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## Archive of Issues

Russia Izhevsk
Year
2017
Volume
27
Issue
1
Pages
54-59
 Section Mathematics Title One problem of group pursuit with fractional derivatives and phase constraints Author(-s) Petrov N.N.a Affiliations Udmurt State Universitya Abstract In the finite-dimensional Euclidean space, we consider the problem of persecution of one evader by the group of pursuers, which is described by the system $$D^{(\alpha)}z_i = a z_i + u_i - v,$$ where $D^{(\alpha)}f$ is the Caputo derivative of order $\alpha \in (0, 1)$ of the function $f$. It is further assumed that the evader does not leave the convex polyhedron with nonempty interior. The evader uses piecewise-program strategies, and the pursuers use piecewise-program counterstrategies. The set of admissible controls is a convex compact, the target sets are the origin of coordinates, and $a$ is a real number. In terms of the initial positions and the parameters of the game, sufficient conditions for the solvability of the pursuit problem are obtained. Keywords differential game, group pursuit, phase restrictions, pursuer, evader UDC 517.977 MSC 49N75, 91A23 DOI 10.20537/vm170105 Received 1 February 2017 Language Russian Citation Petrov N.N. One problem of group pursuit with fractional derivatives and phase constraints, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 1, pp. 54-59. References Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p. Petrosyan L.A. Differentsial'nye igry presledovaniya (Differential games of pursuit), Leningrad: Leningrad State University, 1977, 222 p. Rikhsiev B.B. Differentsial'nye igry s prostym dvizheniem (Differential games with simple motion), Tashkent: Fan, 1989, 232 p. Chikrii A.A. Conflict-controlled processes, Springer Netherlands, 1997, 404 p. DOI: 10.1007/978-94-017-1135-7 Grigorenko N.L. Matematicheskie metody upravleniya neskol'kimi dinamicheskimi protsessami (Mathematical methods of control over multiple dynamic processes), Moscow: Moscow State University, 1990, 197 p. Blagodatskikh A.I., Petrov N.N. Konfliktnoe vzaimodeistvie grupp upravlyaemykh ob''ektov (Conflict interaction of groups of controlled objects), Izhevsk: Udmurt State University, 2009, 266 p. Eidel'man S.D., Chikrii A.A. Dynamic game problems of approach for fractional-order equations, Ukrainian Mathematical Journal, 2000, vol. 52, issue 11, pp. 1787-1806. DOI: 10.1023/A:1010439422856 Chikrii A.A., Matichin I.I. Game problems for fractional-order linear systems, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 268, suppl. 1, pp. 54-70. DOI: 10.1134/S0081543810050056 Chikrii A.A., Matichin I.I. On linear conflict-controlled processes with fractional derivatives, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2011, vol. 17, no. 2, pp. 256-270 (in Russian). Petrov N.N. To a nonstationary group pursuit problem with phase constraints, Automation and Remote Control, 2014, vol. 75, issue 8, pp. 1525-1531. DOI: 10.1134/S0005117914080153 Blagodatskikh A.I., Petrov N.N. Group pursuit with state constraints in Pontryagin’s almost periodic example, Differential Equations, 2015, vol. 51, issue 3, pp. 391-398. DOI: 10.1134/S001226611503009X Petrov N.N., Solov'eva N.A. Multiple capture in Pontryagin’s recurrent example with phase constraints, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 293, suppl. 1, pp. 174-182. DOI: 10.1134/S0081543816050163 Caputo M. Linear models of dissipation whose $Q$ is almost frequency independent-II, Geophysical Journal International, 1967, vol. 13, issue 5, pp. 529-539. DOI: 10.1111/j.1365-246X.1967.tb02303.x Chikrii A.A., Matichin I.I. On an analogue of the Cauchy formula for linear systems of any fractional order, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, 2007, no. 1, pp. 50-55 (in Russian). Popov A.Yu., Sedletskii A.M. Distribution of roots of Mittag-Leffler functions, Journal of Mathematical Sciences, 2013, vol. 190, issue 2, pp. 209-409. DOI: 10.1007/s10958-013-1255-3 Dzhrbashyan M.M. Integral'nye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti (Integral transformations and representations of functions in the complex domain), Moscow: Nauka, 1966, 672 p. Full text