Existence of a Value and a Saddle Point in Positional Differential Games for Neutral-Type Systems

  • M. I. Gomoyunov
  • N. Yu. Lukoyanov
  • A. R. Plaksin


For a conflict-controlled dynamical system described by functional differential equations of neutral type in Hale’s form, we consider a differential game with a performance index that estimates the motion history realized up to the terminal time and includes an integral estimation of realizations of the players’ controls. The game is formalized in the class of pure positional strategies. The main result is the proof of the existence of a value and a saddle point in this game.


neutral type systems control theory differential games. 


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  1. 1.
    R. Isaacs, Differential Games (Wiley, New York, 1965; Mir, Moscow, 1967).MATHGoogle Scholar
  2. 2.
    N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].MATHGoogle Scholar
  3. 3.
    Yu. S. Osipov, “On the theory of differential games of systems with aftereffect,” J. Appl. Math. Mech. 35 (2), 262–272 (1971).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Yu. S. Osipov, “Position control in parabolic systems,” J. Appl. Math. Mech. 41 (2), 187–193 (1977).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    A. I. Subbotin and A. G. Chentsov, Guarantee Optimization in Control Problems (Nauka, Moscow, 1981) [in Russian].MATHGoogle Scholar
  6. 6.
    N. N. Krasovskii, Control of a Dynamical System (Nauka, Moscow, 1985) [in Russian].Google Scholar
  7. 7.
    J. K. Hale and M. A. Cruz, “Existence, uniqueness and continuous dependence for hereditary systems,” Ann. Mat. Pura Appl. 85 (1), 63–81 (1970).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    A. V. Kryazhimskii, “On the theory of positional differential games of approach–evasion,” Dokl. Akad. Nauk SSSR 239 (4), 779–782 (1978).MathSciNetGoogle Scholar
  9. 9.
    A. V. Kryazhimskii, “On stable position control in differential games,” J. Appl. Math. Mech. 42 (6), 1055–1060 (1980).MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. I. Maksimov, “A differential game of guidance for systems with a deviating argument of neutral type,” in Problems of Dynamic Control (IMM UNTs AN SSSR, Sverdlovsk, 1981), pp. 33–45 [in Russian].Google Scholar
  11. 11.
    A. F. Filippov, Differential Equations with Discontinuous Righthand Sides (Nauka, Moscow, 1985; Springer, Berlin, 1988).Google Scholar
  12. 12.
    R. Bellman and K. L. Cooke, Differential–Difference Equations (Academic, New York, 1963; Mir, Moscow, 1967).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • M. I. Gomoyunov
    • 1
    • 2
  • N. Yu. Lukoyanov
    • 1
    • 2
  • A. R. Plaksin
    • 1
    • 2
  1. 1.Krasovskii Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Ural Federal UniversityYekaterinburgRussia

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