Optimal control of the Navier — Stokes system with a space variable in a network-like domain

The research of the problem of optimal control of the Navier — Stokes evolutionary differential system, considered in Sobolev spaces, the elements of which are functions with carriers in an $n$-dimensional network-like domain, is presented. Such domain consists of a finite number of subdomains, mutually adjacent to certain parts of the surfaces of their boundaries according to the graph type. For functions that are elements of these spaces, the conditions for the existence of traces on the surfaces of the joining are presented and the conditions of adjacency subdomains to which these functions satisfy are described. In applied questions of the analysis of the processes of transport of continuous media, the conditions of adjacency describe the regularities of the flow of fluid through the boundaries of the adjacent domains. The paper presents the results of following main research questions: 1) weak solvability of the initial boundary value problem for the Navier — Stokes system and obtaining the conditions for the existence of a weak solution to this problem; 2) the formation and solution of optimal control problems of various types of Navier — Stokes system. The fundamental approach to the analysis of the weak solvability of the initial boundary value problem is its reduction to the differential-difference problem (semi-digitization of the original system by a time variable) and subsequent use of a priori estimates for weak solutions of the obtained boundary value problems. The obtained a priori estimates are used to prove the theorem of the existence of a weak solution of the original differential system and indicate the way of the actual construction of this solution. A universal approach to solving the problems of optimal distributed and starting control of the Navier — Stokes evolutionary system is presented. The latter essentially expands the possibilities of analyzing non-stationary network-like processes of applied hydrodynamics (for example, processes of transporting various types of liquids through network or main line pipelines) and optimal control of these processes.