Solvability of stationary boundary control problems for heat convection equations
type: Item
instance of: scholarly article
title: Solvability of stationary boundary control problems for heat convection equations
zbMATH Open document ID: 0919.49004
DOI: 10.1007/BF02672906
published in: Siberian Mathematical Journal
publication date: 1999-05-11T00:00:00Z
review text: The following optimal control problem is studied for the motion of a viscous heat conducting fluid in \(\Omega\): \[ \begin{aligned} -&\nu\Delta u + (u\cdot\text{grad})u + \text{grad} p + \beta GT =f,\\ -&\lambda\Delta T + u\cdot\text{grad} T = f,\qquad \text{div} u=0,\\ u=g \;&\text{on} \;\Gamma_1,\qquad u=u_0 \;\text{on} \;\Gamma\setminus \Gamma_1,\qquad T=g\;\text{on} \;\Gamma_2,\\ &\lambda\left(\frac{\partial T}{\partial n} + \alpha T\right)=\psi\;\text{on} \;\Gamma\setminus\Gamma_2, \end{aligned} \] where \(\Omega\) is a bounded domain in \(\mathbb R^d\), \(d=2,3\), with Lipschitz boundary \(\Gamma\), which satisfies the condition \(\Gamma\in C^{0,1}\) and \(\Gamma\) has finitely many connected components \(\Gamma^{(1)},\dots,\Gamma^{(n)}\). The functions \(u\), \(\rho\), and \(T\) are the velocity, pressure, and temperature; the function \(g\) describes the Dirichlet boundary control for the velocity \(u\) on the part \(\Gamma_1\) of \(\Gamma\). The parts \(\Gamma_1\) and \(\Gamma_2\) may coincide or differ; moreover, any of them may coincide with \(\Gamma\). In the article under consideration, the optimal control problem is studied for the case \(\beta\neq 0\). This enables us to state some new extremal problems of hydrodynamics; in particular, the problem of controlling the temperature distribution in the domain by choosing the boundary regime for velocity and, conversely, the problem of controlling the flow of a viscous heat conducting fluid by the distribution of the boundary temperature. The aim of the article under review is to analyze optimal control problems. This requires proving existence for weak solutions to the problem, solvability of the optimal boundary control problem for a weakly lower semicontinuous quality functional, and justifying applicability of the Lagrange multiplier principle by using the apparatus of the theory of extremal problems. Using the Lagrange principle, the author derives an optimality system for the sought and adjoint state…
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 1285995
zbMATH Keywords: heat convection equations
zbMATH Keywords: stationary boundary control problem
zbMATH Keywords: weak solution
zbMATH Keywords: Lagrange principle
author: G. V. Alekseev
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02672906
OpenAlex ID: W2027657946
description: scientific article; zbMATH DE number 1285995