Mathematical aspects of optimal control of transference processes in spatial networks

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.


Introduction
This article is devoted to mathematical description of two essential transference networks processes: 1) established process with pronounced laminar character; 2) process with inverse relationship, which dinamics has convective character. The first can be found in the study of economic laws in commercial networks in the analysis of the movement of goods, money, human migration, the transfer of one -parameter continuous media, etc. [1][2][3]. The second process occurs in the study of dynamics of multi-phase medium like hydro-and aero networks, radio and TV network [4][5][6][7][8].
At the same time, the approache to the solution of the actual problem of network transference processes is developed, that is the optimal control of such processes, and its subsequent analysis. The mathematical formalism of differential systems of partial differential equations with distributed parameters on a graph (network) or network-like domain is used, the state of the differential system is determined by its weak solution.

Processes with established laminar character
These include regularities or phenomena observed on one-parameter carriers, which include spatial oriented connected graphs  with edges  parametrized by a single interval [0,1] . , representing a system of differential equations with distributed parameters on each edge   . The The function ( , ) v x t is the boundary control action on the system (1) (the boundary control system

Definition 1. A weak solution of the initial-boundary value problem (1) -(3) is a function
T y x t V a  that satisfies an integral identity.
for any [0, ] tT  and for any function T Wa  -Sobolev spaces whose elements satisfy conditions (2) and have generalized first order derivatives, ( , ) t y  is a bilinear form, defined by the relation The initial-boundary value problem (1) -(3) has at least one weak solution in space 1,0 ( , ) T Va  . Optimal boundary control with delay. As is known, any dynamic processes are accompanied by the effect of a time delay (see, for example, the works [3,5] and bibliography there), which is a reason to consider equation (1) as a system of equations with a constant delay (0, ) hT  : We obtain the initial boundary-value problem (4) - (6), whose weak solution ( , ) y x t determines the state ( )( , ) y v x t of system (4) in space 1,0 , hT Va  , ( , ) v x t -the boundary effect on system (4). We present equation (4) in a form more convenient for analysis [5]. Let  Consider the system (7), the state of which is defined as the solution ( )( , ) y v x t of the initial-boundary value problem (7), (8) in space 1,0 ( , ) T Va  .

Theorem 2
The problem of optimal boundary control of system (7) has a unique solution v   U , i.e.
Synthesis of optimal boundary control. For system (7)

Theorem 3
In order for an element () ux  U to be the optimal control of system (7), and therefore (4), it is necessary and sufficient that the following relations are satisfied: for any [0, ] tT  and for any function