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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2019 Volume 25, Number 3, Pages 188–199 (Mi timm1658)

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


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplement Issues), 2020, 309, suppl. 1, S105–S115

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© Steklov Math. Inst. of RAS, 2025