Multiple Capture of Given Number of Evaders in Linear Recurrent Differential Games

  • Nikolay N. Petrov
  • Nadezhda A. Solov’evaEmail author


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.


Differential game Pursuer Evader Recurrent function 

Mathematics Subject Classification

49N70 49N75 



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.


  1. 1.
    Isaacs, R.: Differential Games. Wiley, New York (1965)zbMATHGoogle Scholar
  2. 2.
    Pontryagin, L.S.: Selected Scientific Works. Nauka, Moscow (1988)zbMATHGoogle Scholar
  3. 3.
    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)Google Scholar
  4. 4.
    Nahin, P.J.: Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton University Press, Princeton (2012)zbMATHGoogle Scholar
  5. 5.
    Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York (1988)CrossRefGoogle Scholar
  6. 6.
    Lewin, J.: Differential Games Theory and Methods for Solving Game Problems with Singular Surfaces. Springer, London (1994)Google Scholar
  7. 7.
    Leitmann, G.: Cooperative and Noncooperative Many-Player Differential Games. Springer, Vienna (1974)Google Scholar
  8. 8.
    Petrosyan, L.A.: Differential Games of Pursuit. World Scientific, New York (1993)CrossRefzbMATHGoogle Scholar
  9. 9.
    Subbotin, A.I., Chentsov, A.G.: Optimization of a Guarantee in Problems of Control. Nauka, Moscow (1981). (in Russian)zbMATHGoogle Scholar
  10. 10.
    Chikrii, A.A.: Conflict Controlled Processes. Naukova dumka, Kiev (1992). (in Russian)Google Scholar
  11. 11.
    Grigorenko, N.L.: Mathematical Methods of Control a Few Dynamic Processes. Moscow State University, Moscow (1990). (in Russian)Google Scholar
  12. 12.
    Blagodatskikh, A.I., Petrov, N.N.: Conflict Interaction of Groups of Controlled Objects. Udmurt State University, Izhevsk (2009). (in Russian)Google Scholar
  13. 13.
    Satimov, N.Y., Rikhsiev, B.B.: Methods of Solving the Problem of Avoiding Encounter in Mathematical Control Theory. Fan, Tashkent (2000). (in Russian)zbMATHGoogle Scholar
  14. 14.
    Alexander, S., Bishop, R., Christ, R.: Capture pursuit games on unbounded domain. L’Enseignement Math. 55(1/2), 103–125 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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)MathSciNetGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hagedorn, P., Breakwell, J.V.: A differential game with two pursuers and one evader. J. Optim. Theory Appl. 18(2), 15–29 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Blagodatskikh, A.I.: Simultaneous multiple capture in a simple pursuit problem. J. Appl. Math. Mech. 73(1), 36–40 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Bopardikar, S.D., Suri, S.: \(k\)-Capture in multiagent pursuit evasion, or the lion and the hyenas. Theor. Comput. Sci. 522, 13–23 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Petrov, N.N.: Multiple capture in Pontryagin’s example with phase constraint. J. Appl. Math. Mech. 61(5), 725–732 (1997)MathSciNetCrossRefGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Petrov, N.N., Solov’eva, N.A.: Multiple capture in Pontryagin’s recurrent example. Autom. Remote Control 77(5), 854–860 (2016)CrossRefzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Blagodatskikh, A.I.: Simultaneous multiple capture in a conflict-controlled process. J. Appl. Math. Mech. 77(3), 314–320 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Petrov, N.N., Prokopenko, V.A.: One problem of pursuit of a group of evader. Differ. Uravn. 23(4), 725–726 (1987). (in Russian)zbMATHGoogle Scholar
  29. 29.
    Sakharov, D.V.: On two differential games of simple group pursuit. Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki. 1, 50–59 (2012). (in Russian)CrossRefzbMATHGoogle Scholar
  30. 30.
    Zubov, V.I.: The theory of recurrent functions. Sib. Math. J. 3(4), 532–560 (1962). (in Russian)MathSciNetGoogle Scholar
  31. 31.
    Hall, M.: Combinatorial Theory. Blaisdell Publishing Company, Waltham, Toronto, London (1967)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussia

Personalised recommendations