The Problem of Approach of Controlled Objects in Dynamic Game Problems with a Terminal Payoff Function

Abstract

To solve the problem of convergence of controlled objects in dynamic game problems with the terminal payoff function, the author proposes a method that systematically uses the Fenchel–Moreau ideas as applied to 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 payoff function and, using the involution of the conjugation operator for a convex closed function, a guaranteed estimate of the terminal value of the payoff function is obtained, 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 generated, and guaranteed times are compared. The results are illustrated by a model example.