Towards the theory of linear dynamic nonantagonistic games

The article studies the problems of the theory of dynamic games of several persons with non-zero sum, when the value of the game is the system of functionals of distance type. The peculiarity of the work lies in the fact that to describe the evolution of objects there may be used three cases of linear systems of Volterra type: integro-differential system of equations with managing impacts outside of the integral, integro-differential system of equations with control actions under the sign of the interval and the system of integral equations. Solution of the problem lies in the construction of equilibrium, the Nash equilibrium, the set of optimal strategies for specified types of dynamical systems and the selected features. The problem is solved by constructing some modification of the well known extreme construction of the academician N. N. Krasovskiy, which is based on a new definition of the position of the game that uses a full memory of the control inputs, which significantly complicates the entire study. The corresponding theorems have been proven.