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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
3
Pages
83-89
 Section Mathematics Title Group pursuit in recurrent Pontryagin example Author(-s) Solov'eva N.A.a Affiliations Udmurt State Universitya Abstract A non-stationary differential game (a generalized example of L.S. Pontryagin) with $n$ pursuers and one evader is considered in the space $\mathbb R^k$ $(k \geqslant 2)$. All players have equal dynamic and inertial capabilities. The game is described by a system of the form $$Lz_{i}=z_{i}^{(l)}+a_{1}(t)z_{i}^{(l-1)}+ \dots +a_{l}(t)z_{i} =u_{i}-v, \quad u_{i},v\in V,$$ $$z_{i}^{(s)}(t_0) = z_{is}^0,\quad i=1,2, \ldots, n,\ s=0,1, \ldots, l-1.$$ The set $V$ of admissible player controls is strictly convex compact set with smooth boundary, $a_{1}(t),\dots, a_{l}(t)$ are continuous on $[t_0, \infty)$ functions, the terminal sets are the origin of coordinates. Pursuers use quasi-strategies. It is assumed that functions $\xi_{i}(t)$ being the solution of Cauchy problem $$Lz_{i}=0,\quad z_{i}^{(s)}(t_0) = z_{is}^0,$$ are recurrent. Properties of recurrent functions are given. In terms of initial positions and game parameters the sufficient conditions of the pursuit problem solvability are obtained. The proof is carried out using the method of resolving functions. An example illustrating the obtained conditions is given. Keywords differential game, group pursuit, capture problem, Pontryagin's example, recurrent function UDC 517.977 MSC 91A06, 91A23, 91A24, 49N70, 49N75 DOI 10.20537/vm140308 Received 22 August 2014 Language Russian Citation Solov'eva N.A. Group pursuit in recurrent Pontryagin example, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 3, pp. 83-89. References Pontryagin L.S. Izbrannye nauchnye trudy (Selected scientific works), vol. 2, Moscow: Nauka, 1988, 575 p. Chikrii A.A. Konfliktno upravlyaemye protsessy (Conflict controlled processes), Kiev: Naukova Dumka, 1992, 380 p. 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. Pshenichnyi B.N. Simple pursuit by several objets, Kibernetika, 1976, no. 3, pp. 145-146 (in Russian). Grigorenko N.L. The game of simple pursuit-evasion for groups of pursuers and one evader, Moscow University Computational Mathematics and Cybernetics, 1983, no. 1, pp. 41-47 (in Russian). Blagodatskikh A.I. Simultaneous multiple capture in a simple pursuit problem, Journal of Applied Mathematics and Mechanics, 2009, vol. 73, no. 1, pp. 36-40. Petrov N.N. To the non-stationary problem of the group pursuit with phase restrictions, Matematicheskaya Teoriya Igr i Ee Prilozheniya, 2010, vol. 2, no. 4, pp. 74-83 (in Russian). Petrov N.N. Multiple capture in Pontryagin's problem with phase restrictions, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, no. 5, pp. 725-732. Petrov N.N. “Soft” capture in Pontryagin's example with many participants, Journal of Applied Mathematics and Mechanics, 2003, vol. 67, no. 5, pp. 671-680. Bannikov A.S., Petrov N.N. On non-stationary problem of group pursuit with phase restrictions, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 271, no. 1, pp. 41-52. Blagodatskikh A.I. Simultaneous multiple capture of evaders in a simple group pursuit problem, Vestnik Udmurtskogo Universiteta. Matematika, 2007, no. 1, pp. 17-24 (in Russian). Blagodatskikh A.I. Almost periodic processes with conflict control with many participants, Journal of Computer and Systems Sciences International, 2007, vol. 46, no. 2, pp. 244-247. Blagodatskikh A.I. Group pursuit in Pontryagin’s nonstationary example, Differential Equations, 2008, vol. 44, no. 1, pp 40-46. Zubov V.I. The theory of recurrent functions, Sib. Mat. Zh., 1962, vol. 3, no. 4, pp. 532-560 (in Russian). Full text