Advertisement

Dynamic Games and Applications

, Volume 7, Issue 4, pp 609–633 | Cite as

Zero-Sum Pursuit-Evasion Differential Games with Many Objects: Survey of Publications

  • Sergey S. KumkovEmail author
  • Stéphane Le Ménec
  • Valerii S. Patsko
Article

Abstract

If a pursuit game with many persons can be formalized in the framework of zero-sum differential games, then general methods can be applied to solve it. But difficulties arise connected with very high dimension of the phase vector when there are too many objects. Just due to this problem, special formulations and methods have been elaborated for conflict interaction of groups of objects. This paper is a survey of publications and results on group pursuit games.

Keywords

Differential games Conflict interacting groups of objects Group pursuit problems Constant-bearing method Maximal stable sets 

Notes

Acknowledgements

This work has been partially supported by Russian Foundation for Basic Research, Project No. 15-01-07909.

References

  1. 1.
    Abramyants TG, Ivanov MN, Maslov EP, Yakhno VP (2004) A detection evasion problem. Autom Rem Control 65(10):1523–1530CrossRefzbMATHGoogle Scholar
  2. 2.
    Abramyants TG, Maslov EP, Rubinovich EY (1980) An elementary differential game of alternative pursuit. Autom Rem Control 40(8):1043–1052zbMATHGoogle Scholar
  3. 3.
    Abramyants TG, Maslov EP, Yakhno VP (2007) Evasion from detection in the three-dimensional space. J Comput Syst Sci Int 46(5):675–680MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bakolas E, Tsiotras P (2010) On the relay pursuit of a maneuvering target by a group of pursuers. In: 50th IEEE conference on decision and control and European control conference (CDC-ECC’2011). IEEE, pp 7431–7436Google Scholar
  5. 5.
    Bakolas E, Tsiotras P (2011) Optimal pursuit of moving targets using dynamic Voronoi diagrams. In: 49th IEEE conference on decision and control (CDC’2010). IEEE, pp 4270–4275Google Scholar
  6. 6.
    Bardi M, Falcone M, Soravia P (1999) Numerical methods for pursuit-evasion games via viscosity solutions. In: Bardi M, Parthasaraty T, Raghavan TES (eds) Annals of the international society of dynamic games, vol 4. Birkhauser, Boston, pp 105–175Google Scholar
  7. 7.
    Blagodatskih AI, Petrov NN (2009) Conflict interaction controlled objects groups. Udmurt State University, Izhevsk (in Russian)Google Scholar
  8. 8.
    Blagodatskikh AI (2008) Group pursuit in Pontryagin’s nonstationary example. Differ Equ 44(1):40–46MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Botkin ND, Hoffmann KH, Turova VL (2011) Stable numerical schemes for solving Hamilton–Jacobi–Bellman–Isaacs equations. SIAM J Sci Comput 33(2):992–1007MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Botkin ND, Ryazantseva EA (1992) Algorithm of constructing the solvability set for linear differential games of high dimension. Trudy Inst Mat i Mekh 2:128–134 (in Russian)zbMATHGoogle Scholar
  11. 11.
    Breakwell JV, Hagedorn P (1979) Point capture of two evaders in succession. JOTA 27(1):89–97MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bryson AE, Ho YC (1975) Applied optimal control. Optimization, estimation and control. Hemisphere Publishing Corporation, Washington, DCGoogle Scholar
  13. 13.
    Chen M, Fisac J, Sastry S, Tomlin C (2015) Safe sequential path planning of multi-vehicle systems via double-obstacle Hamilton–Jacobi–Isaacs variational inequality. In: Proceedings of the 14th European control conference, pp 3304–3309Google Scholar
  14. 14.
    Chernous’ko FL (1976) A problem of evasion from many pursuers. J Appl Math Mech 40(1):11–20MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chikrii AA (1978) On the evasion problem in a linear differential game. Autom Rem Control 38(9):1291–1295Google Scholar
  16. 16.
    Chikrii AA (1992) Conflict controlled processes. Naukova dumka, Kiev (in Russian)zbMATHGoogle Scholar
  17. 17.
    Chikrii AA (1997) Conflict-controlled processes, mathematics and its applications, vol 405. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  18. 18.
    Dvurechensky PE, Ivanov GE (2014) Algorithms for computing Minkowski operators and their application in differential games. Comput Math Math Phys 54(2):235–264MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fisac JF, Sastry S (2015) The pursuit-evasion-defense differential game in dynamic constrained environments. In: Proceedings of the IEEE 54th annual conference on decision and control (CDC), pp 4549–4556Google Scholar
  20. 20.
