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**Differential game: “Life line” for non-stationary geometric constraints on controls.**
*(English)*
Zbl 1491.91028

Summary: We consider the differential game with “life line” of R. Isaacs that occupies a special place as an example of differential game with phase constraint. In the present paper, the problem of one pursuer and one evader is studied, in which case controls of players are subjected to non-stationary geometric constraints of different types. The notion of strategy of parallel pursuit (briefly \(\Pi \)-strategy) was introduced and used to solve the quality problem for “The game with a life line” by L. A. Petrosjan. Dynamics of changing of the attainability domains of the players is studied by the properties of theory of multi-valued mapping and a simple proof of the main lemma is given. This work develops and extends the works of Isaacs, Petrosjan, Pshenichnyi, Azamov and other researchers, including the authors.

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\textit{B. T. Samatov} et al., Lobachevskii J. Math. 43, No. 1, 237--248 (2022; Zbl 1491.91028)

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### References:

[1] | Isaacs, R., Differential Games (1965), New York: Wiley, New York · Zbl 0125.38001 |

[2] | Petrosjan, L. A., Differential Games of Pursuit. Series on Optimization (1993), Singapore: World Scientific, Singapore |

[3] | Pshenichnyi, B. N., Simple pursuit by several objects, Cybern. Syst. Anal., 12, 484-485 (1976) |

[4] | Pontryagin, L. S., Selected Works (2004), Moscow: MAKS Press, Moscow · Zbl 1200.01049 |

[5] | Azamov, A. A., On the quality problem for simple pursuit games with constraint, Serdica Bulg. Math. Publ. Sofia, 12, 38-43 (1986) · Zbl 0629.90107 |

[6] | A. A. Azamov and B. T. Samatov, ‘‘The \(\Pi \)-Strategy: Analogies and applications,’’ in Proceedings of the 4th International Conference on Game Theory and Management, St. Petersburg (2010), pp. 33-47. · Zbl 1272.91032 |

[7] | Blagodatskikh, A. I.; Petrov, N. N., Conflict Interaction of Groups of Controlled Objects (2009), Izhevsk: Udmurt Gos. Univ., Izhevsk · Zbl 1233.49002 |

[8] | Grigorenko, N. L., Mathematical Methods of Control for Several Dynamic Processes (1990), Moscow: Mosk. Gos. Univ., Moscow |

[9] | Samatov, B. T., On a pursuit-evasion problem under a linear change of the pursuer resource, Sib. Adv. Math., 23, 294-302 (2013) · Zbl 1340.91017 |

[10] | Samatov, B. T., The pursuit-evasion problem under integral-geometric constraints on pursuer controls, Autom. Remote Control, 74, 1072-1081 (2013) · Zbl 1301.49103 |

[11] | Samatov, B. T., The \(\Pi \), J. Appl. Math. Mech., 78, 258-263 (2014) · Zbl 1432.49054 |

[12] | Samatov, B. T., Problems of group pursuit with integral constraints on controls of the players I, Cybern. Syst. Anal., 49, 756-767 (2013) · Zbl 1298.91045 |

[13] | Samatov, B. T., Problems of group pursuit with integral constraints on controls of the players II, Cybern. Syst. Anal., 49, 907-921 (2013) · Zbl 1457.91086 |

[14] | Samatov, B. T.; Ibragimov, G. I.; Khodjibayeva, I. V., Pursuit-evasion differential games with the Gronwall type constraints on controls, Ural Math. J., 6, 95-107 (2020) · Zbl 1461.91052 |

[15] | Samatov, B. T.; Umaraliyeva, N. T.; Uralova, S. I., Differential games with the Langenhop type constrains on controls, Lobachevskii J. Math., 42, 2942-2951 (2021) · Zbl 1480.91049 |

[16] | Bannikov, A. S.; Petrov, N. N., “Linear non-stationary differential pursuit games with several evaders,” Vestn. Udmurt. Univ, Mat. Mekh. Komp. Nauki, 25, 3-12 (2014) · Zbl 1299.91008 |

[17] | Vinogradova, M. N., “On the capture of two evaders in a non-stationary pursuit-evasion problem with phase restrictions,” Vestn. Udmurt. Univ, Mat. Mekh. Komp. Nauki, 25, 12-20 (2015) · Zbl 1331.49055 |

[18] | Petrov, N. N., To the non-stationary problem of the group pursuit with phase restrictions, Mat. Teor. Igr Prilozh., 2, 74-83 (2010) · Zbl 1228.91012 |

[19] | Kotlyachkova, E. V., About non-stationary problem of simple pursuit in the class of impulse strategies, Izv. IMI Udmur. Univ., 45, 106-113 (2015) · Zbl 1331.49054 |

[20] | Bannikov, A. S., A non-stationary problem of group pursuit, J. Comput. Syst. Sci. Int., 48, 527-532 (2009) · Zbl 1308.49032 |

[21] | Yuldashev, T. K., Nonlinear optimal control of thermal processes in a nonlinear inverse problem, Lobachevskii J. Math., 41, 124-136 (2020) · Zbl 1450.35295 |

[22] | Vassilina, G. K., Optimal control problem of stochastic systems, Lobachevskii J. Math., 42, 641-648 (2021) · Zbl 1464.49012 |

[23] | Malikov, A. I., State and unknown inputs observers for time-varying nonlinear systems with uncertain disturbances, Lobachevskii J. Math., 40, 769-775 (2019) · Zbl 1421.93028 |

[24] | Lapin, A. V.; Khasanov, M. G., State-constrained optimal control of an elliptic equation with its right-hand side used as control function, Lobachevskii J. Math., 32, 453-462 (2011) · Zbl 1255.49045 |

[25] | Gaifullin, A. M.; Bosnyakov, I. S.; Sviridenko, Yu. N.; Suprunenko, S. N., An approach to ensuring vortex safety of an aircraft, Lobachevskii J. Math., 41, 1184-1189 (2020) · Zbl 1451.76020 |

[26] | Ibragimov, G. I., The optimal pursuit problem reduced to an infinite system of differential equations, J. Appl. Math. Mech., 77, 470-476 (2013) · Zbl 1432.49056 |

[27] | 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. Inst. Math., 24, 200-214 (2018) · Zbl 1432.91025 |

[28] | Vinogradova, M. N., On the capture of two escapees in the non-stationary problem of simple pursuit, Mat. Teor. Igr Prilozh., 4, 21-31 (2012) · Zbl 1273.91065 |

[29] | Lu, Zuliang, Adaptive mixed finite element methods for nonlinear optimal control problems, Lobachevskii J. Math., 32, 1-15 (2011) · Zbl 1255.49046 |

[30] | Lapin, A. V.; Khasanov, M. G., Iterative solution methods for mesh approximation of control and state constrained optimal control problem with observation in a part of the domain, Lobachevskii J. Math., 35, 241-258 (2014) · Zbl 1316.65063 |

[31] | Bannikov, A. S., Some non-stationary problems of group pursuit, Izv. IMI Udmur. Univ., 41, 3-46 (2013) · Zbl 1305.49051 |

[32] | Bannikov, A. S.; Petrov, N. N., On a nonstationary problem of group pursuit, Proc. Steklov Inst. Math., 3, 41-52 (2010) · Zbl 1237.49056 |

[33] | Andrianova, A. A., One approach for solving optimization problems with apriori estimates of approximation of admissible set, Lobachevskii J. Math., 34, 368-376 (2013) · Zbl 1290.90069 |

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