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## Archive of Issues

Russia Izhevsk
Year
2017
Volume
27
Issue
3
Pages
389-395
 Section Mathematics Title To a nonlinear pursuit problem with discrete control Author(-s) Shchelchkov K.A.a Affiliations Udmurt State Universitya Abstract A two-person differential game is considered. The game is described by the following system of differential equations $\dot x = f(x, u) + g(x, v)$, where $x \in \mathbb R^k$, $u \in U$, $v \in V$. The pursuer's admissible control set is a finite subset of phase space. The evader's admissible control set is a compact subset of phase space. The pursuer's purpose is a translation of phase coordinates to zero. The evader's purpose is to prevent implementation of pursuer's purpose. Sufficient conditions on game parameters for the existence of zero neighborhood from which a capture occurs, that is translation of phase coordinates to zero, have been received. Also, it is proved that a period of time necessary for the pursuer to translate phase coordinates to zero tends to zero with the approaching of the initial position to zero. It happens regardless of the evader's control. Keywords differential game, pursuer, evader, nonlinear system UDC 517.977 MSC 49N70, 49N75 DOI 10.20537/vm170308 Received 26 June 2017 Language Russian Citation Shchelchkov K.A. To a nonlinear pursuit problem with discrete control, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 3, pp. 389-395. References Isaacs R. Differential games, New York: John Wiley and Sons, 1965, 416 p. Translated under the title Differentsial'nye igry, Moscow: Mir, 1967, 480 p. Blaquiere A., Gerard F., Leitmann G. Quantitative and qualitative differential games, New York: Academic Press, 1969, 172 p. Krasovskii N.N. Igrovye zadachi o vstreche dvizhenii (Game problems on meeting motions), Moscow: Nauka, 1970, 420 p. Friedman A. Differential games, New York: John Wiley and Sons, 1971, 350 p. Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p. Hajek O. Pursuit games, New York: Academic Press, 1975, 266 p. Leitmann G. Cooperative and non-cooperative many players differential games, Udine: Springer-Verlag Wien, 1974, 77 p. DOI: 10.1007/978-3-7091-2914-2 Petrosyan L.A. Differentsial'nye igry presledovaniya (Differential pursuit games), 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, 270 p. Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya (Optimization of guarantee in control problems), Moscow: Nauka, 1981, 288 p. Pontryagin L.S. Izbrannye nauchnye trudy. Tom 2 (Selected scientific works. Vol. 2), Moscow: Nauka, 1988, 575 p. Chikrii A.A. Conflict-controlled processes, Springer Netherlands, 1997, xx + 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. Satimov N.Yu., Rikhsiev B.B. Metody resheniya zadachi ukloneniya ot vstrechi v matematicheskoi teorii upravleniya (Methods of solution of evasion problems in mathematical control theory), Tashkent: Fan, 2000, 176 p. Nikol'skii M.S. A certain nonlinear pursuit problem, Kibernetika, 1973, no. 2, pp. 92-94 (in Russian). Pshenichnyi B.N., Shishkina N.B. Sufficient conditions of finiteness of the pursuit time, J. Appl. Math. Mech., 1985, vol. 49, issue 4, pp. 399-404. DOI: 10.1016/0021-8928(85)90043-7 Dvurechensky P.E., Ivanov G.E. Algorithms for computing Minkowski operators and their application in differential games, Comput. Math. Math. Phys., 2014, vol. 54, issue 2, pp. 235-264. DOI: 10.1134/S0965542514020055 Ushakov V.N., Ershov A.A. On the solution of control problems with fixed terminal time, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2016, vol. 26, issue 4, pp. 543-564 (in Russian). DOI: 10.20537/vm160409 Petrov N.N. On the controllability of autonomous systems, Differ. Uravn., 1968, vol. 4, no. 4, pp. 606-617 (in Russian). Petrov N.N. Local controllability of autonomous systems, Differ. Uravn., 1968, vol. 4, no. 7, pp. 1218-1232 (in Russian). Full text