    Ganebny SA, Kumkov SS, Le Ménec S, Patsko VS (2012) Model problem in a line with two pursuers and one evader. Dyn Games Appl 2:228–257MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Grigorenko NL (1989) On a quasilinear problem of pursuit by several objects. Sov Math Dokl 20:1365–1368zbMATHGoogle Scholar
  22. 22.
    Grigorenko NL (1989) The pursuit problem in \(n\)-person differential games. Math USSR-Sb 63(1):35–45CrossRefzbMATHGoogle Scholar
  23. 23.
    Grigorenko NL (1990) Mathematical methods of control of multiple dynamic processes. Moscow State University, Moscow (in Russian)Google Scholar
  24. 24.
    Grigorenko NL, Kiselev YN, Lagunova NV, Silin DB, Trin’ko NG (1996) Solution methods for differential games. Comput Math Model 7(1):101–116MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hagedorn P, Breakwell JV (1976) A differential game with two pursuers and one evader. JOTA 18(1):15–29MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Isaacs R (1965) Differential games. Wiley, New YorkzbMATHGoogle Scholar
  27. 27.
    Krasovskii NN, Subbotin AI (1974) Positional differential games. Nauka, Moscow (in Russian)zbMATHGoogle Scholar
  28. 28.
    Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New YorkCrossRefGoogle Scholar
  29. 29.
    Kumkov SS, Le Ménec S, Patsko VS (2013) Model formulation of pursuit problem with two pursuers and one evader. In: Advances in aerospace guidance, navigation and control. Springer, Berlin, Heidelberg, pp 121–137Google Scholar
  30. 30.
    Kumkov SS, Le Ménec S, Patsko VS (2014) Level sets of the value function in differential games with two pursuers and one evader. Interval analysis interpretation. Math Comput Sci 8:443–454MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kumkov SS, Patsko VS (2001) Construction of singular surfaces in linear differential games. In: Altman E, Pourtallier O (eds) Annals of the international society of dynamic games: advances in dynamic games and applications, vol 6. Birkhauser, Boston, pp 185–202CrossRefGoogle Scholar
  32. 32.
    Kumkov SS, Patsko VS, Shinar J (2005) On level sets with “narrow” throats in linear differential games. IGTR 7(3):285–312MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kurzhanski AB (2015) On a team control problem under obstacles. Proc Steklov Inst Math 291:128–142CrossRefGoogle Scholar
  34. 34.
    Kurzhanski AB (2016) Problem of collision avoidance for a team motion with obstacles. Proc Steklov Inst Math 293:120–136CrossRefzbMATHGoogle Scholar
  35. 35.
    Levchenkov AY, Pashkov AG (1990) Differential game of optimal approach of two inertial pursuers to a noninertial evader. JOTA 65(3):501–518MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Le Ménec S (2011) Linear differential game with two pursuers and one evader. In: Breton M, Szajowski K (eds) Annals of the international society of dynamic games, vol 11. Birkhauser, Boston, pp 209–226CrossRefGoogle Scholar
  37. 37.
    Mikhalev DK, Ushakov VN (2007) Two algorithms for approximate construction of the set of positional absorption in the game problem of pursuit. Autom Rem Control 68(11):2056–2070CrossRefzbMATHGoogle Scholar
  38. 38.
    Mishchenko EF, Nikol’skii MS, Satimov NY (1980) The problem of avoiding encounter in \(n\)-person differential games. Proc Steklov Inst Math 143:111–136zbMATHGoogle Scholar
  39. 39.
    Nikol’skii MS (1983) On the alternating integral of Pontryagin. Sb Math 44(1):125–132CrossRefGoogle Scholar
  40. 40.
    Pashkov AG, Terekhov SD (1987) A differential game of approach with two pursuers and one evader. JOTA 55(2):303–311MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Patsko VS, Turova VL (2001) Level sets of the value function in differential games with the homicidal chauffeur dynamics. IGTR 3(1):67–112MathSciNetzbMATHGoogle Scholar
  42. 42.
    Petrosjan LA (1965) A family of differential survival games in the space \(R^{n}\). Sov Math Dokl 6:377–380Google Scholar
  43. 43.
    Petrosyan LA (1966) The multi-person lifeline games of pursuit. Proc Armen Acad Sci Ser Math 1:331–340 (in Russian)Google Scholar
  44. 44.
    Petrosyan LA (1977) Differential games of pursuit. Leningrad State University, Leningrad (in Russian)zbMATHGoogle Scholar
  45. 45.
    Petrosyan LA (1993) Differential games of pursuit. World Scientific Publisher, LondonCrossRefzbMATHGoogle Scholar
  46. 46.
