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Extremum Strategies of Approach of Controlled Objects in Dynamic Game Problems with 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. The method is reduced to the systematic use of the Fenchel–Moreau 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 involutivity 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 principle does not hold. Two schemes of the method of resolving functions with extremum strategies of approach of controlled objects are constructed and the guaranteed times are compared.

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Correspondence to A. A. Chikrii.

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Translated from Kibernetyka ta Systemnyi Analiz, No. 2, March–April, 2022, pp. 42–57.

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Chikrii, A.A., Rappoport, J.S. Extremum Strategies of Approach of Controlled Objects in Dynamic Game Problems with Terminal Payoff Function. Cybern Syst Anal 58, 197–209 (2022). https://doi.org/10.1007/s10559-022-00451-4

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  • DOI: https://doi.org/10.1007/s10559-022-00451-4

Keywords

  • terminal payoff function
  • quasilinear differential game
  • multi-valued mapping
  • measurable selector
  • extremum strategy
  • resolving function