UDC 517.977
1 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
 jeffrappoport@gmail.com
|
|
TO SOLVING THE PROBLEM OF APPROACH OF CONTROLLED
OBJECTS IN DYNAMIC GAME PROBLEMS
Abstract. The problem of a guaranteed result in game problems of approach of controlled objects is considered.
A method for solving such problems is proposed. It involves constructing some scalar functions that qualitatively characterize
the course of approach of controlled objects and the efficiency of the decisions made. Such functions are called resolving functions.
In contrast to the main scheme of the method, the case is considered where the classical Pontryagin condition does not hold.
In this situation, instead of the Pontryagin selector, which does not exist, some shift functions are considered and with their help
special multivalued mappings are introduced. They generate upper and lower resolving functions, which are used to formulate
the sufficient conditions for the game completion in a certain guaranteed time. An example is given to illustrate the approach
of controlled objects with a simple motion, in order to obtain upper and lower resolving functions in explicit form, which allows making
a conclusion about the possibility of ending the game when the Pontryagin condition does not hold.
Keywords: quasilinear differential game, multi-valued mapping, measurable selector, stroboscopic strategy, resolving function.
FULL TEXT
REFERENCES
- Chikrii A.A. Conflict controlled processes. Boston; London; Dordrecht: Springer Science and Business Media. 2013. 424 p.
- Chikrii A.A. An analytical method in dynamic pursuit games. Proc. Steklov Institute of Mathematics. 2010. Vol. 271. P. 69–85.
- Chikrii A.A., Chikrii V.K. Image structure of multivalued mappings in game problems of motion control. Journal of Automation and Information Sciences. 2016. Vol. 48, N 3. P. 20–35.
- Chikrii A.A. Upper and lower resolving functions in game dynamics problems. Tr. IMM UB RAS. 2017. Vol. 23, N 1. P. 293–305. https://doi.org/10.21538/0134-4889- 2017-23-1-293-305.
- Krasovsky N.N., Subbotin A.I. Positional differential games [in Russian]. Moscow: Nauka, 1974. 455 p.
- Pontryagin L.S. Selected scientific works [in Russian]. Moscow: Nauka, 1988. Vol. 2. 576 p.
- Subbotin A.I., Chentsov A.G. Optimization of guarantees in control problems [in Russian]. Moscow: Nauka, 1981. 288 p.
- Hajek O. Pursuit games. New York: Acad. Press, 1975. Vol. 12. 266 p.
- Aubin J.-P., Frankowska H. Set-valued analysis. Boston; Basel; Berlin: Birkhauser, 1990. 461 p.
- Rockafellar R. Convex analysis [Russian translation]. Moscow: Mir, 1973. 470 p.
- Ioffe A.D., Tikhomirov V.M. Theory of extreme problems [in Russian]. Moscow: Nauka, 1974. 480 p.
- Chikrii A.A., Rappoport I.S. The method of resolving functions in the theory of conflict-controlled processes. Kibernetika i sistemnyj analiz. 2012. Vol. 48, N 5. P. 40–64.
- Chikrii A.A., Eidelman S.D. Game problems for fractional quasilinear systems. Journal Computers and Mathematics with Applications. New York: Pergamon, 2002. Vol. 44. P. 835–851.
- Chikrii A.A., Matychyn I.I. On linear conflict-controlled processes with fractional derivatives. Trudy Instituta Mathematiki i Mechaniki URo RAN. 2011. Vol. 17, N 2. P. 256–270.
- Chikrii A.A., Dzyubenko K.G. Bilinear Markov processes of searching for moving objects. Problemy upravleniya i informatiki. 1997. N 1. P. 92–107.
- Chikrii A.A., Kalashnikova S.F. Pursuit of a group of evaders by a single controlled object. Cybernetics. 1987. Vol. 23, N 4. P. 437–445.
- Chikrii A.A., Matychyn I.I. Game problems for fractional-order linear systems. Proc. Steklov Institute of Mathemaics. 2010. Vol. 268, suppl 1. P. s54–s70.
- Pilipenko Yu.V., Chikrii A.A. Oscillatory conflict-controlled processes. App. mathematics and mechanics. 1993. Vol. 57, N 3. P. 3–14.
- Albus J., Meystel A., Chikrii A.A., Belousov A.A., Kozlov A.J. Analytical method for solution of the game problem of soft landing for moving object. Cybernetics and Systems Analysis. 2001. Vol. 37, N 1. P. 75–91.
- Chikrii A.A., Eidelman S.D. Game control problems for quasilinear systems with fractional Riemann–Liouville derivatives. Kibernetika i sistemnyj analiz. 2001. N 6. P. 66–99.
- Chikrii A.A., Matychyn I.I. Riemann–Liouville, Caputo, and sequentiol frectional derivatives in differential games. Advances in Dynamic Games: Theory, Applications, and Numerical Methods for Differential and Stochastic Games. 2011. Vol. 11. P. 61–82.
- Chikrii A.A., Eidelman S.D. Generalized Mittag-Leffler matrix functions in game problems for fractional-order evolution equations. Kibernetika i sistemnyj analiz. 2000. N 3. P. 3–32.
- Filippov A.F. On some questions of the theory of optimal control. Herald of the Moscow State University. Ser. mathematics, mechanics, astronomy, physics, chemistry. 1959. N 2. P. 25–32.
- Polovinkin E.S. Elements of the theory of multivalued mappings [in Russan]. Moscow: MIPT Publishing House, 1982. 127 p.
