Globalization has given a powerful impetus to the development of international commercial activity and logistics management systems taking full advantage of cross-border networking. The solution lies at the intersection of information technologies, technical means of machine-to-machine (M2M) interaction, mobile high-speed networks, geolocation, cloud services, and a number of international standards. The current trend towards creating digital logistics platforms has set a number of serious challenges for developers. The most important requirement is the condition of sustainability of the obtained solutions with respect to disturbances in the conditions of logistics activities caused not only by market uncertainty but also by a whole set of unfavorable factors accompanying the transportation process. Within the framework of the presented research, the problem of obtaining the conditions for the stability of solutions obtained on the basis of mathematical models is set. At the same time, the processes of transferring not only discrete but also continuous material flows through complex structured networks are taken into account. This study contains the results of the analysis of the stability of solutions of differential systems of various types that simulate the transfer processes in network media. Initial boundary value problems for evolutionary equations and differential-difference systems are relevant in logistics, both for the discrete transportation of a wide range of goods and for the quasi-continuous transportation of, for example, liquid hydrocarbons. The criterion for the work of a logistics operator is the integral functional. For the mathematical description of the transport process of continuous and discrete media, a wide class of integrable functions are used, which adequately describe the transport of media with a complex internal rheological structure.

The new kind of practical issues was born due to modern trend on globalization and transition to the concept of industry 4.0. These are the processes in commercial networks representing the movement of goods and finance, which have specific economic indicators; migration of populations and labor resources; transference of one-parameter continuous media (diffusion effect). The issues with higher level of complexity occur during the research of multi-phase environments dynamics in network-like media and mathematical modeling of such processes. The authors divide these two problems and offer the formalized description for the laminar processes and dynamic convective processes with feedback. There was developed an approach to solve the problems of optimal control for these processes. The research is based on methods for analyzing systems of partial differential equations with distributed parameters on a graph or network-like domain.

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.

This paper considers a differential system whose state is described by a parabolic initial boundary value problem with distributed parameters on a graph. For the above system, the author studies the problem of boundary control in the class of generalized solutions. Control action and observations are simultaneously boundary. And finally, the existence conditions of a unique control action and the expressions characterizing control actions are obtained.

We consider generalized solutions to boundary-value problems for elliptic equations on an arbitrary geometric graph and their corresponding eigenfunctions. We construct analogs of Sobolev spaces that are dense in L ₂. We obtain conditions for the Fredholm solvability of boundary-value problems of various types, describe their spectral properties and conditions for the expansion in generalized eigenfunctions. The results presented here are fundamental in studying boundary control problems of oscillations of multiplex jointed structures consisting of strings or rods, as well as in studying the cell metabolism.