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## Archive of Issues

Russia Izhevsk
Year
2016
Volume
26
Issue
1
Pages
46-57
 Section Mathematics Title Multiple capture of rigidly coordinated evaders Author(-s) Blagodatskikh A.I.a Affiliations Udmurt State Universitya Abstract The present paper deals with the problem of pursuit of a group of rigidly coordinated evaders in a nonstationary conflict-controlled process with equal opportunities $$\begin{array}{llllllllcccc} P_i & : & \dot x_i = A(t)x_i + u_i,& u_i \in U(t), & x_i(t_0) = X_i^0, & i = 1,2, \dots, n, \\ E_j & : & \dot y_j = A(t)y_j + v, & v \in U(t) , & y_j(t_0) = Y_j^0 , & j = 1,2, \dots, m. \\ \end{array}$$ We say that a multiple capture in the problem of pursuit holds if the specified number of pursuers catch evaders, possibly at different times $$x_\alpha (\tau_\alpha) = y_{j_\alpha}(\tau_\alpha), \quad \alpha \in \Lambda, \quad \Lambda \subset \{1,2, \dots, n\}, \quad |\Lambda| = b\quad (n \geqslant b \geqslant 1), \quad j_\alpha \subset \{1,2, \dots, m\}.$$ The problem of nonstrict simultaneous multiple capture requires that capture moments coincide $$x_\alpha (\tau) = y_{j_\alpha}(\tau), \quad \alpha \in \Lambda.$$ The problem of a simultaneous multiple capture requires that lowest capture moments coincide $$x_\alpha (\tau) = y_{j_\alpha}(\tau), \quad x_\alpha(s) \ne y_{j_\alpha}(s), \quad s \in [t_0, \tau), \quad \alpha \in \Lambda.$$ In this paper we obtain necessary and sufficient conditions for simultaneous multiple capture and nonstrict simultaneous multiple capture. Keywords capture, multiple capture, simultaneous multiple capture, pursuit, evasion, differential games, conflict-controlled processes UDC 517.977.8, 519.837.4 MSC 49N70, 49N75 DOI 10.20537/vm160104 Received 20 February 2016 Language Russian Citation Blagodatskikh A.I. Multiple capture of rigidly coordinated evaders, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 46-57. References Isaacs R. Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization, New York: John Wiley and Sons, 1965, 384 p. Translated under the title Differentsial'nye igry, Moscow: Mir, 1967, 479 p. Pontryagin L.S. A linear differential evasion game, Proceedings of the Steklov Institute of Mathematics, 1971, vol. 112, pp. 27-60. Krasovskii N.N., Subbotin A.I. Positsionnye differentsial'nye igry (Positional differential games), Moscow: Fizmatlit, 1974, 456 p. Petrosyan L.A. Differentsial'nye igry presledovaniya (Differential games of pursuit), Leningrad: Leningrad State University, 1977, 222 p. Chernous'ko F.L., Melikyan A.A. Igrovye zadachi upravleniya i poiska (Control and search game problems), Moscow: Nauka, 1978, 272 p. Petrosyan L.A. “Life-line” pursuit games with several players, Izvestiya Akademii Nauk Armyanskoi SSR. Matematika, 1966, vol. 1, no. 5, pp. 331-340 (in Russian). Pshenichnyi B.N. Simple pursuit by several objects, Kibernetika, 1976, no. 3, pp. 145-146 (in Russian). 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. Chikrii A.A. Konfliktno upravlyaemye protsessy (Conflict controlled processes), Kiev: Naukova Dumka, 1992, 380 p. Petrov N.N. Multiple capture in Pontryagin's example with phase constraints, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, no. 5, pp. 725-732. 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. 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. Blagodatskikh A.I. Simultaneous multiple capture in a conflict-controlled process, Journal of Applied Mathematics and Mechanics, 2013, vol. 77, no. 3, pp. 314-320. Demidovich B.P. Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the mathematical stability theory), Moscow: Nauka, 1967, 472 p. Full text