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The authors consider the optimization problem related to the integral representation of the deviation of positive linear operators on the classes of (ψ, β )-differentiable functions in the integral metric. The Poisson integral is taken as a positive linear operator, being the solution of the Laplace equation in polar coordinates with the corresponding initial conditions given on the boundary of the unit disc. The Poisson integral refers to operators with a delta-like kernel; therefore, it is the best apparatus for solving many problems of applied mathematics, namely: optimization methods and calculus of variations, mathematical control theory, theory of dynamical systems and game problems of dynamics, applied nonlinear analysis and moving objects search. The classes of (ψ, β )-differentiable functions are generalizations of the well-known Sobolev, Weyl–Nagy, etc. classes in optimization problems, on which the asymptotic properties of solutions of Laplace equations in the unit disc are analyzed. The problem solved in the paper will make it possible to generate high-quality mathematical models of many natural and social processes. [ABSTRACT FROM AUTHOR]

The issues of constantly changing geopolitical situation in the world and the desire of leading powers to obtain or strengthen the new and existing spheres of influence, as well as the urge of other countries to maintain or regain their former positions, leads to the use of new methods of hybrid warfare. One of the methods employed for information environment (i.e. network system) risk assessment is the probability of an attack on it. Information Security Assessment is determined by the values of the interaction between the attacking and defensive sides. This raises the challenge of the need to assess and predict the actions of both the defensive and the attacking sides. The process of interaction between attackers and defenders can be viewed as a game play, understood as the process of strategic interaction between the opposing or cooperating players under certain constraints and their plans of action. Deriving from this interpretation of the game, resolving the issues of hybrid war on the basis of the theory of games, predicting the actions of the attacking side, and determining the decisions by the defensive side is proposed. As in any theory of applied mathematics, game theory provides some formal structure. It is assumed that all possible actions of each player can be accurately established and for each combination of such actions the expected outcome of the competition can be quantified. The goal of each player is to maximize their expectation of winning. The player can assume that each of his opponents is busy with a similar optimization. The paper considers various possible situations of interaction between the attacking and defending sides. Functional expressions are given to determine the optimal strategies of the opposing sides and to find the guaranteed maximum level of the cost of an attack. Methods for solving these expressions are proposed. In particular, these include the following: methods for reducing matrix and continuous games to linear programming problems, methods for reducing continuous games to matrix games, algorithms for the approximate solution of continuous games by the random search method, and methods for modeling games. [ABSTRACT FROM AUTHOR]