Abstract
The article deals with the linear pursuit problem with n pursuers and m evaders with equal opportunities for all participants and geometric restrictions on the control of players. The evaders use program strategies, and each pursuer catches no more than one evader. The goal of the pursuers is to catch a given number of evaders, and each evader needs to be caught no less than a certain number of pursuers. In this paper, sufficient conditions are obtained for multiple capture of a given number of evaders.
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References
Isaacs, R.: Differential Games. Wiley, New York (1965)
Pontryagin, L.S.: Selected Scientific Works. Nauka, Moscow (1988)
Cristiani E., Falkone M.: Fully-discrete schemes for the value function of pursuit-evasion games with state constraints. In: Bernhard P., Gaitsgory V., Pourtallier O. (eds) Advances in Dynamic Games and Applications. Annals of the International Society of Dynamic Games, vol 10. Birkhauser, Boston, 177–206 (2009)
Nahin, P.J.: Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton University Press, Princeton (2012)
Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York (1988)
Lewin, J.: Differential Games Theory and Methods for Solving Game Problems with Singular Surfaces. Springer, London (1994)
Leitmann, G.: Cooperative and Noncooperative Many-Player Differential Games. Springer, Vienna (1974)
Petrosyan, L.A.: Differential Games of Pursuit. World Scientific, New York (1993)
Subbotin, A.I., Chentsov, A.G.: Optimization of a Guarantee in Problems of Control. Nauka, Moscow (1981). (in Russian)
Chikrii, A.A.: Conflict Controlled Processes. Naukova dumka, Kiev (1992). (in Russian)
Grigorenko, N.L.: Mathematical Methods of Control a Few Dynamic Processes. Moscow State University, Moscow (1990). (in Russian)
Blagodatskikh, A.I., Petrov, N.N.: Conflict Interaction of Groups of Controlled Objects. Udmurt State University, Izhevsk (2009). (in Russian)
Satimov, N.Y., Rikhsiev, B.B.: Methods of Solving the Problem of Avoiding Encounter in Mathematical Control Theory. Fan, Tashkent (2000). (in Russian)
Alexander, S., Bishop, R., Christ, R.: Capture pursuit games on unbounded domain. L’Enseignement Math. 55(1/2), 103–125 (2009)
Alias, I.A., Ibragimov, G.I., Rakmanov, A.: Evasion differential games of infinitely many evaders from infinitely many pursuers in Hilbert space. Dyn. Games Appl. 6(2), 1–13 (2016)
Ganebny, S.A., Kumkov, S.S., Le Menec, S., Patsko, V.S.: Model problem in a line with two pursuers and one evader. Dyn. Games Appl. 2, 228–257 (2012)
Hagedorn, P., Breakwell, J.V.: A differential game with two pursuers and one evader. J. Optim. Theory Appl. 18(2), 15–29 (1976)
Kuchkarov, A.S., Ibragimov, G.I., Khakestari, M.: On a linear differential game of optimal approach of many pursuers with one evader. J. Dyn. Control Syst. 19(1), 1–15 (2013)
Stipanovic, D.M., Melikyan, A., Hovakimyan, N.: Guaranteed strategies for nonlinear multi-player pursuit-evasion games. Int. Game Theory Rev. 12(1), 1–17 (2000)
Grigorenko, N.L.: Simple pursuit evasion game with a group of pursuers and one evader. Vestnik Moskov. Univ. Ser XV Vychisl. Matematika i Kibernetika. 1, 41–47 (1983). (in Russian)
Blagodatskikh, A.I.: Simultaneous multiple capture in a simple pursuit problem. J. Appl. Math. Mech. 73(1), 36–40 (2009)
Bopardikar, S.D., Suri, S.: \(k\)-Capture in multiagent pursuit evasion, or the lion and the hyenas. Theor. Comput. Sci. 522, 13–23 (2014)
Petrov, N.N.: Multiple capture in Pontryagin’s example with phase constraint. J. Appl. Math. Mech. 61(5), 725–732 (1997)
Petrov, N.N., Solov’eva, N.A.: Multiple capture in Pontryagin’s recurrent example with phase constraints. Proc. Steklov Inst. Math. 293(1), 174–182 (2016)
Petrov, N.N., Solov’eva, N.A.: Multiple capture in Pontryagin’s recurrent example. Autom. Remote Control 77(5), 854–860 (2016)
Petrov, N.N., Solov’eva, N.A.: A multiple capture of an evader in linear recursive differential games. Trudy Inst. Mat. Mekh. UrO RAN. 23(1), 212–218 (2017)
Blagodatskikh, A.I.: Simultaneous multiple capture in a conflict-controlled process. J. Appl. Math. Mech. 77(3), 314–320 (2013)
Petrov, N.N., Prokopenko, V.A.: One problem of pursuit of a group of evader. Differ. Uravn. 23(4), 725–726 (1987). (in Russian)
Sakharov, D.V.: On two differential games of simple group pursuit. Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki. 1, 50–59 (2012). (in Russian)
Zubov, V.I.: The theory of recurrent functions. Sib. Math. J. 3(4), 532–560 (1962). (in Russian)
Hall, M.: Combinatorial Theory. Blaisdell Publishing Company, Waltham, Toronto, London (1967)
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This work was supported by grant 1.5211.2017/8.9 from the Ministry of Education and Science of the Russian Federation within the framework of the basic part of the state project in the field of science and grant 18-51-41005 from Russian Foundation for Basic Research.
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Petrov, N.N., Solov’eva, N.A. Multiple Capture of Given Number of Evaders in Linear Recurrent Differential Games. J Optim Theory Appl 182, 417–429 (2019). https://doi.org/10.1007/s10957-019-01526-7
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DOI: https://doi.org/10.1007/s10957-019-01526-7