    Petrosyan LA, Dutkevich YG (1972) Games with a “lifeline”. The case of \(l\)-capture. SIAM J Control 10(1):40–47MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Petrov NN (1988) A group pursuit problem with phase constraints. J Appl Math Mech 52(6):803–806MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Petrov NN, Shuravina IN (2009) On the “soft” capture in one group pursuit problem. J Comput Syst Sci Int 48(4):521–526MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Pittsyk MV, Chikrii AA (1982) On a group pursuit problem. J Appl Math Mech 46(5):584–589MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Polovinkin ES, Ivanov GE, Balashov MV, Konstantiov RV, Khorev AV (2001) An algorithm for the numerical solution of linear differential games. Sb Math 192(10):1515–1542MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Pontryagin LS (1967) Linear differential games, 1. Sov Math Dokl 8:769–771zbMATHGoogle Scholar
  52. 52.
    Pontryagin LS (1967) Linear differential games, 2. Sov Math Dokl 8:910–912zbMATHGoogle Scholar
  53. 53.
    Pontryagin LS (1971) A linear differential evasion game. Proc Steklov Inst Math 112:27–60MathSciNetzbMATHGoogle Scholar
  54. 54.
    Pontryagin LS (1981) An algorithm for the numerical solution of linear differential games. Math USSR-Sb 40(3):285–303CrossRefzbMATHGoogle Scholar
  55. 55.
    Pontryagin LS, Mishchenko EF (1969) A problem on the escape of one controlled object from another. Sov Math Dokl 10:1488–1490zbMATHGoogle Scholar
  56. 56.
    Pschenichnyi BN (1976) Simple pursuit by several objects. Cybern Syst Anal 12(3):484–485MathSciNetGoogle Scholar
  57. 57.
    Pschenichnyi BN, Chikrii AA, Rappoport IS (1981) An efficient method of solving differential games with many pursuers. Sov Math Dokl 23(1):104–109Google Scholar
  58. 58.
    Pschenichnyi BN, Sagaidak MI (1970) Differential games of prescribed duration. Cybernetics 6(2):72–80CrossRefGoogle Scholar
  59. 59.
    Rappoport IS, Chikrii AA (1997) Guaranteed result in a differential game of group pursuit with terminal payoff function. J Appl Math Mech 61(4):567–576MathSciNetCrossRefGoogle Scholar
  60. 60.
    Shevchenko II (1997) Successive pursuit with a bounded detection domain. JOTA 95(1):25–48MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Shima T (2005) Capture conditions in a pursuit-evasion game between players with biproper dynamics. JOTA 126(3):503–528MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Shima T, Shinar J (2002) Time-varying linear pursuit-evasion game models with bounded controls. J Guid Control Dyn 25(3):425–432CrossRefGoogle Scholar
  63. 63.
    Shinar J, Glizer VY, Turetsky V (2013) The effect of pursuer dynamics on the value of linear pursuit-evasion games with bounded controls. In: Krivan V, Zaccour G (eds) Annals of the international society of dynamic games, vol 13. Birkhauser, Boston, pp 313–350CrossRefGoogle Scholar
  64. 64.
    Shinar J, Medinah M, Biton M (1984) Singular surfaces in a linear pursuit-evasion game with elliptical vectograms. JOTA 43(3):431–456MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Stipanović DM, Tomlin C, Leitmann G (2012) Monotone approximations of minimum and maximum functions and multi-objective problems. Appl Math Opt 66(3):455–473MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Subbotin AI (1995) Generalized solutions of first order PDEs. The dynamical optimization perspective. Birkhäuser, BostonCrossRefzbMATHGoogle Scholar
  67. 67.
    Subbotin AI, Patsko VS (eds) (1984) Algorithms and programs for solving linear differential games. Institute of Mathematics and Mechanics, Ural Scientific Center, Academy of Sciences of USSR, Sverdlovsk (in Russian)Google Scholar
  68. 68.
    Taras’ev AM, Ushakov VN, Khripunov AP (1988) On a computational algorithm for solving game control problems. J Appl Math Mech 51(2):167–172MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Ushakov VN (1981) On the problem of constructing stable bridges in a differential pursuit-evasion game. Eng Cybern 18(4):16–23Google Scholar
  70. 70.
    Zarkh MA, Ivanov AG (1992) Construction of the value function in the linear differential game with the fixed terminal time. Trudy Inst Mat i Mekh 2:140–155 (in Russian)zbMATHGoogle Scholar
  71. 71.
    Zarkh MA, Patsko VS (1988) Numerical solution of a third-order directed game. Eng Cybern 26(4):92–99MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sergey S. Kumkov
    • 1
    Email author
  • Stéphane Le Ménec
    • 2
  • Valerii S. Patsko
    • 1
  1. 1.Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, and Institute of Mathematics and Computer SciencesUral Federal UniversityEkaterinburgRussia
  2. 2.Airbus Group/MBDA FranceLe Plessis Robinson CedexFrance

Personalised recommendations