Analysis of optimal control problems for the 2-D stationary Boussinesq equations
type: Item
instance of: scholarly article
title: Analysis of optimal control problems for the 2-D stationary Boussinesq equations
zbMATH Open document ID: 0951.49006
author: Hyung-Chun Lee
author: O. Yu. Imanuvilov
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-08-28T00:00:00Z
review text: The authors use the system \[ \begin{aligned} - \nu \Delta {\mathbf u} + ({\mathbf u} \cdot \nabla){\mathbf u} &= - \nabla p + \alpha T{\mathbf g} + {\mathbf f}, \\ \nabla \cdot {\mathbf u} &= 0,\\ - \kappa \Delta T + ({\mathbf u} \cdot \nabla)T &= Q \end{aligned} \] in a plane domain \(\Omega\) with boundary \(\Gamma\) as a model of viscous incompressible thermally convected flow. \((({\mathbf u}, p, T) =\) (velocity, pressure, temperature)). The boundary conditions are \[ {\mathbf u} = 0 \;\;\text{on} \;\Gamma, \quad T = h \;\;\text{on} \;\Gamma_D, \quad {\partial T \over \partial {\mathbf n}} = g \;\;\text{on} \;\Gamma_N , \] where \(\Gamma_N\) is an open subset of \(\Gamma\) and \(\Gamma_D = \Gamma \setminus \Gamma_N .\) The controls are \(Q\) and \(g,\) and the authors study minimization of two quadratic functionals, both involving the integrals of \(|Q|^2\) and \(|g|^2.\) In the first functional, the additional term is the integral of \(|\nabla \times {\mathbf u}|^2\) (thus the objective is to minimize vorticity); in the second, the additional term is the integral of \(|T - T_d|^2\) where \(T_d\) is a target temperature field. The authors show existence of minimizers and obtain an optimality system using the method of Lagrange multipliers.
reviewed by: Hector O. Fattorini
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76B75
zbMATH DE Number: 1423630
zbMATH Keywords: optimal control
zbMATH Keywords: control of fluid flow
zbMATH Keywords: incompressible flows
zbMATH Keywords: thermally convected flows
zbMATH Keywords: existence of minimizers
zbMATH Keywords: optimality system
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6651
OpenAlex ID: W2025847446
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DOI: 10.1006/JMAA.1999.6651
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description: scientific article; zbMATH DE number 1423630