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## Archive of Issues

Russia Izhevsk
Year
2018
Volume
28
Issue
1
Pages
111-118
 Section Mathematics Title A nonlinear pursuit problem with discrete control and incomplete information Author(-s) Shchelchkov K.A.a Affiliations Udmurt State Universitya Abstract A two-person differential game is considered. The game is described by the 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 to capture the evader, viz. system translation to any given neighborhood of zero. Sufficient conditions for the solvability of a capture problem in the piecewise open-loop strategies class are obtained. In addition, it is proved that the capture time tends to zero with the initial position approaching to zero. It happens independent of the evader's actions. Keywords differential game, pursuer, evader, nonlinear system UDC 517.977 MSC 49N70, 49N75 DOI 10.20537/vm180110 Received 26 January 2018 Language Russian Citation Shchelchkov K.A. A nonlinear pursuit problem with discrete control and incomplete information, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 1, pp. 111-118. References Isaacs R. Differential games, New York: John Wiley and Sons, 1965, 416 p. Blaquiere A., Gerard F., Leitmann G. Quantitative and qualitative differential games, New York: Academic Press, 1969, 172 p. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo rezul’tata (Control of dynamic systems. Problem of the minimum guaranteed result), Moscow: Nauka, 1985, 518 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 noncooperative many-player differential games, Austria, Vienna: Springer-Verlag, 1974, 77 p. Pshenichnyi B.N., Ostapenko V.V. Differentsial'nye igry (Differential games) Kiev: Naukova dumka, 1992, 259 p. Chernousko F.L., Melikyan A.A. Igrovye zadachi upravleniya i poiska (Game problems of control and search), 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 (Selected scientific works. Vol. 2), Moscow: Nauka, 1988, 575 p. Chikrii A.A. Conflict-controlled processes, Boston-London-Dordrecht: Kluwer Academic Publishers, 1997, 404 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. Satimov N.Yu., Rikhsiev B.B. Metody resheniya zadachi ukloneniya ot vstrechi v matematicheskoi teorii upravleniya (Methods of solving the problem of avoiding encounter in mathematical control theory), Tashkent: Fan, 2000, 176 p. Nikol'skii M.S. A 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, Journal of Applied Mathematics and Mechanics, 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, Computational Mathematics and Mathematical Physics, 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, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 4, pp. 543-564 (in Russian). DOI: 10.20537/vm160409 Petrov N.N. On a controllability of autonomous systems, Differ. uravn., 1968, vol. 4, no. 4, pp. 606-617 (in Russian). 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. DOI: 10.20537/vm170308 Full text