Ukrainian Mathematical Journal

, Volume 52, Issue 11, pp 1787–1806

Dynamic Game Problems of Approach for Fractional-Order Equations

  • S. D. Éidel'man
  • A. A. Chikrii
Article

Abstract

We propose a general method for the solution of game problems of approach for dynamic systems with Volterra evolution. This method is based on the method of decision functions and uses the apparatus of the theory of set-valued mappings. Game problems for systems with Riemann–Liouville fractional derivatives and regularized Dzhrbashyan–Nersesyan derivatives (fractal games) are studied in more detail on the basis of matrix Mittag-Leffler functions introduced in this paper.

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REFERENCES

  1. 1.
    L. S. Pontryagin, Selected Works [in Russian], Vol. 2, Nauka, Moscow (1988).Google Scholar
  2. 2.
    N. N. Krasovskii, Game Problems on Control of Movements [in Russian], Nauka, Moscow (1970).Google Scholar
  3. 3.
    R. Isaacs, Differential Games.A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization [Russian translation], Mir, Moscow (1967).Google Scholar
  4. 4.
    O. Hajek, Pursuit Games, Academic Press, New York (1975).Google Scholar
  5. 5.
    A. A. Chikrii, Conflict-Controlled Processes, Kluwer, Dordrecht (1997).Google Scholar
  6. 6.
    A. Friedman, Differential Games, Wiley, New York (1971).Google Scholar
  7. 7.
    N. N. Krasovskii and A. I. Subbotin, Positional Differential Games [in Russian], Nauka, Moscow (1974).Google Scholar
  8. 8.
    B. N. Pshenichnyi and V. V. Ostapenko, Differential Games [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  9. 9.
    A. A. Chikrii, “Quasilinear controlled processes under conflict. Dynamical systems. 2,” J.Math.Sci., 80, No.1, 1489–1518 (1996).Google Scholar
  10. 10.
    A. A. Chikrii, “Differential games with several pursuers,” Proc.Internat.Banach Math.Center, 14, 81–107 (1985).Google Scholar
  11. 11.
    N. L. Grigorenko, Mathematical Methods of Control over Several Dynamic Processes [in Russian], Moscow University, Moscow (1990).Google Scholar
  12. 12.
    A. A. Chikrii, “Set-valued mappings and their selectors in game problems of control,” Probl.Upravl.Inform., No. 1, 2, 3–14 (1994).Google Scholar
  13. 13.
    A. A. Chikrii, “Minkowski functionals in the theory of pursuit,” Dokl.Ros.Akad.Nauk, 329, No.3, 243–248 (1993).Google Scholar
  14. 14.
    B. N. Pshenichnyi, A. A. Chikrii, and I. S. Rappoport, “Group pursuit in differential games,” Techische Hochschul Leipzig, No. 2, 13–27 (1982).Google Scholar
  15. 15.
    B. N. Pshenichnyi, A. A. Chikrii, and I. S. Rappoport, “An efficient method for the solution of differential games with many pursuers,” Dokl.Akad.Nauk SSSR, 23, No.1, 104–109 (1981).Google Scholar
  16. 16.
    A. A. Chikrii and I. S. Rappoport, “Guaranteed result in differential games with terminal payoff,” Ann.Int.Sci.Dyn.Games, New Trends Dyn.Games Appl., 3, 323–330 (1995).Google Scholar
  17. 17.
    S. D. Éidel'man, A. A. Chikrii, and A. G. Rurenko, “Game problems for fractional systems,” Dokl.Akad.Nauk Ukr., No. 1, 92–96 (1999).Google Scholar
  18. 18.
    S. D. Éidel'man, A. A. Chikrii, and A. G. Rurenko, “Game problems for dynamic systems having quasilregular Volterra evolution,” in: Proc.Int.Congr.Math., Berlin (1998), p. 338.Google Scholar
  19. 19.
    S. D. Éidel'man, A. A. Chikrii, and A. G. Rurenko, “Quasilinear integral games of approach,” in: IEEE Int.Symp.Intelligence Control, Gaithersburg (1998), pp. 152–158.Google Scholar
  20. 20.
    S. D. Éidel'man, A. A. Chikrii, and A. G. Rurenko, “Quasilinear integral games with summable kernels possessing polar peculiarity,” in: Proceedings of the 8th International Symposium on Dynamic Games and Applications, Maastricht (1998), pp. 158–163.Google Scholar
  21. 21.
    S. D. Éidel'man, A. A. Chikrii, and A. G. Rurenko, “Quasilinear integral games of approach,” Dokl.Akad.Nauk Ukr., No. 7, 92–98 (1998).Google Scholar
  22. 22.
    S. D. Éidel'man, A. A. Chikrii, and A. G. Rurenko, “Linear integro-differential games of approach,” Probl.Upravl.Inform., No. 2, 5–19 (1998).Google Scholar
  23. 23.
    A. A. Chikrii and G. Ts. Chikrii, “Group pursuit in differential-difference games,” Differents.Uravn., 20, No.5, 802–810 (1984).Google Scholar
  24. 24.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Their Applications [in Russian], Tekhnika, Minsk (1987).Google Scholar
  25. 25.
    M. M. Dzhrbashyan and A. B. Nersesyan, “Fractional derivatives and the Cauchy problem for differential equations of fractional order,” Izv.Akad.Nauk Arm.SSR, 3, No.1, 3–29 (1968).Google Scholar
  26. 26.
    A. N. Kochubei, “Cauchy problem for evolution equations of fractional order,” Differents.Uravn., 25, No.8, 1359–1367 (1989).Google Scholar
  27. 27.
    J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York (1984).Google Scholar
  28. 28.
    K. Kuratowski, Topology, Vol. 1, Academic Press, New York (1966).Google Scholar
  29. 29.
    A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).Google Scholar
  30. 30.
    F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, (1983).Google Scholar
  31. 31.
    J.-P. Aubin and H. Frankovska, Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar
  32. 32.
    V. I. Lyashko and P. V. Prokopovich, “Normal integrands in differential pursuit games,” Dokl.Nats.Akad.Nauk Ukr., No. 5, 98–101 (1997).Google Scholar
  33. 33.
    R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton (1970).Google Scholar
  34. 34.
    M. M. Dzhrbashyan, Integral Transformations and Representations of Functions in a Complex Domain [in Russian], Nauka, Moscow (1966).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • S. D. Éidel'man
    • 1
  • A. A. Chikrii
    • 2
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev
  2. 2.Institute of CyberneticsUkrainian Academy of SciencesKiev

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