Abstract

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.

中文翻译：

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非线性规划平均问题中卡拉西奥多里定理和极大值原理的推广

摘要

考虑函数随时间的平均值与其对所需变量值集的平均值之间的关系。研究优化问题，其准则和约束包括函数或变量平均值函数的平均。结果表明，这些问题的最优性条件具有极大值原理的形式，并且它们在时域上的最优解是分段常数函数。证明了卡拉西奥多里关于函数凸包定理的推广。通过根据变量的平均值对部分变量和函数进行平均来获得非线性规划问题的最优性条件。