Рассмотрена распространенная в приложениях задача оптимального граничного управления параболической системой с запаздыванием и распределенными параметрами на графе. Состояние системы определяется слабым решением начально-краевой задачи для параболического уравнения в пространстве соболевского типа, управляющее воздействие на систему и наблюдение за ее состоянием осуществляются в граничных узлах графа на всем временном промежутке. Сопряженное состояние системы обусловливается также слабым решением начально-краевой задачи с запаздыванием и распределенными параметрами на графе с финальным условием. Получены необходимые и достаточные условия существования оптимального управления с использованием сопряженного состояния системы, решена задача синтеза оптимального управления для случая отсутствия ограничений на управляющие воздействия и получен аналог известных для конечномерного случая результатов Калмана. Используемый метод применим ко многим задачам оптимизации дифференциальных систем, состояние которых определяется слабыми решениями эволюционных уравнений на сетях. Представленные результаты являются основополагающими при исследовании задач граничного управления динамикой ламинарных течений многофазных сред. Библиогр. 20 назв. Ключевые слова: начально-краевая задача, распределенные параметры на графе, слабые решения, оптимальное граничное управление, синтез управления.

This work is a natural extension of research into optimal control problems of evolution equations with distributed parameters on a geometrical graph (network) of one of the authors in the direction of increasing the dimension of a spatial variable and the functions describing the state of the study of the Navier-Stokes equations. At the same time is examined a simple case of the absence of convective effect (laminar flow of an incompressible viscous fluid)linearized system of Navier-Stokes equations in a net-like domain. It proves unique solvability of the initial boundary value problem in the weak formulation which is based on the Faedo-Galerkin method using a special basis (the set of generalized eigenfunctions of the special spectral problem) and a priori estimates of norms solutions such as power inequalities. The proof is constructive: to construct a sequence of approximate solutions that converges weakly to the exact solution of the problem. Problems are analyzed with distributed and a start control with a final observation, widespread in applications, that provides the necessary and sufficient conditions for the existence of optimal controls in terms of the conjugate states of the respective systems. Sufficient attention is paid to the synthesis of the optimal control action, and analogues of established finite-dimensional case for Kalman results have been obtained. Although, the use of this method is demonstrated by examples of optimal control theory, this method has a highly susceptible to generalization and applicable to a wide class of linear problems. Refs 15.

The counterparties’ economic relations have been mainly transferred to the online sector. This reflects the concept of digital interaction between manufacturers, intermediaries and consumers. At the software engineering stage, it is necessary to use one of the main formalisms - sales funnel, which has transformed into a conversion funnel under the concept of a digital economy. In the presented article, a mathematical model is developed that formalizes in detail all the stages of commercial interaction. The research results provide a set of software engineering algorithms for the expert systems designed for commercial services managers.

The modern automatic warehouse of industrial production is equipped with sensors to determine the type, size, weight, expire date of each SKU. There are also robots for moving SKUs, control algorithms, and an interface for interacting with external services. When developing such a cyber physical system, it is necessary to optimize its main parameters. In engineering, only 2-3% of the whole time is spent on the processing of parts. The substantial time is spent on interoperational storage and transportation. The productivity of the automatic warehouse determines the economic performance of the enterprise as a whole. In addition, the assortment and emergent stock of products stored in the warehouse should ensure stable work under conditions of a change in the production program. This paper is devoted to the algorithm that underlies the digital interaction of an automatic warehouse as a cyber-physical system.

The modern concept of the developing engineering production is based on a digital interaction. At the same time, management of all related processes is possible due to a digital shadow at each stage of work. This primarily related to supply, marketing and interoperational units within the enterprise. The solution of the automation problems of the industrial production, matching to the priorities of the Industry 4.0 program is ought to begin with the organization of the inter-machine interaction M2M. This article presents the results of a study devoted to a search for optimal algorithms for managing the flow of goods and cargoes of the engineering production in modern supply chain management concepts such as 3PL logistics, pick-by-line, cross-docking, as well as ensuring the implementation of managerial decision-making systems based on outrunning economic indicators.