Abstract
In the present paper, we have carried out a fairly complete study of the pursuit problem when the players move by inertia, i.e. when the players carry out their movements with the help of acceleration vectors. The main tool for solving this pursuit problem is the parallel approach strategy (\(\Pi \)-strategy), which, under an arbitrary action of the evader, allows the best approach of the players. It is known that for the simple pursuit problem, the set of meeting points of the players is the ball of Apollonius. In the present paper, it is shown that in the case of inertial motions of players, the set of meeting points of the players is a linear combination of two such Apollonius balls, the first of which is built from the initial states, and the second from the initial states of the velocities.
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References
Isaacs, R.: Differential Games. Wiley, New York (1965)
Petrosjan, L.A.: Differential Games of Pursuit. Series on Optimization, Vol. 2. World Scientific Poblishing, Singapore (1993). https://doi.org/10.1114/1670
Azamov, A.: On the quality problem for simple pursuit games with constraint. Publ. Sofia: Serdica Bulgariacae Math. 12(1), 38–43 (1986)
Azamov, A.A., Samatov, B.T.: The \(\Pi \)-strategy: analogies and applications. Contrib. Game Theor. Manage. 4, 33–46 (2011)
Sun, W., Tsiotras, P., Lolla, T., Subramani, D.N., Lermusiaux, P.F.J.: Multiple-pursuit/one-evader pursuit-evasion game in in dynamical flow fields. J. Guid. Control Dyn. 40(7), 1627–1637 (2017). https://doi.org/10.2514/1.G002125
Garcia, E., Casbeer, D.W., Pachter, M.: Optimal strategies of the differential game in a circular region. IEEE Control Syst. Lett. 4(2), 492–497 (2019). https://doi.org/10.1109/LCSYS.2019.2963173
Liang, L., Deng, F., Peng, Z., Li, X., Zha, W.: A differential game for cooperative target defense. Automatica 102(April), 58–71 (2019). https://doi.org/10.1016/j.automatica.2018.12.034
Weintraub, I.E., Pachter, M., Garcia, E.: An introduction to pursuit–evasion differential games. In: American Control Conference (ACC), July 01–03 (2020). https://doi.org/10.23919/ACC45564.2020.9147205
Dorothy, M., Maity, D., Shishika, D., Von Moll, A.: One Apollonius Circle is Enough for Many Pursuit-Evasion Games. https://doi.org/10.48550/arXiv.2111.09205
Samatov, B.T., Horilov, M.A., Akbarov, AKh.: Differential game: “life line’’ for non-stationary geometric constraints on controls. Lobachevskii J. Math. 43(1), 237–248 (2022). https://doi.org/10.1134/S1995080222040187
Pshenichnii, B.N.: Simple pursuit by several objects. Cybern. Syst. Anal. 12(5), 484–485 (1976). https://doi.org/10.1007/BF01070036
Satimov, NYu.: Methods of Solving the Pursuit Problems in the Theory of Differential Games. Izd-vo NBRUz, Tashkent (2019)
Samatov, B.T.: Problems of group pursuit with integral constraints on controls of the players II. Cybern. Syst. Anal. 49(6), 907–921 (2013). https://doi.org/10.1007/s10559-013-9581-5
Grigorenko, N.L.: Mathematical Methods of Control for Several Dynamic Processes. Izdat. Gos. Univ, Moscow (1990)
Chikrii, A.A.: Conflict-Controlled Processes. Kluwer Academic Publishers, Dordrecht (1997). https://doi.org/10.1007/978-94-017-1135-7
Munts, N.V., Kumkov, S.S.: On the coincidence of the minimax solution and the value function in a time-optimal game with a lifeline. Proc. Steklov Inst. Math. 305, S125–S139 (2019). https://doi.org/10.1134/S0081543819040138
Samatov, B.T., Sotvoldiyev, A.I.: Intercept problem in dynamic flow field. Uzbek Math. J. 2, 103–112 (2019). https://doi.org/10.29229/uzmj.2019-2-12
Samatov, B.T., Ibragimov, G.I., Hodjibayeva, I.V.: Pursuit-evasion differential games with the Grönwall type constraints on controls. Ural Math. J. 6(2), 95–107 (2020). https://doi.org/10.15826/umj.2020.2.010
Samatov, B.T., Umaraliyeva, N.T., Uralova, S.I.: Differential games with the Langenhop type constraints on controls. Lobachevskii J. Math. 42(12), 2942–2951 (2021). https://doi.org/10.1134/S1995080221120295
Samatov, B.T.: The pursuit-evasion problem under integral-geometric constraints on pursuer controls. Autom. Remote Control 74(7), 1072–1081 (2013). https://doi.org/10.1134/S0005117913070023
Samatov, B.T.: The \(\Pi \)-strategy in a differential game with linear control constraints. J. Appl. Math. Mech. 78(3), 258–263 (2014). https://doi.org/10.1016/j.jappmathmech.2014.09.008
Bakolas, E.: Optimal guidance of the isotropic rocket in the presence of wind. J. Optim. Theor. Appl. 162(3), 954–974 (2014). https://doi.org/10.1007/s10957-013-0504-4
Samatov, B.T., Soyibboev, U.B.: Differential game with a lifeline for the inertial movements of players. Ural Math. J. 7(2), 94–109 (2021). https://doi.org/10.15826/umj.2021.2.007
Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal’noye upravleniye [Optimal Control]. Nauka, Moscow (1979)
Blagodatskikh, V.I.: Introduction to Optimal Control (Linear Theory). Visshaya Shkola, Moscow (2001)
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Samatov, B., Soyibboev, U. (2024). Applications of the \(\Pi \)-Strategy When Players Move with Acceleration. In: Azimov, D. (eds) Proceedings of the IUTAM Symposium on Optimal Guidance and Control for Autonomous Systems 2023. IUTAM 2023. IUTAM Bookseries, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-031-39303-7_10
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