Generalization of the Carathéodory theorem and the maximum principle in averaged problems of non-linear programming

The relationship between the averaging of functions over time and its averaging over the set of values of the required variables is considered. Optimization problems are studied, the criterion and constraints of which include the averaging of functions or functions of the average values of variables. It is shown that the optimality conditions for these problems have the form of the maximum principle, and their optimal solution in the time domain is a piecewise constant function. A generalization of Carathéodory’s theorem on convex hulls of a function is proved. Optimality conditions are obtained for non-linear programming problems with averaging over a part of the variables and functions depending on the average values of the variables.