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

Nikolay Nikandrovich Petrov

doi:10.3390/math9111171

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.

Highlights

The theory of differential two-player games, originally considered by Rufus Isaacs [1], has grown to be a profound and substantial theory in which various approaches to analysis of conflict situations [2,3,4,5,6,7,8,9,10] are developed
Games involving a group of pursuers and one or several evaders [11,12,13,14] are a natural generalization of the differential two-player pursuit–evasion games
These games are of interest since they cannot be solved using two-player game theory

Summary

Introduction

The theory of differential two-player games, originally considered by Rufus Isaacs [1], has grown to be a profound and substantial theory in which various approaches to analysis of conflict situations [2,3,4,5,6,7,8,9,10] are developed. Games involving a group of pursuers and one or several evaders [11,12,13,14] are a natural generalization of the differential two-player pursuit–evasion games. The problem of multiple capture of an evader without phase restrictions in a differential game with fractional derivatives is treated in [22]. This paper deals with the problem of multiple capture of an evader in a differential game with fractional derivatives and phase restrictions. 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 law of motion of evader E has the form

This yields

Since a