Approaching Coalitions of Evaders on the Average

  • Igor Shevchenko
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 9)


In the game Φ N with simple motions, the pursuer P and the coalition E N = E 1, E 2, …, E N of evaders move in a plane with constant speeds 1, β12, …,β N . The average distance from a point to a set of points is defined as a weighted sum of the corresponding Euclidean distances with given positive constant weights. P strives to minimize the distance to E N and terminates the game when the distance shortening is not guaranteed.

First, we describe several conditions that are met by the states on the terminal manifold MФ N of ФN depending on the index of evaders caught there. Then, we study Ф2 in detail. This game is a game of alternative pursuit since there are three different terminal sub-manifolds: P catches E 1 (E 2) on M 1 Ф2 (M 2 Ф2) and all players are apart on M θ Ф2. We set up and study associated games Ф1 22 2) and Фθ 2 with the payoffs equal to the average distance to E 2 at instants when the state reaches M 1 Ф2 (M 2 Ф2) and M θ Ф2 correspondingly. It is shown that Ф2 is strategically equivalent to the associated game with the minimal value.


Average Distance Playing Space Optimal Trajectory Differential Game Optimal Direction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Abramyantz T., Maslov E., Open-Loop coplanar pursuit of a multiple-target, Podstawy Sterowania, 11(2), 149–160, 1981.MathSciNetGoogle Scholar
  2. [2]
    Abramyants T.G., Volkovinskii M.I., Maslov E.P., Petrovskii A.M., Aplanar pursuit game with a limited number of corrections of the paths of the pursuer, Avtomat. Telemekh., 7, 31–39, 1972.MathSciNetGoogle Scholar
  3. [3]
    Berkowitz L.D., Avariational approach to differential games, Annals of Math-ematics Study, 52, 127–174, 1964.Google Scholar
  4. [4]
    Bernhard P., Singular surfaces in differential games: An introduction. In P. Hargedorn, H.W. Knobloch, G.H. Olsder (Eds.) Differential Games and Applications, Springer Lecture Notes in Information and Control Sciences, Berlin: Springer, 3, 1–33, 1977.CrossRefGoogle Scholar
  5. [5]
    Bernhard P., Differential games: Introduction. In M.G Singh (Ed.) System and Control Encyclopedia, Pergamon Press, Oxford, 1009–1010, 1987.Google Scholar
  6. [6]
    Bernhard P., Differential games: Isaacs equation. In M.G. Singh (Ed.) System and Control Encyclopedia, Pergamon Press, Oxford, 1010–1017, 1987.Google Scholar
  7. [7]
    Bernhard P., Closed-loop differential games. In M.G. Singh (Ed.) System and Control Encyclopedia, Pergamon Press, Oxford, 1004–1009, 1987.Google Scholar
  8. [8]
    Gabasov R., Kirillova F.M., Some applications of functional analysis in the theory of optimal systems, Izv. Akad. Nauk SSSR, Tekh. Kibern., 4, 3–13, 1966.MathSciNetGoogle Scholar
  9. [9]
    Isaacs R., Differential Games, Wiley, New York, 1965.MATHGoogle Scholar
  10. [10]
    Kurzhanskii A.B., Differential games of approach in systems with delay, Differents. Uravn., VII(8), 1389–1409, 1971.MathSciNetGoogle Scholar
  11. [11]
    Kurzhanskii A.B., Osipov Yu. S., On some problems of open-loop pursuit in linear systems, Izv. Akad. Nauk SSSR, Tekh. Kibern., 3, 18–29, 1970.Google Scholar
  12. [12]
    Maslov E.P., Rubinovich E.Ya., Differential games with a multiple target, ItogiNauki Tekhn., Tekh. Kibern., 32, 32–59, 1991.Google Scholar
  13. [13]
    Ol’shanskii V.K., Rubinovich E. Ya., Simple differential games of pursuit of a system of two evaders, Avtomat. Telemekh., 1, 24–34, 1974.MathSciNetGoogle Scholar
  14. [14]
    Shevchenko I.I., On open-loop pursuit-evasion, Methods for Investigation of Nonlinear Control Systems, Nauka, Moscow, 80–83, 1983.Google Scholar
  15. [15]
    Shevchenko I.I., Approaching a coalition of evaders, Avtomat. Telemekh., 1, 47–55, 1986.MathSciNetGoogle Scholar
  16. [16]
    Shevchenko I.I., Geometry of Alternative Pursuit, Far Eastern State Univer-sity Publishing, Vladivostok, 2003.Google Scholar
  17. [17]
    Shevchenko I.I., Minimizing the distance to one evader while chasing another, Computers and Mathematics with Applications, 47(12), 1827–1855, 2004.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • Igor Shevchenko
    • 1
    • 2
  1. 1.Pacific Fisheries Research CenterVladivostokRussia
  2. 2.Informatics DepartmentFar Eastern State UniversityVladivostokRussia

Personalised recommendations