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 decisions. 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 special multivalued mappings are introduced with their help. 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.
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References
A. A. Chikrii, Conflict Controlled Processes, Springer Science and Business Media, Boston–London–Dordrecht (2013).
A. A. Chikrii, “An analytical method in dynamic pursuit games,” Proc. Steklov Institute of Mathematics, Vol. 271, 69–85 (2010).
A. A. Chikrii and V. K. Chikrii, “Image structure of multivalued mappings in game problems of motion control,” J. Autom. Inform. Sci., Vol. 48, No. 3, 20–35 (2016).
A. A. Chikrii, “Upper and lower resolving functions in dynamic game problems,” in: Tr. IMM UrO Russian Acad. Sci., Vol. 23, No. 1, 293–305 (2017). https://doi.org/10.21538/0134-4889-2017-23-1-293-305.
N. N. Krasovskii and A. I. Subbotin, Positional Differential Games [in Russian], Nauka, Moscow (1974).
L. S. Pontryagin, Selected Scientific Works, Vol. 2 [in Russian], Nauka, Moscow (1988).
A. I. Subbotin and A. G. Chentsov, Guarantee Optimization in Control Problems [in Russian], Nauka, Moscow (1981).
O. Hajek, Pursuit Games, Vol. 12, Acad. Press, New York (1975).
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston–Basel–Berlin (1990).
R. T. Rockafellar, Convex Analysis, Princeton University Press (1973).
A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems [in Russian], Nauka, Moscow (1974).
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). https://doi.org/10.1007/s10559-012-9430-y.
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Translated from Kibernetyka ta Systemnyi Analiz, No. 5, September–October, 2021, pp. 120–131.
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Rappoport, I.S. Solving the Problem of Approach of Controlled Objects in Dynamic Game Problems. Cybern Syst Anal 57, 775–786 (2021). https://doi.org/10.1007/s10559-021-00402-5
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DOI: https://doi.org/10.1007/s10559-021-00402-5