OPTIMAL NUMBER OF PURSUERS IN THE GAME ON THE 1-SKELETON OF TESSERACT: OPTIMAL NUMBER OF PURSUERS IN THE GAME ON THE 1-SKELETON OF TESSERACT

Гафуржан Ибрагимов; Захриддин Муминов

doi:10.52754/16948645_2023_2_169

Авторы

Исмаилович Ташкентский государственный экономический университет
Эшкобилович Ташкентский государственный экономический университет

DOI:

https://doi.org/10.52754/16948645_2023_2_169

Ключевые слова:

тессеракт, дифференциальная игра, стратегия, преследователь, убегающий

Аннотация

В статье изучаются дифференциальные игры преследования и уклонения внутри четырехмерного куба т.е. тессеракта, где все игроки перемещаются по ребрам. Задача состоит в том, чтобы найти оптимальное количество преследователей в игре, построить стратегии преследователей в игре преследования и стратегию уклонения в игре уклонения.

Библиографические ссылки

Isaacs, R.: Differential games. John Wiley & Sons, New York, NY, USA. (1965)

Pontryagin, L.S.: Selected works. Nauka, Moscow. (1988)

Krasovskii, N.N. and Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York. (1988) DOI: https://doi.org/10.1007/978-1-4612-3716-7

Azamov, A.A.: On Pontryagin’s second method in linear differential games of pursuit. Math. USSR-Sb., 46(3), 429–437 (1983) DOI: https://doi.org/10.1070/SM1983v046n03ABEH002944

Berkovitz, L.D.: Differential game of generalized pursuit and evasion. SIAM J. Contr. 24(3): 361–373 (1986) DOI: https://doi.org/10.1137/0324021

Elliot, R.J., Kalton, N.J.: The Existence of Value for Differential Games. American Mathematical Soc.: Providence, RI, USA. (1972) DOI: https://doi.org/10.1090/memo/0126

Fleming, W.H.: The convergence problem for differential games. J. Math. Anal. Appl. 3, 102–116 (1961) DOI: https://doi.org/10.1016/0022-247X(61)90009-9

Friedman, A.: Differential Games. John Wiley and Sons, New York (1971)

Hajek, O.: Pursuit Games. Math. Sci. Engrg.. Academic Press, New York (1975)

Pontryagin, L.S., Mishchenko, E.F.: The problem of evading the encounter in linear differential games, Differencial’nye Uravnenija, 7, 436–445 (1971)

Petrosyan, L.A.: Differential Games of Pursuit. World Scientific, Singapore, London (1993) DOI: https://doi.org/10.1142/1670

Satimov, N.Y., Rikhsiev, B.B.: Methods of Solving of Evasion Problems in Mathematical Control Theory. Fan, Tashkent, Uzbekistan (2000)

Grigorenko, N.L.: Mathematical methods of control of several dynamic processes. Moscow: MSU Press (1990) (in Russian).

Ibragimov, G.I.: A game of optimal pursuit of one object by several. J. Appl. Maths Mechs, 62(2), 187–192 (1998) DOI: https://doi.org/10.1016/S0021-8928(98)00024-0

Blagodatskikh, A.I., Petrov, N.N.: Simultaneous Multiple Capture of Rigidly CoordinatedEvaders. Dynamic Games and Applications 9, 594–613, (2019). doi:10.1007/s13235-019-00300-8 DOI: https://doi.org/10.1007/s13235-019-00300-8

Azamov, A.A., Kuchkarov, A.Sh. Holboyev, A.G.: The pursuit-evasion game on the 1-skeleton graph of the regular polyhedron. III. Mat. Teor. Igr Pril., 11(4), 5–23 (2019) DOI: https://doi.org/10.1134/S0005117919010144

Scott, W.L., Leonard, N.E.: Optimal evasive strategies for multiple interacting agents with motion constraints. Automatica J. IFAC, 94, 26–34 (2018) DOI: https://doi.org/10.1016/j.automatica.2018.04.008

Yan, R., Shi, Z., Zhong, Y.: Cooperative strategies for two-evader-one-pursuer reach-avoid differential games. International Journal of Systems Science. 52(9), 1894–1912, (2021). doi:10.1080/00207721.2021.1872116. DOI: https://doi.org/10.1080/00207721.2021.1872116

Alias I.A., Ibragimov G.I., Rakhmanov A.T.: Evasion Differential Game of Infinitely Many Evaders from Infinitely Many Pursuers in Hilbert Space. Dynamic Games and Applications. 6(2): 1–13 (2016). doi:10.1007/s13235-016-0196-0, DOI: https://doi.org/10.1007/s13235-016-0196-0

Ibragimov, G.I., Ferrara, M., Ruziboev M., Pansera B.A.: Linear evasion differential game of one evader and several pursuers with integral constraints. International Journal of Game Theory. 50, 729–750.(2021). doi:10.1007/s00182-021-00760-6 DOI: https://doi.org/10.1007/s00182-021-00760-6

