Method of Resolving Functions in the Group Pursuit Problem with a Terminal Pay Off Function and Integral Constraints on Controls

Abstract

A method is proposed for solving game dynamics problems with a terminal pay off function and integral constraints on controls which consists in systematical using the ideas of Fenhel−Moreau as applied to the general scheme of resolving functions method. The essence of the proposed method lies in the fact that the resolving function can be expressed in terms of the function conjugate to the pay off function and using the involute of the conjugation operator for a convex closed function to obtain a guaranteed estimate of the terminal value of the pay off function which is represented by the paying off value at the initial time instant and the integral of the resolving function. The main feature of the method is the cumulative principle used in the current summation of the resolving function for estimating the game quality until a certain threshold value is reached. The paper considers linear differential games of group pursuit with a terminal pay off function and integral constraints on controls. Sufficient conditions for the game completion in a finite guaranteed time in a class of quasi-strategies are formulated. Two schemes of the resolving functions method are proposed that ensure without additional assumptions the game completion in the final guaranteed time in the class of stroboscopic strategies. The results of comparing the guaranteed times of different schemes of the resolving functions method are presented.