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Section
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Mathematics
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Title
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The problem of simple group pursuit with phase constraints in time scales
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Author(-s)
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Petrov N.N.a
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Affiliations
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Udmurt State Universitya
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Abstract
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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.
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Keywords
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differential game, group pursuit, pursuer, evader, phase restriction, time scale
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UDC
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517.977
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MSC
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49N79, 49N70, 91A24
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DOI
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10.35634/vm200208
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Received
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1 February 2020
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Language
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Russian
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Citation
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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.
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References
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- 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
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