UDC 517.977
1 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
 jeffrappoport@gmail.com
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THE PROBLEM OF APPROXIMATION OF CONTROLLED OBJECTS
IN DYNAMIC GAME PROBLEMS WITH A TERMINAL PAYOFF FUNCTION
Abstract. A method is proposed for solving the problem of convergence
of controlled objects in dynamic game problems with the terminal payoff function, which consists in the systematic use
of Fenchel–Moreau ideas as applied to the general scheme of the method of resolving functions. The essence of the proposed
method is that the resolving function can be expressed in terms of the function conjugate to payoff function and, using the involution
of the conjugation operator for a convex closed function, we obtain a guaranteed estimate of the terminal value of the payoff function,
which can be presented in terms of the payoff value at the initial instant of time and integral of the resolving function.
The concepts of upper and lower resolving functions of two types are introduced and sufficient conditions for a guaranteed result in a differential game with a terminal payoff function are obtained for the case where the Pontryagin condition does not hold. Two schemes of the method of resolving functions are considered, the corresponding control strategies are constructed, and guaranteed times are compared. The results are illustrated by a model example.
Keywords: terminal payoff function, quasilinear differential game, multi-valued mapping, measurable selector, stroboscopic strategy, resolving function.
FULL TEXT
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