Ibragimov, G.I., Salleh, Y.: Simple motion evasion differential game of many pursuers and one evader with integral constraints on control functions of players. Journal of Applied Mathematics, Article ID 748096, (2012). doi:10.1155/2012/748096 DOI: https://doi.org/10.1155/2012/748096

Kumkov, S.S., Le Ménec, S., Patsko, V.P.: Zero-sum pursuit-evasion differential games with many objects: Survey of publications. Dynamic games and applications. 7, 609–633, (2017). doi:10.1007/s13235-016-0209-z DOI: https://doi.org/10.1007/s13235-016-0209-z

Belousov A. A.: O lineinykh differentsialnykh igrakh presledovaniya s integralnymi ogranicheniyami. In-t matematiki im. V. A. Steklova RAN, MGU im. M. V. Lomonosova, M., 321–322 (2008)

Chikrii, A. A., Belousov, A. A.: On linear differential games with convex integral constraints. Trudy Inst. Mat. i Mekh. UrO RAN, 19(4), 308–319 (2013)

Ibragimov, G.I., Ferrara, M., Kuchkarov, A.Sh., and Pansera, B.A.: Simple motion evasion differential game of many pursuers and evaders with integral constraints. Dynamic Games and Applications. 8, 352–378 (2018). doi:10.1007/s13235-017-0226-6 DOI: https://doi.org/10.1007/s13235-017-0226-6

Satimov, N.Yu., Ibragimov, G.I.: One class of simultaneous pursuit games, Izv. Vyssh. Uchebn. Zaved. Mat. 5, 46–-55 (2012) DOI: https://doi.org/10.3103/S1066369X12050052

Ibragimov, G.I., Salimi, M., Amini, M.: Evasion from many pursuers in simple motion differential game with integral constraints, European Journal of Operational Research 218(2), 505–511 (2012) DOI: https://doi.org/10.1016/j.ejor.2011.11.026

Ibragimov, G., Alias, I.A., Tukhtasinov, M, Hasim, R.M.: A pursuit problem described by infinite system of differential equations with coordinate-wise integral constraints on control functions, Malaysian Journal of Mathematical Sciences, 9(1), 67–76, (2015)

Ferrara, M., Ibragimov, G.I., Salimi, M.: Pursuit-evasion game of many players with coordinate-wise integral constraints on a convex set in the plane. Atti della Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche, Matematiche e Naturali, 95(2), 1–6, (2017)

Alias, I.A., Jaman, K., Ibragimov, G.: Pursuit differential game of many pursuers and one evader in a convex hyperspace, Mathematical Modeling and Computing, 9(1), 9–17 (2022) DOI: https://doi.org/10.23939/mmc2022.01.009

Azamov, A.A.: Lower bound for the advantage coefficient in the graph search problem. Differential equations, 44(12) 1764–1767 (2008) DOI: https://doi.org/10.1134/S0012266108120136

Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching, Theoret. Comput. Sci., 399, 236–245 (2008) DOI: https://doi.org/10.1016/j.tcs.2008.02.040

Ibragimov, G.I., Luckraz, Sh.: On a Characterization of Evasion Strategies for Pursuit-Evasion Games on Graphs. Journal of Optimization Theory and Applications. 175, 590–596, (2017). doi:10.1007/s10957-017-1155-7 DOI: https://doi.org/10.1007/s10957-017-1155-7

Bonato, A., Golovach, P., Hahn, G., Kratochvil, J.: The capture time of a graph. Discrete Mathematics, 309(18), 5588–5595 (2009) DOI: https://doi.org/10.1016/j.disc.2008.04.004

Azamov, A., Ibaydullaev, T.: A pursuit-evasion differential game with slow pursuers on the edge graph of simplexes I. Mathematical Game Theory and Applications. 12(4), 7–23 (2020). doi:10.17076/mgta_2020_4_23 DOI: https://doi.org/10.17076/mgta_2020_4_23

Andreae, T.; Hartenstein, F.; Wolter, A.: A two-person game on graphs where each player tries to encircle his opponent’s men. Theoret. Comput. Sci. (Math Games), 215, 305–323 (1999) DOI: https://doi.org/10.1016/S0304-3975(98)00203-5

Azamov, A.A., Kuchkarov, A.Sh., Holboyev, A.G.: The pursuit-evasion game on the 1-skeleton graph of the regular polyhedron. II. Mat. Teor. Igr Pril. 8(4), 3–13 (2016) DOI: https://doi.org/10.1134/S0005117917040166

Azamov, A.A., Kuchkarov, A.Sh., Holboyev, A.G.: The pursuit-evasion game on the 1-skeleton graph of the regular polyhedron. I. Mat. Teor. Igr Pril., 7(3), 3–15 (2015)

Azamov, A.A., Ibaydullaev, T., Ibragimov, G.I., Alias, I.A.: Optimal number of pursuers in the differential games on the 1-skeleton of orthoplex. Symmetry (Game Theoretical Symmetry Dynamic Processes), 13(11), 2170 (2021). doi:10.3390/sym13112170 DOI: https://doi.org/10.3390/sym13112170

OPTIMAL NUMBER OF PURSUERS IN THE GAME ON THE 1-SKELETON OF TESSERACT