parameters, sufficient conditions for the capture are obtained. Using the method of resolving functions as a basic research tool, we derive sufficient conditions for the solvability of the approach problem in some guaranteed time.

Détail

Keywords: differential game, pursuer, evader, fractional derivatives, phase restrictions.