Issue 186, article 3

Posted on

by admin

DOI:https://doi.org/10.15407/kvt186.04.015

KVT, 2016, Issue 186, pp.15-30

UDC 517.977

SET-VALUED MAPPINGS AND ITS SELECTIONS IN GAME DYNAMIC PROBLEMS

Chikrii A.A.

Glushkov Institute of Cybernetics NAS of Ukraine, Kiev, Ukraine

g.chikrii@gmail.com

Introduction. Mathematical theory of control under conflict and uncertainty provides a wide range of fundamental methods to study controlled evolutionary processes of various nature. These are, first of all, the classic methods of L.S. Pontryagin and N.N. Krasovskii. This paper is closely related to the mentioned investigations. It is devoted to research of non-stationary game dynamic problems on the basis of the L.S. Pontryagin first direct method and the method of resolving functions.

The purpose of the paper is to derive sufficient conditions for the game termination for some guaranteed time in favor of the first player and to provide the control realizing this result.

Results. Here, in the development of the method of resolving functions general scheme, the upper and the lower resolving functions of two types are introduced in the form of selections of special set-valued mappings. This made it possible to deduce conditions for the game termination in the class of quasi- and stroboscopic strategies.

Conclusions. The in-depth analysis of properties of the set-valued mappings and their selections, around which measurable controls are chosen by virtue of the Filippov-Castaing theorem, is provided. A comparison of the guaranteed times of the above-mentioned methods is given.

Keywords: Conflict controlled processes, set — valued map, Pontryagin’s condition, Aumann’s integral, resolving function.

Download full text (ru)!

Reference

  1. Pontryagin L.S. Selected scientific papers. M.: Nauka, 1988, 576p. (in Russian)
  2. Krasovskii N.N. Game Problems on the Encounter of Motions, M.: Nauka, 1970. 420 p. (in Russian)
  3. Chikrii A.A. Conflict controlled processes. Boston; London; Dordrecht: Springer Science and Busines Media, 2013, 424 p.
  4. Chikrii A.A. An analytic method in dynamic games. Trudy Mat. Inst. RAN im. V.A. Steklova, 2010. Vol.271. pp. 76–92. (in Russian)
  5. Aubin J.-P., Frankowska H. Set — valued analysis. Boston; Basel; Berlin: Birkhauser, 1990, 461 p.
  6. Hajek O. Pursuit games. New York: Academic Press, 1975, Vol. 12, 266 p.
  7. Lappo — Danilevsky I.A. Application of the matrix functions to the theory of systems of ordinary differential equations. M.: SPHTTL, 1957, 235 p. (in Russian)
  8. Pschenichnyi B.N Convex analysis and extremal problems. M.: Nauka, 1980, 320 p. (in Russian)

Received 13.09.2016