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

Trudy Inst. Mat. i Mekh. UrO RAN, 2022 Volume 28, Number 3, Pages 129–141 (Mi timm1932)

This article is cited in 2 papers

On a Linear Group Pursuit Problem with Fractional Derivatives

A. I. Machtakovaab, N. N. Petrovab

a Udmurt State University, Izhevsk
b N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: A problem of pursuit of one evader by a group of pursuers is considered in a finite-dimensional Euclidean space. The dynamics is described by the system
$$ D^{(\alpha_i)}z_i=A_iz_i+B_iu_i-C_iv, \quad u_i\in U_i,\quad v\in V, $$
where $D^{(\alpha)}f$ is the Caputo derivative of order $\alpha$ of a function $f$. The sets of admissible controls of the players are convex and compact. The terminal set consists of cylindrical sets $M_i$ of the form $M_i=M_i^1+M_i^2$, where $M_i^1$ is a linear subspace of the phase space and $M_i^2$ is a convex compact set from the orthogonal complement of $M_i^1$. We propose two approaches to solving the problem, which ensure the termination of the game in a certain guaranteed time in the class of quasi-strategies. In the first approach, the pursuers construct their controls so that the terminal sets “cover” the evader's uncertainty region. In the second approach, the pursuers construct their controls using resolving functions. The theoretical results are illustrated by model examples.

Keywords: differential game, group pursuit, pursuer, evader, fractional derivative.

UDC: 517.977

MSC: 49N79, 49N70, 91A24

Received: 30.05.2022
Revised: 07.07.2022
Accepted: 11.07.2022

DOI: 10.21538/0134-4889-2022-28-3-129-141


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplement Issues), 2022, 319, suppl. 1, S175–S187

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© Steklov Math. Inst. of RAS, 2025