Advertisement

On the unimprovability of full-memory strategies in the risk minimization problem

  • D. A. Serkov
Article

Abstract

Methods from the theory of guaranteeing positional control are used to study the risk minimization problem, i.e., the problem of optimal control under dynamic disturbances in a formalization based on the Savage criterion. A control system described by an ordinary differential equation is considered. The values of control actions and disturbance at each moment lie in known compact sets. Realizations of the disturbance are also subject to an unknown functional constraint from a given set of functional constraints. Realizations of the control are formed by full-memory positional strategies. The quality functional, which is defined on motions of the control system, is assumed to be continuous on the corresponding space of continuous functions. New conditions that provide the unimprovability of the class of fullmemory positional strategies under program constraints and L 2-compact constraints on the disturbance are presented.

Keywords

full-memory strategy Savage criterion functionally limited disturbance 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].zbMATHGoogle Scholar
  2. 2.
    N. N. Krasovskii, Control of a Dynamical System (Nauka, Moscow, 1985) [in Russian].Google Scholar
  3. 3.
    A. I. Subbotin and A. G. Chentsov, Guarantee Optimization in Control Problems (Nauka, Moscow, 1981) [in Russian].zbMATHGoogle Scholar
  4. 4.
    N. N. Krasovskii, Game Problems on the Encounter of Motions (Nauka, Moscow, 1970) [in Russian].zbMATHGoogle Scholar
  5. 5.
    N. N. Barabanova and A. I. Subbotin, “On continuous evasion strategies in game problems on the encounter of motions,” J. Appl. Math. Mech. 34(5), 765–772 (1970).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    N. N. Barabanova and A. I. Subbotin, “On classes of strategies in differential games of evasion of contact,” J. Appl. Math. Mech. 35(3), 349–356 (1971).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    A. V. Kryazhimskii, “The problem of optimization of the ensured result: Unimprovability of full-memory strategies,” in Constantin Carathéodory: An International Tribute, Ed. by T. M. Rassias (World Sci., Teaneck, NJ, 1991), Vol. 1, pp. 636–675.CrossRefGoogle Scholar
  8. 8.
    J. Niehans, “Zur Preisbildung bei ungewissen Erwartungen,” Schweiz. Z. Volkswirtsch. Stat. 84(5), 433–456 (1948).Google Scholar
  9. 9.
    L. J. Savage, “The theory of statistical decision,” J. Amer. Stat. Assoc. 46, 55–67 (1951).CrossRefzbMATHGoogle Scholar
  10. 10.
    A. V. Kryazhimskii and Yu. S. Osipov, “Modelling of a control in a dynamic system,” Engrg. Cybernetics 21(2), 38–47 (1984).MathSciNetGoogle Scholar
  11. 11.
    Yu. S. Osipov and A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions (Gordon and Breach, London, 1995).zbMATHGoogle Scholar
  12. 12.
    D. A. Serkov, “Risk-optimal control under functional constraints on the disturbance,” Mat. Teor. Igr Prilozh. 5(1), 74–103 (2013).zbMATHGoogle Scholar
  13. 13.
    D. A. Serkov, “Optimization of guaranteed results under functional restrictions on the dynamic disturbance,” Dokl. Math. 87(3), 310–313 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    J. Warga, Optimal Control of Differential and Functional Equations (Academic, New York, 1972; Nauka, Moscow, 1977).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Institute of Radioelectronics and Informational TechnologiesUral Federal UniversityYekaterinburgRussia

Personalised recommendations