Approach of single-type objects, evolution of which is described by Volterra systems

Background. The paper discusses some problems of optimal control, namely, the theory of dynamic games when the game dynamics is described by linear integral and integrodifferential vector Volterra equations. The aim of the article is to solve problems of optimization of distance-type functionals. Materials and methods. To solve these problems, the author built a modification of the famous extreme construction of academician N. N. Krasovskiy developed for ordinary differential systems. The centerpiece of this modification is a new definition of the game position for which it is necessary to calculate the total memory to manage stress that greatly complicates the entire study compared with the case of ordinary differential systems. Results and conclusions. The paper presents significant new results that complement and extend the general theory of dynamic games. They consist in the spread of classical methods of academician N. N. Krasovskiy on more complex objects - Volterra dynamic systems. Thus, the author has proved the possibility of extending the field of application of these methods.