On the Numerical Solution of Differential Games for Neutral-Type Linear Systems

Abstract

The paper deals with a zero-sum differential game, in which the dynamics of a conflict-controlled system is described by linear functional differential equations of neutral type and the quality index is the sum of two terms: the first term evaluates the history of motion of the system realized up to the terminal time, and the second term is an integral–quadratic evaluation of the corresponding control realizations of the players. To calculate the value and construct optimal control laws in this differential game, we propose an approach based on solving a suitable auxiliary differential game, in which the motion of a conflict-controlled system is described by ordinary differential equations and the quality index evaluates the motion at the terminal time only. To find the value and the saddle point in the auxiliary differential game, we apply the so-called method of upper convex hulls, which leads to an effective solution in the case under consideration due to the specific structure of the quality index and the geometric constraints on the control actions of the players. The efficiency of the approach is illustrated by an example, and the results of numerical simulations are presented. The constructed optimal control laws are compared with the optimal control procedures with finitedimensional approximating guides, which were developed by the authors earlier.