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Simultaneous Multiple Capture of Rigidly Coordinated Evaders

  • Aleksandr I. Blagodatskikh
  • Nikolai N. PetrovEmail author
Article
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Abstract

Differential games of two players represent a very serious mathematical theory. Conflict-controlled processes with many objects (at least from one of the opposing sides) are a natural generalization of differential games of two players. Mathematical problems involving the conflict interaction between two groups of controlled objects are the most difficult to investigate. The specific nature of these problems requires new methods of research. The problem of pursuit of a group of rigidly coordinated evaders in a nonstationary conflict-controlled process with equal capabilities is examined. We say that a multiple capture in the problem of pursuit holds if a certain number of pursuers catch evaders possibly at different instants. In the nonstrict simultaneous multiple capture, there is a requirement of coinciding instants of capture. Simultaneous multiple capture means that the smallest instants of capture coincide. In this paper, sufficient and necessary conditions for simultaneous multiple capture of rigidly coordinated evaders are obtained for the case where pursuers use piecewise-program counterstrategies. Control of the pursuers which can guarantee simultaneous multiple capture not later than at a finite instant is constructed explicitly. A number of examples are considered.

Keywords

Capture Multiple capture Simultaneous multiple capture Pursuit Evasion Differential games Conflict-controlled processes 

Mathematics Subject Classification

49N70 49N75 91A06 

Notes

Acknowledgements

The work of the second author was supported by the Russian Foundation for Basic Research (Grant 18-51-41005).

References

  1. 1.
    Alexander S, Bishop R, Christ R (2009) Capture pursuit games on unbounded domain. Enseign Math 55(1/2):103–125MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alias IA, Ibragimov GI, Rakmanov A (2016) Evasion differential games of infinitely many evaders from infinitely many pursuers in Hilbert space. Dyn Games Appl 6(2):1–13MathSciNetGoogle Scholar
  3. 3.
    Blagodatskikh AI (2009) Simultaneous multiple capture in a simple pursuit problem. J Appl Math Mech 73(1):36–40MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blagodatskikh AI, Petrov NN (2009) Conflict interaction of groups of controlled objects. Udmurt State University, Izhevsk (in Russian) Google Scholar
  5. 5.
    Blagodatskikh AI (2013) Simultaneous multiple capture in a conflict-controlled process. J Appl Math Mech 77(3):314–320MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blagodatskikh AI (2016) Multiple capture of rigidly coordinated evaders. Bull Udmurt Univ Math Mech Comput Sci 26(1):46–57 (in Russian) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Blaquiere A, Gerard F, Leitmann G (1969) Quantitative and qualitative differential games. Academic Press, New YorkzbMATHGoogle Scholar
  8. 8.
    Bopardikar SD, Suri S (2014) \(k\)-Capture in multiagent pursuit evasion, or the lion and hyenas. Theor Comput Sci 522:13–23MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chikrii AA (1992) Conflict controlled processes. Naukova Dumka, Kiev (in Russian) Google Scholar
  10. 10.
    Demidovich BP (1967) Lectures on the mathematical stability theory. Nauka, Moscow (in Russian) zbMATHGoogle Scholar
  11. 11.
    Friedman A (1971) Differential games. Wiley, New YorkzbMATHGoogle Scholar
  12. 12.
    Ganebny SA, Kumkov SS, Le Ménec S, Patsko VS (2012) Model problem in a line with two pursuers and one evader. Dyn Games Appl 2:228–257Google Scholar
  13. 13.
    Grigorenko NL (1990) Mathematical methods of control over multiple dynamic processes. Moscow State University, Moscow (in Russian) Google Scholar
  14. 14.
    Hagedorn P, Breakwell JV (1976) A differential game with two pursuers and one evader. J Optim Theory Appl 18(2):15–29MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hajek O (1975) Pursuit games. Academic Press, New YorkzbMATHGoogle Scholar
  16. 16.
    Huang H, Zhang W, Ding J, Stipanovic DM, Tomlin CJ (2011) Guaranteed decentralized pursuit-evasion in the plane with multiple pursuers. In: Proceedings of the IEEE conference on decision and control. pp 4835–4840Google Scholar
  17. 17.
    Isaacs R (1965) Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization. Wiley, New YorkzbMATHGoogle Scholar
  18. 18.
    Krasovskii NN, Subbotin AI (1974) Positional differential games. Nauka, Moscow (in Russian) zbMATHGoogle Scholar
  19. 19.
    Kuchkarov AS, Ibragimov GI, Khakestari M (2013) On a linear differential game of optimal approach of many pursuers with one evader. J Dyn Control Syst 19(1):1–15MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Leitmann G (1974) Cooperative and non-cooperative many player differential games. Springer, ViennaCrossRefzbMATHGoogle Scholar
  21. 21.
    Petrosyan LA (1977) Differential games of pursuit. Leningrad State University, Leningrad (in Russian) zbMATHGoogle Scholar
  22. 22.
    Petrov NN (1997) Multiple capture in Pontryagin’s example with phase constraints. J Appl Math Mech 61(5):725–732MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pontryagin LS (1971) A linear differential evasion game. Proc Steklov Inst Math 112:27–60 (in Russian) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Pshenichnyi BN (1976) Simple pursuit by several objects. Kibernetika 3:145–146MathSciNetGoogle Scholar
  25. 25.
    Satimov N, Mamatov MS (1983) On problems of pursuit and evasion away from meeting in differential games between the group of pursuers and evaders. Doklady Akademii Nauk Uzbekskoj SSR 4:3–6 (in Russian) zbMATHGoogle Scholar
  26. 26.
    Stipanovic DM, Melikyan A, Hovakimyan N (2010) Guaranteed strategies for nonlinear multi-player pursuit-evasion games. Int Game Theory Rev 12(1):1–17MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussia

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