Cybernetics and Systems Analysis

, Volume 55, Issue 5, pp 828–839 | Cite as

Sufficient Conditions of Approach of Controlled Objects in Dynamic Game Problems. I

  • I. S. RappoportEmail author


The problem of approach of control objects is solved on the basis of the method of resolving functions. Sufficient conditions for game termination in a guaranteed finite time are proposed for the case where the Pontryagin condition is not satisfied. The upper and lower resolving functions of different types are introduced and are used to develop two schemes of the method of resolving functions that ensure termination of the differential game in the class of quasi-strategies and counter-controls.


quasilinear differential game multivalued mapping measurable selector stroboscopic strategy resolving function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. A. Chikrii and I. S. Rappoport, “Method of resolving functions in the theory of conflict-controlled processes,” Cybern. Syst. Analysis, Vol. 48, No. 4, 512–531 (2012).MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. A. Chikrii, “Upper and lower resolving functions in dynamic game problems,” in: Tr. IMM UrO RAN, Vol. 23, No. 1, 293–305 (2017).Google Scholar
  3. 3.
    N. N. Krasovskii and A. I. Subbotin, Positional Differential Games [in Russian], Nauka, Moscow (1974).zbMATHGoogle Scholar
  4. 4.
    L. S. Pontryagin, Selected Scientific Works [in Russian], Vol. 2, Nauka, Moscow (1988).Google Scholar
  5. 5.
    M. S. Nikol’skii, First Direct Pontryagin’s Method in Differential Games [in Russian], Izd. MGU, Moscow (1984).Google Scholar
  6. 6.
    A. I. Subbotin and A. G. Chentsov, Guarantee Optimization in Control Problems [in Russian], Nauka, Moscow (1981).zbMATHGoogle Scholar
  7. 7.
    O. Hajek, Pursuit Games, Vol. 12, Academic Press, New York (1975).zbMATHGoogle Scholar
  8. 8.
    J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston–Basel–Berlin (1990).zbMATHGoogle Scholar
  9. 9.
    R. T. Rockafellar, Convex Analysis, Princeton Univ. Press (1970).Google Scholar
  10. 10.
    A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems [in Russian], Nauka, Moscow (1974).Google Scholar
  11. 11.
    A. A. Chikrii, “An analytical method in dynamic pursuit games,” Proc. of the Steklov Institute of Math., Vol. 271, 69–85 (2010).MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. A. Chikrii, “Multivalued mappings and their selections in game control problems,” J. Autom. Inform. Sci., Vol. 27, No. 1, 27–38 (1995).MathSciNetGoogle Scholar
  13. 13.
    M. V. Pittsyk and A. A. Chikrii, “On group pursuit problem,” J. of Applied Math. and Mechanics, Vol. 46, No. 5, 584–589 (1982).MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. A. Chikrii and K. G. Dzyubenko, “Bilinear Markov processes of searching for moving targets,” J. of Autom. Inform. Sci., Vol. 33, No. 5–8, 62–74 (2001).Google Scholar
  15. 15.
    A. A. Chikrii and S. D. Eidelman, “Game problems for fractional quasilinear systems,” J. Computers and Math. with Applications, Pergamon, New York, Vol. 44, 835–851 (2002).Google Scholar
  16. 16.
    A. A. Chikrii, “Game dynamic problems for systems with fractional derivatives,” Springer Optimization and ItsApplications, Vol. 17, 349–387 (2008).Google Scholar
  17. 17.
    Yu. B. Pilipenko and A. A. Chikrii, “Oscillatory conflict-control processes,” J. of Applied Math. and Mechanics, Vol. 57, No. 3, 407–417 (1993).MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. A. Chikrii, “Quasilinear controlled processes under conflict,” J. of Mathematical Sci., Vol. 80, No. 3, 1489–1518 (1996).MathSciNetCrossRefGoogle Scholar
  19. 19.
    A. A. Chikrii and S. D. Eidelman, “Control game problems for quasilinear systems with Riemann–Liouville fractional derivatives,” Cybern. Syst. Analysis, Vol. 37, No. 6, 836–864 (2001).MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. A. Chikrii, “Optimization of game interaction of fractional-order controlled systems,” Optimization Methods and Software, Vol. 23, No. 1, 39–72 (2008).MathSciNetCrossRefGoogle Scholar
  21. 21.
    I. S. Rappoport, “Stroboscopic strategy in the method of resolving functions for game control problems with terminal payoff function,” Cybern. Syst. Analysis, Vol. 52, No. 4, 577–587 (2016).CrossRefGoogle Scholar
  22. 22.
    A. A. Chikrii, Conflict Controlled Processes, Springer Sci. and Business Media, Dordrecht–Boston–London (2013).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of UkraineKyivUkraine

Personalised recommendations