Dynamic Games and Applications

, Volume 8, Issue 2, pp 352–378 | Cite as

Simple Motion Evasion Differential Game of Many Pursuers and Evaders with Integral Constraints

  • Gafurjan Ibragimov
  • Massimiliano Ferrara
  • Atamurat Kuchkarov
  • Bruno Antonio Pansera


We study a simple motion evasion differential game of many pursuers and evaders. Control functions of players are subjected to integral constraints. If the state of at least one evader does not coincide with that of any pursuer forever, then evasion is said to be possible in the game. The aim of the group of evaders is to construct their strategies so that evasion can be possible in the game and the aim of the group of pursuers is opposite. The problem is to find a sufficient condition of evasion. If the total energy of pursuers is less than or equal to that of evaders, then it is proved that evasion is possible, and moreover, evasion strategies are constructed explicitly.


Differential game Many pursuers Many evaders Integral constraint Evasion Strategy 

Mathematics Subject Classification

Primary 91A23 Secondary 49N75 



The present research was partially supported by the National Fundamental Research Grant Scheme FRGS of Malaysia, 01-01-13-1228FR.


  1. 1.
    Alexander S, Bishop R, Christ R (2009) Capture pursuit games on unbounded domain. L’Enseignement Mathématique 55(1/2):103–125MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alias IA, Ibragimov G, Rakhmanov A (2016) Evasion differential game of infinitely many evaders from infinitely many pursuers in Hilbert space. Dyn Games Appl 6(2):1–13. doi: 10.1007/s13235-016-0196-0 MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bannikov AS, Petrov NN (2010) To non-stationary group pursuit problem. Trudy Inst Math Mech UrO RAN 16(1):40–51zbMATHGoogle Scholar
  4. 4.
    Belousov AA (2010) Method of resolving functions for differential games with integral constraints. Theory Optim Solut 9:10–16Google Scholar
  5. 5.
    Blagodatskikh AI, Petrov NN (2009) Conflict interaction between groups of controlled objects. Udmurt State University Press, Izhevsk (in Russian)Google Scholar
  6. 6.
    Borowko P, Rzymowski W, Stachura A (1988) Evasion from many pursuers in the simple motion case. J Math Anal Appl 135(1):75–80MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chernous’ko FL (1976) A problem of evasion of several pursuers. Prikl Mat Mekh 40(1):14–24zbMATHGoogle Scholar
  8. 8.
    Chikrii AA, Prokopovich PV (1992) Simple pursuit of one evader by a group. Cybern Syst Anal 28(3):438–444. doi: 10.1007/BF01125424 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chodun W (1989) Differential games of evasion with many pursuers. J Math Anal Appl 142(2):370–389MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Conti R (1974) Problemi di Controllo e di Controllo Ottimale. UTET, Torino (in Italian) zbMATHGoogle Scholar
  11. 11.
    Croft HT (1964) Lion and man: a postscript. J Lond Math Soc 39:385–390MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Friedman A (1971) Differential games. Wiley, New YorkzbMATHGoogle Scholar
  13. 13.
    Grigorenko NL (1990) Mathematical methods of control of several dynamic processes. MSU Press, Moscow (in Russian) Google Scholar
  14. 14.
    Hajek O (1975) Pursuit games. Academic Press, New YorkzbMATHGoogle Scholar
  15. 15.
    Huseyin A, Huseyin N, Guseinov KG (2015) Approximation of the sections of the set of trajectories of the control system described by a nonlinear Volterra integral equation. Math Model Anal 20(4):502–515MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ibragimov G, Satimov N (2012). A multiplayer pursuit differential game on a closed convex set with integral constraints. In: Abstract and applied analysis. Article ID 460171: 12 p. doi: 10.1155/2012/460171
  17. 17.
    Ibragimov GI, Salimi M, Amini M (2012) Evasion from many pursuers in simple motion differential game with integral constraints. Eur J Oper Res 218(2):505–511MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Isaacs R (1965) Differential games. Wiley, New YorkzbMATHGoogle Scholar
  19. 19.
    Ivanov RP (1980) Simple pursuit-evasion on a compact convex set. Doklady Akademii Nauk SSSR 254(6):1318–1321MathSciNetGoogle Scholar
  20. 20.
    Krasovskii NN (1968) Theory of control of motion: linear systems. Nauka, MoscowGoogle Scholar
  21. 21.
    Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New YorkCrossRefGoogle Scholar
  22. 22.
    Kuchkarov AS, Risman MH, Malik AH (2012) Differential games with many pursuers when evader moves on the surface of a cylinder. ANZIAM J 53(E):E1–E20MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kuang FH (1986) Sufficient conditions of capture in differential games of \(m\) pursuer and one evader. Kibernetika 6:91–97Google Scholar
  24. 24.
    Kuchkarov A, Ibragimov G, Ferrara M (2016) Simple motion pursuit and evasion differential games with many pursuers on manifolds with Euclidean metric. Discrete Dyn Nat Soc. doi: 10.1155/2016/1386242 MathSciNetGoogle Scholar
  25. 25.
    Mishchenko EF, Nikol’skii MS, Satimov NY (1977) Evoidance encounter problem in differential games of many persons. Trudy MIAN USSR 143:105–128Google Scholar
  26. 26.
    Petrosyan LA (1993) Differential games of pursuit. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  27. 27.
    Pontryagin LS (1988) Selected works. Nauka, MoscowzbMATHGoogle Scholar
  28. 28.
    Pshenichnii BN (1976) Simple pursuit by several objects. Cybern Syst Anal 12(3):145–146MathSciNetGoogle Scholar
  29. 29.
    Pshenichnii BN, Chikrii AA, Rappoport JS (1981) An efficient method of solving differential games with many pursuers. Dokl Akad Nauk SSSR 256:530–535 (in Russian) MathSciNetGoogle Scholar
  30. 30.
    Satimov NY, Rikhsiev BB, Khamdamov AA (1983) On a pursuit problem for \(n\) person linear differential and discrete games with integral constraints. Math USSR Sbornik 46(4):456–469CrossRefzbMATHGoogle Scholar
  31. 31.
    Satimov NY, Fazilov AZ, Hamdamov AA (1984) On pursuit and evasion problems for multi-person linear differential and discrete games with integral constraints. Dif Uravneniya 20(8):1388–1396Google Scholar
  32. 32.
    Vagin D, Petrov N (2001) The problem of the pursuit of a group of rigidly coordinated evaders. J Comput Syst Sci Int 40(5):749–753MathSciNetzbMATHGoogle Scholar
  33. 33.
    Zak VL (1978) On a problem of evading many pursuers. J Appl Maths Mekhs 43(3):492–501MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Institute for Mathematical ResearchUniversiti Putra MalaysiaSerdangMalaysia
  2. 2.Department of Law and EconomicsUniversity Mediterranea of Reggio CalabriaReggio CalabriaItaly
  3. 3.ICRIOSBocconi UniversityMilanItaly
  4. 4.Institute of MathematicsNational University of UzbekistanTashkentUzbekistan

Personalised recommendations