Game Problem Guidance for Integro-Differential System of Volterra Type for Three Persons

The problem of guidance of a dynamic object in space ${\Bbb R}^n$ on a closed set $M$ is considered. In this problem three players take part, and two of them make up the coalition that seeks to bring moving point $x(t)$ to the set of at the moment o, and a third player tries to avoid the meeting, $x(t)$ with the set $M$.
Feature of our work is to describe the evolution of the object of nonlinear integral differential system, which gives to the controlled system new essential properties: memory and the effect of delay on control inputs, which complicates the study, compared with the case where the evolution of the object is described by ordinary differential systems. To solve the problem we assume the existence of a stable bridge in the space of continuous functions, containing pieces of solutions of the initial system when using players' coalition of their extreme strategies defined in the work for any admissible management of the opposite side. It is assumed that a stable bridge dropped on the target set $M$ in a fixed moment of time $\theta$.
We prove that the constructed in the work of the extreme strategy coalition holds the solution (the movement) of the system at stable bridge, and solves the problem of guidance.