This article is cited in
3 papers
Multiple Capture of a Given Number of Evaders in a Problem with Fractional Derivatives and a Simple Matrix
N. N. Petrova,
A. Ya. Narmanovb a Udmurt State University, Mathematical Department
b National University of Uzbekistan named after Mirzo Ulugbek,
Abstract:
A problem of pursuing a group of evaders by a group of pursuers with
equal capabilities of all the participants is considered in a finite-dimensional
Euclidean space. The system is described by the equation
$$
D^{(\alpha)}z_{ij}=az_{ij}+u_i-v_j, \ \ u_i, v_j \in V,
$$
where
$D^{(\alpha)}f$ is the Caputo fractional derivative of order
$\alpha$
of the function
$f$, the set of admissible controls
$V$ is strictly convex and
compact, and
$a$ is a real number. The aim of the group of pursuers is to
capture at least
$q$ evaders; each evader must be captured by at least
$r$
different pursuers, and the capture moments may be different. The terminal
set is the origin. Assuming that the evaders use program strategies and
each pursuer captures at most one evader, we obtain sufficient conditions
for the solvability of the pursuit problem in terms of the initial positions.
Using the method of resolving functions as a basic research tool, we derive
sufficient conditions for the solvability of the approach problem with one
evader at some guaranteed instant. Hall's theorem on a system of distinct
representatives is used in the proof of the main theorem.
Keywords:
differential game, group pursuit, multiple capture, pursuer, evader, fractional derivative.
UDC:
517.977
MSC: 49N79,
49N70,
91A24 Received: 06.05.2019
Revised: 19.06.2019
Accepted: 24.06.2019
DOI:
10.21538/0134-4889-2019-25-3-188-199