Game with Two Pursuers and One Evader: Case of Weak Pursuers

  • Sergey KumkovEmail author
  • Valerii Patsko
  • Stéphane Le Ménec
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 13)


This paper deals with a zero-sum differential game, in which the first player controls two pursuing objects, whose aim is to minimize the minimum of the misses between each of them and the evader at some given instant. The case is studied when the pursuers have equal dynamic capabilities, but are less powerful than the evader. The first player’s control based on its switching lines is analyzed. Results of numeric application of this control are given.


Pursuit differential games Fixed termination instant Positional control Switching lines 



We are grateful to the unknown reviewer for very helpful remarks.

This work was supported by Program of Presidium RAS “Dynamic Systems and Control Theory” under financial support of UrB RAS (project No.12-Π-1-1002) and also by the Russian Foundation for Basic Research under grant no.12-01-00537.


  1. Abramyantz TG, Maslov EP (2004) A differential game of pursuit of a group target. Izv Akad Nauk Teor Sist Upr 5:16–22 (in Russian)Google Scholar
  2. Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, BostonCrossRefzbMATHGoogle Scholar
  3. Blagodatskih AI, Petrov NN (2009) Conflict interaction controlled objects groups. Udmurt State University, Izhevsk (in Russian)Google Scholar
  4. Botkin ND, Patsko VS (1983) Positional control in linear differential game. Eng Cybern 21(4):69–76MathSciNetzbMATHGoogle Scholar
  5. Botkin ND, Kein VM, Patsko VS (1984) The model problem of controlling the lateral motion of an aircraft during landing. J App Math Mech 48(4):395–400CrossRefGoogle Scholar
  6. Chikrii AA (1997) Conflict-controlled processes. Mathematics and its applications, vol 405. Kluwer Academic, DordrechtGoogle Scholar
  7. Ganebny SA, Kumkov SS, Le Ménec S, Patsko VS (2012a) Model problem in a line with two pursuers and one evader. Dyn Game Appl 2(2):228–257CrossRefzbMATHGoogle Scholar
  8. Ganebny SA, Kumkov SS, Le Ménec S, Patsko VS (2012b) Study of linear game with two pursuers and one evader: Different strength of pursuers. In: Cardaliaguet P, Cressman R (eds) Advances in dynamic games. Annals of the international society of dynamic games, vol 12, chap 14. Birkhäuser, Boston, pp 269–292Google Scholar
  9. Grigorenko NL (1991) The problem of pursuit by several objects. In: Differential games — developments in modelling and computation (Espoo, 1990). Lecture notes in control and information sciences, vol 156. Springer, Berlin, pp 71–80Google Scholar
  10. Hagedorn P, Breakwell JV (1976) A differential game with two pursuers and one evader. J Optimiz Theory App 18(1):15–29MathSciNetCrossRefzbMATHGoogle Scholar
  11. Isaacs R (1965) Differential games. Wiley, New YorkzbMATHGoogle Scholar
  12. Krasovskii NN (1985) Control of dynamic system. Nauka, Moscow (in Russian)Google Scholar
  13. Krasovskii NN, Subbotin AI (1974) Positional differential games. Nauka, Moscow (in Russian)zbMATHGoogle Scholar
  14. Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New YorkCrossRefzbMATHGoogle Scholar
  15. Levchenkov AY, Pashkov AG (1990) Differential game of optimal approach of two inertial pursuers to a noninertial evader. J Optimiz Theory App 65:501–518MathSciNetCrossRefzbMATHGoogle Scholar
  16. Le Ménec S (2011) Linear differential game with two pursuers and one evader. In: Breton M, Szajowski K (eds) Advances in dynamic games. Theory, applications, and numerical methods for differential and stochastic games. Annals of the international society of dynamic games, vol 11. Birkhäuser, Boston, pp 209–226Google Scholar
  17. Patsko VS (2006) Switching surfaces in linear differential games. J Math Sci 139(5):6909–6953MathSciNetCrossRefzbMATHGoogle Scholar
  18. Patsko VS, Botkin ND, Kein VM, Turova VL, Zarkh MA (1994) Control of an aircraft landing in windshear. J Optimiz Theory App 83(2):237–267MathSciNetCrossRefzbMATHGoogle Scholar
  19. Petrosjan LA (1977) Differential games of pursuit. Leningrad University, Leningrad (in Russian)Google Scholar
  20. Petrosjan LA, Tomski GV (1983) Geometry of simple pursuit. Nauka, Sibirsk. Otdel., Novosibirsk (in Russian)Google Scholar
  21. Pschenichnyi BN (1976) Simple pursuit by several objects. Kibernetika 3:145–146 (in Russian)Google Scholar
  22. Rikhsiev BB (1989) Differential games with simple moves. Fan, Tashkent (in Russian)Google Scholar
  23. Shima T, Shinar J (2002) Time-varying linear pursuit-evasion game models with bounded controls. J Guid Control Dynam 25(3):425–432CrossRefGoogle Scholar
  24. Shinar J, Shima T (2002) Non-orthodox guidance law development approach for intercepting maneuvering targets. J Guid Control Dynam 25(4):658–666CrossRefGoogle Scholar
  25. Stipanović DM, Melikyan AA, Hovakimyan N (2009) Some sufficient conditions for multi-player pursuit-evasion games with continuous and discrete observations. In: Bernhard P, Gaitsgory V, Pourtallier O (eds) Advances in dynamic games and applications. Annals of the international society of dynamic games, vol 10. Springer, Berlin, pp 133–145Google Scholar
  26. Stipanović DM, Tomlin CJ, Leitmann G (2012) Monotone approximations of minimum and maximum functions and multi-objective problems. Appl Math Optim 66:455–473MathSciNetCrossRefzbMATHGoogle Scholar
  27. Subbotin AI, Chentsov AG (1981) Guaranteed optimization in control problems. Nauka, Moscow (in Russian)Google Scholar
  28. Tarasyev AM, Uspenskiy AA, Ushakov VN (1995) Approximation schemes and finite-difference operators for constructing generalized solutions of Hamilton-Jacobi equations. J Comput Syst Sci Int 33(6):127–139MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Sergey Kumkov
    • 1
    Email author
  • Valerii Patsko
    • 1
  • Stéphane Le Ménec
    • 2
  1. 1.Institute of Mathematics and MechanicsEkaterinburgRussia
  2. 2.EADS / MBDA FranceLe Plessis-Robinson CedexFrance

Personalised recommendations