In this paper questions of building models of information exchange systems with discrete and distributed delay and with delayed feedback by methods of the theory of linear functional differential equations are investigated. When solving the said equations, it is necessary to consider restrictions caused by the uncertainties in the system under modeling, such as the absence of the exact data on the parameters of the model elements, their natural spread and temporal variations, thus requiring the solution of an identification problem. The models with continuous aftereffect introduced in this work take a fuller account of reflected signal characteristics in closed space, which increases the reliability of modeling results in comparison with the known differential models. At the same time, there is a problem of finding functions that characterize the value distribution of the echo delay. In this work, these functions (kernels) are approximated by a number of exponents, which simplifies the equations and allows the assumption that the aftereffect can be concentrated on both final and infinite intervals. The echo components caused by closed-space resonances are modeled by transfer functions of the corresponding linear links. In numerical modeling, a single-channel model represented by a second-order resonance link and a pulse-shaped kernel described by a sum of two decreasing exponents is considered. The analysis of stability of the models of systems with delayed feedback is conducted by a frequency method. In the paper an approach to estimating the correlation and spectral functions of signals and noise components based on the parametric representation of the latter is considered. The paper also considers issues relating to the practical significance of the research results.
Рассмотрены проблемы моделирования процессов и систем экономики, поставлена задача оптимального управления, определены представляющая экономику система нелинейных дифференциальных уравнений и оптимизируемый целевой функционал; выбран вариационный метод решения поставленной задачи, описан принцип инвариантности нелинейных систем и предоставляемые им возможности моделирования динамических, в том числе экономических, систем. Дается критический анализ состояния в области экономического моделирования. Рассмотрены в терминах преобразования Лапласа, особенности и свойства принципа инвариантности линейных систем. Получены функции чувствительности нелинейных систем и определяющие их производные Гато и Фреше. Показано применение теории чувствительности в задачах оптимального управления нелинейными системами на примере достаточно простой задачи.Ключевые слова: экономическая модель, вариационное исчисление, оптимальное управление, принцип инвариантности, чувствительность динамических систем Для цитирования: Ермолаев В. А., Проскуряков А. Ю. Об инвариантности и чувствительности параметров математических моделей экономических систем // Развитие территорий. 2023. № 2. С. 08-14.
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