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## Archive of Issues

Russia Izhevsk
Year
2020
Volume
30
Issue
2
Pages
249-258
 Section Mathematics Title The problem of simple group pursuit with phase constraints in time scales Author(-s) Petrov N.N.a Affiliations Udmurt State Universitya Abstract In the finite-dimensional Euclidean space $\mathbb R^k,$ the problem of pursuit of one evader by a group of pursuers with equal opportunities for all participants is considered, which is described in a given time scale $T$ by a system of the form $$z_i^{\Delta} = u_i - v,$$ where $f^{\Delta}$ is the $\Delta$-derivative of the function $f$ in the time scale $T$. The set of admissible controls is a ball of unit radius with the center at the origin. Terminal sets are the coordinate origin. Additionally, it is assumed that the evader does not leave the convex polyhedral set with a nonempty interior during the game. Sufficient conditions for the solvability of the pursuit and evasion problems are obtained. In the research, the method of resolving functions is used as the basic one. Keywords differential game, group pursuit, pursuer, evader, phase restriction, time scale UDC 517.977 MSC 49N79, 49N70, 91A24 DOI 10.35634/vm200208 Received 1 February 2020 Language Russian Citation Petrov N.N. The problem of simple group pursuit with phase constraints in time scales, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 2, pp. 249-258. References Aulbach B., Hilger S. Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems: Contributions to the International Seminar ISAM-90, held in Gaussig (GDR), March 19-23, 1990, Berlin: Akademie-Verlag, 1990, pp. 9-20. Hilger S. Analysis on measure chains - a unified approach to continuous and discrete calculus, Results in Mathematics, 1990, vol. 18, issue 1, pp. 18-56. https://doi.org/10.1007/BF03323153 Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions, New York: Hindawi Publishing Corporation, 2006. Bohner M., Peterson A. Advances in dynamic equations on time scales, Boston: Birkhauser, 2003. Martins N., Torres D.F.M. Necessary conditions for linear noncooperative $N$-player delta differential games on time scales, Discussiones Mathematica. Differential Inclusions, Control and Optimization, 2011, vol. 31, no. 1, pp. 23-37. https://doi.org/10.7151/dmdico.1126 Guseinov G.Sh. Integration on time scales, Journal of Mathematical Analysis and Applications, 2003, vol. 285, no. 1, pp. 107-127. https://doi.org/10.1016/S0022-247X(03)00361-5 Cabada A., Vivero D.R. Expression of the Lebesgue $\Delta$-integral on time scales as a usual Lebesgue integral; application to the calculus of $\Delta$-antiderivatives, Mathematical and Computer Modelling, 2006, vol. 43, no. 1-2, pp. 194-207. https://doi.org/10.1016/j.mcm.2005.09.028 Pshenichnyi B.N. Simple pursuit by several objects, Cybernetics, 1976, vol. 12, issue 3, pp. 484-485. https://link.springer.com/article/10.1007/BF01070036 Grigorenko N.L. The game of simple pursuit-evasion of pursuit group and one evader, Moscow University Computational Mathematics and Cybernetics, 1983, no. 1, pp. 41-47 (in Russian). Ivanov R.P. Simple pursuit-evasion on a compactum, Dokl. Akad. Nauk SSSR, 1980, vol. 254, no. 6, pp. 1318-1321 (in Russian). http://mi.mathnet.ru/eng/dan43986 Alias I., Ramli R., Ibragimov G., Narzullaev A. Simple motion pursuit differential game of many pursuers and one evader on convex compact set, International Journal of Pure and Applied Mathematics, 2015, vol. 102, no. 4, pp. 733-745. https://doi.org/10.12732/ijpam.v102i4.11 Petrov N.N. A certain simple pursuit problem with phase constraints, Automation and Remote Control, 1992, vol. 53, no. 5, pp. 639-642. https://zbmath.org/?q=an:0798.90145 Bannikov A.S. About a problem of positional capture of one evader by group of pursuers, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2011, issue 1, pp. 3-7 (in Russian). http://mi.mathnet.ru/eng/vuu201 Azamov A.A. Osnovaniya teorii diskretnykh igr (Foundations of theory of discrete games), Tashkent: Niso Poligraf, 2011. Krivonos Yu.G., Matichin I.I., Chikrii A.A. Dinamicheskie igry s razryvnymi traektoriyami (Dynamic games with discontinuous paths), Kiev: Naukova Dumka, 2005. Blagodatskikh A.I., Petrov N.N. Konfliktnoe vzaimodeistvie grupp upravlyaemykh ob''ektov (Conflict interaction of groups of controlled objects), Izhevsk: Udmurt State University, 2009. Bannikov A.S. Some non-stationary problems of group pursuit, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2013, issue 1 (41), pp. 3-46. http://mi.mathnet.ru/eng/iimi247 Petrov N.N. Group pursuit problem in a differential game with fractional derivatives, state constraints, and simple matrix, Differential Equations, 2019, vol. 55, no. 6, pp. 841-848. https://doi.org/10.1134/S0012266119060119 Full text