Source representation strategy for optimal boundary control problems with state constraints
type: Item
instance of: scholarly article
title: Source representation strategy for optimal boundary control problems with state constraints
zbMATH Open document ID: 1159.49032
DOI: 10.4171/ZAA/1380
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2009-05-05T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: Summary: A state-constrained optimal boundary control problem governed by a linear elliptic equation is considered. In order to obtain the optimality conditions for the solutions to the model problem, a Slater assumption has to be made that restricts the theory to the two-dimensional case. This difficulty is overcome by a source representation of the control and combined with a Lavrentiev type regularization. Optimality conditions for the regularized problem are derived, where the corresponding Lagrange multipliers have \(L^2\)-regularity. By the spectral theorem for compact and normal operators, the convergence result of \textit{F. Tröltzsch} and \textit{I. Yousept} in [Comput. Optim. Appl. 42, 43--66 (2009; Zbl 1153.65343)] is extended to a higher dimensional case. Moreover, the convergence for vanishing regularization parameter of the adjoint state associated with the regularized problem is shown. Finally, the uniform boundedness of the regularized Lagrange multipliers in \(L^1(\Omega)\) is verified by a maximum principle argument.
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 49M05
Mathematics Subject Classification ID: 65K10
zbMATH DE Number: 5550894
zbMATH Keywords: optimal boundary control problem
zbMATH Keywords: pointwise state constraints
zbMATH Keywords: elliptic equations
zbMATH Keywords: source representation
zbMATH Keywords: Lavrentiev type regularization
zbMATH Keywords: spectral theorem
author: Fredi Tröltzsch
author: Irwin Yousept
MaRDI profile type: Publication
cites work: Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls
cites work: Primal-Dual Strategy for Constrained Optimal Control Problems
cites work: On the structure of Lagrange multipliers for state-constrained optimal control problems
cites work: Primal-dual strategy for state-constrained optimal control problems
cites work: Control of an Elliptic Problem with Pointwise State Constraints
cites work: Boundary Control of Semilinear Elliptic Equations with Pointwise State Constraints
cites work: Q3470840
cites work: The Primal-Dual Active Set Strategy as a Semismooth Newton Method
cites work: Feasible and Noninterior Path‐Following in Constrained Minimization with Low Multiplier Regularity
cites work: Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems
cites work: Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces
cites work: Semi-smooth Newton methods for state-constrained optimal control problems
cites work: The Primal-Dual Active Set Method for Nonlinear Optimal Control Problems with Bilateral Constraints
cites work: Q3905599
cites work: A virtual control concept for state constrained optimal control problems
cites work: Q5529214
cites work: Q5566063
cites work: On two numerical methods for state-constrained elliptic control problems
cites work: Optimal control of PDEs with regularized pointwise state constraints
cites work: Q3094428
cites work: Q5311324
cites work: A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints
OpenAlex ID: W2066281568
description: scientific article; zbMATH DE number 5550894