Optimal control of the linearized Navier–Stokes system in a netlike 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.