UDC 517.977
1 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
 g.chikrii@gmail.com
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2 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
jeffrappoport@gmail.com
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EXTREMUM STRATEGIES OF APPROACH OF CONTROLLED OBJECTS
IN DYNAMIC GAME PROBLEMS WITH A TERMINAL PAYOFF FUNCTION
Abstract. The authors propose a method for solving the problem of approach of controlled objects in dynamic game problems with a terminal payoff function, which is reduced to the systematic use of the Fenhel-Moro ideas on the general scheme of the method of resolving functions. The essence of the method is that the resolving function can be expressed in terms of the function conjugate to the payoff function and, using the inclusiveness of the connection operator for a convex closed function, it is possible to obtain a guaranteed estimate of the terminal value of the payoff function represented by the payoff value at the initial instant of time and integral of the resolving function. A feature of the method is the cumulative principle used in the current summation of the resolving function to assess the quality of the game before reaching a certain threshold. The notion of the upper and lower resolving functions of two types is introduced and sufficient conditions of a guaranteed result in the differential game with the terminal payoff function are obtained in the case where Pontryagin's condition is not satisfied. Two schemes of the method of resolving functions with extremum strategies of approach of controlled objects are constructed and the guaranteed times are compared.
Keywords: terminal payoff function, quasilinear differential game, multi-valued mapping, measurable selector, extremum strategy, resolving function.
FULL TEXT
REFERENCES
- Chikrii A. A. Conflict controlled processes. Dordrecht; Boston; London: Springer Science and Business Media, 2013. 424 p.
- Chikrii A.A., Rappoport J.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., Chikrii V. K. Image structure of multialued mappings in game problems of motion control. Journal of Automation and Information Sciences. 2016. Vol. 48, N 3. P. 20–35.
- Krasovskii 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.
- Nikolsky M.S. The first direct method of L.S. Pontryagin in differential games [in Russian]. Moscow: MGU Publishing House, 1984. 65 p.
- Subbotin A.I., Chentsov A.G. 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: Academic Press, 1975. Vol. 12. 266 p.
- Aubin J.-P., Frankowska H. Set-alued analysis. Boston; Basel; Berlin: Birkhauser, 1990. 461 p.
- Rockefellar R. Convex analysis [Russian translation]. Moscow: Mir, 1973. 470 p.
- Ioffe A.D., Tikhomirov V.M. Theory of extremal problems [in Russian]. Moscow: Nauka, 1974. 480 p.
- Rappoport I.S. The method of resolving functions in the theory of conflict-controlled processes with a terminal payoff function. Problemy upravleniya i informatiki. 2016. N 2. P. 49–58. https://doi.org/10.1615/JAutomatInfScien.v48.i5.70.
- Rappoport I.S. On the stroboscopic strategy in the method of resolving functions for game control problems with a terminal payoff function. Kibernetika i sistemnyj analiz. 2016. Vol. 52, N 4. P. 90–102. https://doi.org/10.1007/s10559-016-9860-z.
- Rappoport I.S. Sufficient conditions for a guaranteed result in a differential game with a terminal payoff function. Problemy upravleniya i informatiki. 2018. N 1. P. 72–84.
https://doi.org/10.1615/JAutomatInfScien.v50.i2.20.
