Guaranteed result in a differential game with a terminal payoff function

Abstract

A method for solving game-type control problems with a terminal payoff function is proposed. It consists of applying the ideas of Fenchel-Moreau duality [1] to the general scheme of the method of resolvent functions [2]. The main point of the method is that the resolvent function can be expressed in terms of the conjugate of the payoff function; then, using the involutive property of the conjugation operator for a convex closed function, one obtains a guaranteed estimate for the terminal value of the payoff function, expressed in terms of the initial value of the payoff and the integral of the resolvent function.