
Multiple capture of a given number of evaders in a problem with fractional derivatives and a simple matrix
N. N. Petrov^{a}, A. Ya. Narmanov^{b} ^{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 finitedimensional Euclidean space. The system is described by the equation \begin{gather*} D^{(\alpha)}z_{ij}=az_{ij}+u_iv_j, u_i, v_j \in V, \end{gather*} 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 sets are 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 in some guaranteed time. 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.
Funding Agency 
Grant Number 
Russian Foundation for Basic Research 
185141005 
Ministry of Innovative Development of the Republic of Uzbekistan 
MRU10/17 
The research of the first and second authors was supported by the Russian Federation for Basic Research (project no. 185141005) and by Grant MRU1017 (Uzbekistan), respectively. 
DOI:
https://doi.org/10.21538/013448892019253188199
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UDC:
517.977
MSC: 49N79, 49N70, 91A24 Received: 06.05.2019 Revised: 19.06.2019 Accepted:24.06.2019
Citation:
N. N. Petrov, A. Ya. Narmanov, “Multiple capture of a given number of evaders in a problem with fractional derivatives and a simple matrix”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 3, 2019, 188–199
Citation in format AMSBIB
\Bibitem{PetNar19}
\by N.~N.~Petrov, A.~Ya.~Narmanov
\paper Multiple capture of a given number of evaders in a problem with fractional derivatives and a simple matrix
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2019
\vol 25
\issue 3
\pages 188199
\mathnet{http://mi.mathnet.ru/timm1658}
\crossref{https://doi.org/10.21538/013448892019253188199}
\elib{http://elibrary.ru/item.asp?id=39323548}
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