Analysis and approximation of optimal control problems for first-order elliptic systems in three dimensions
type: Item
instance of: scholarly article
title: Analysis and approximation of optimal control problems for first-order elliptic systems in three dimensions
zbMATH Open document ID: 0928.49023
DOI: 10.1016/S0096-3003(98)00017-4
author: Hyung-Chun Lee
author: Max Gunzburger
published in: Applied Mathematics and Computation
publication date: 1999-12-15T00:00:00Z
review text: The problem consists in minimizing the functionals \[ \begin{aligned}{{\mathcal J}_f}({\mathbf u}, f) &= {1 \over 2} \int_\Omega | {\mathbf u} - {\mathbf U}| ^2 dx + {\delta \over 2} \int_\Omega | f| ^2 dx,\\ {{\mathcal J}_{\mathbf g}}({\mathbf u}, {\mathbf g}) &= {1 \over 2} \int_\Omega | {\mathbf u} - {\mathbf U}| ^2 dx + {\delta \over 2} \int_\Omega | {\mathbf g}| ^2 dx\end{aligned} \] in a 3-dimensional domain \(\Omega,\) the scalar function \(f\) and the vector functions \({\mathbf u}, {\mathbf g}\) satisfying \[ \nabla \cdot {\mathbf u} = f, \quad \nabla \times {\mathbf u} = {\mathbf g} \qquad \text{in} \;\Omega \] with either of the boundary conditions \({\mathbf u} \cdot {\mathbf n} = 0\) or \({\mathbf u} \times {\mathbf n} = 0\) on the boundary \(\Gamma.\) There are in addition compatibility conditions for \(f\) and \({\mathbf g}\) depending on the boundary condition used; in optimal control problems, these compatibility conditions play the role of constraints on the control variables. The authors show existence and uniqueness of solutions to the minimization problem and existence of Lagrange multipliers, and derive a method to compute optimal controls and states from an \textit{optimality system} of partial differential equations. The solution of this system is approximated with finite elements.
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 35J45
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49M15
zbMATH DE Number: 1311093
zbMATH Keywords: steady state optimal control problems
zbMATH Keywords: elliptic systems
zbMATH Keywords: finite elements
reviewed by: Hector O. Fattorini
MaRDI profile type: Publication
cites work: Q4073917
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description: scientific article; zbMATH DE number 1311093