On a Single-Type Differential Game with a Non-convex Terminal Set

  • Igor’ V. Izmest’evEmail author
  • Viktor I. Ukhobotov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)


We consider the problem of controlling a rod attached to a rotor. A rotating flywheel is attached to one end of the rod. The rotor is controlled by the first player. The flywheel is controlled by the second player. The goal of the first player is to bring the rotor to a vertical position at a given time. The goal of the second player is the opposite. This problem is an example of a more general linear differential game with a one-dimensional aim. Using a linear change of variables, this problem is reduced to a single-type one-dimensional differential game with a non-convex terminal set, for which we have found the necessary and sufficient conditions of termination and constructed the corresponding controls of the players.


Control Differential game Terminal set Flywheel 


  1. 1.
    Andrievsky, B.R.: Global stabilization of the unstable reaction-wheel pendulum. Control Big Syst. 24, 258–280 (2009). (in Russian)Google Scholar
  2. 2.
    Beznos, A.V., Grishin, A.A., Lenskiy, A.V., Okhozimskiy, D.E., Formalskiy, A.M.: The control of pendulum using flywheel. In: Workshop on Theoretical and Applied Mechanics, pp. 170–195. Publishing of Moscow State University, Moscow (2009). (in Russian)Google Scholar
  3. 3.
    Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Wiley, New York (1965)zbMATHGoogle Scholar
  4. 4.
    Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Nauka Publ., Moscow (1972). (in Russian)zbMATHGoogle Scholar
  5. 5.
    Krasovskii, N.N., Subbotin, A.I.: Positional Differential Games. Nauka Publ., Moscow (1974). (in Russian)zbMATHGoogle Scholar
  6. 6.
    Kudryavtsev, L.D.: A Course of Mathematical Analysis, vol. 1. Vysshaya shkola Publ., Moscow (1981). (in Russian)zbMATHGoogle Scholar
  7. 7.
    Matviychuk, A.R., Ukhobotov, V.I., Ushakov, A.V., Ushakov, V.N.: The approach problem of a nonlinear controlled system in a finite time interval. J. Appl. Math. Mech. 81(2), 114–128 (2017). Scholar
  8. 8.
    Pshenichnyi, B.N.: Convex Analysis and Extremal Problems. Nauka Publ., Moscow (1980). (in Russian)zbMATHGoogle Scholar
  9. 9.
    Pontryagin, L.S.: Linear differential games of pursuit. Math. USSR-Sbornik 40(3), 285–303 (1981). Scholar
  10. 10.
    Ukhobotov, V.I.: Synthesis of control in single-type differential games with fixed time. Bull. Chelyabinsk Univ. 1, 178–184 (1996). (in Russian)MathSciNetGoogle Scholar
  11. 11.
    Ukhobotov, V.I.: Method of One-Dimensional Design in Linear Differential Games with Integral Constraints: Study Guide. Publishing of Chelyabinsk State University, Chelyabinsk (2005). (in Russian)Google Scholar
  12. 12.
    Ukhobotov, V.I.: The same type of differential games with convex purpose. Proc. Inst. Math. Mech. Ural. Branch Russ. Acad. Sci. 16(5), 196–204 (2010). (in Russian)Google Scholar
  13. 13.
    Ukhobotov, V.I., Izmest’ev I.V.: Single-type differential games with a terminal set in the form of a ring. In: Systems Dynamics and Control Process, Proceedings of the International Conference, Dedicated to the 90th Anniversary of Acad. N.N. Krasovskiy, Ekaterinburg, 15–20 September 2014, pp. 325–332. Publishing House of the UMC UPI, Ekaterinburg, Russia (2015). (in Russian)Google Scholar
  14. 14.
    Ushakov, V.N., Ukhobotov, V.I., Ushakov, A.V., Parshikov, G.V.: On solution of control problems for nonlinear systems on finite time interval. IFAC-PapersOnLine 49(18), 380–385 (2016). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Chelyabinsk State UniversityChelyabinskRussia

Personalised recommendations