Skip to main content Accessibility help
×
Home
Hostname: page-component-55b6f6c457-xdj6x Total loading time: 0.202 Render date: 2021-09-26T09:00:05.581Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

A duality-based approach to elliptic control problems in non-reflexive Banach spaces*

Published online by Cambridge University Press:  24 March 2010

Christian Clason
Affiliation:
Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria. christian.clason@uni-graz.at; karl.kunisch@uni-graz.at
Karl Kunisch
Affiliation:
Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria. christian.clason@uni-graz.at; karl.kunisch@uni-graz.at
Get access

Abstract

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with measures and functions of bounded variation as controls are considered. Existence and uniqueness of the corresponding predual problems are discussed, as is the solution of the optimality systems by a semismooth Newton method. Numerical examples illustrate the structural differences in the optimal controls in these Banach spaces, compared to those obtained in corresponding Hilbert space settings.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. Oxford University Press, New York, USA (2000).
Amrouche, C., Ciarlet, P.G. and Ciarlet, P., Vector, Jr. and scalar potentials, Poincaré's theorem and Korn's inequality. C. R. Math. Acad. Sci. Paris 345 (2007) 603608. CrossRef
H. Attouch, G. Buttazzo and G. Michaille, Variational analysis in Sobolev and BV spaces, MPS/SIAM Series on Optimization 6. Society for Industrial and Applied Mathematics, Philadelphia, USA (2006).
H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, France (1983).
Chavent, G. and Kunisch, K., Regularization of linear least squares problems by total bounded variation. ESAIM: COCV 2 (1997) 359376. CrossRef
I. Ekeland and R. Témam, Convex analysis and variational problems. Society for Industrial and Applied Mathematics, Philadelphia, USA (1999).
Hintermüller, M. and Stadler, G., An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28 (2006) 123. CrossRef
Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2002) 865888. CrossRef
K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications, Advances in Design and Control 15. Society for Industrial and Applied Mathematics, Philadelphia, USA (2008).
Ring, W., Structural properties of solutions to total variation regularization problems. ESAIM: M2AN 34 (2000) 799810. CrossRef
Stadler, G., Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comp. Optim. Appl. 44 (2009) 159181. CrossRef
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189258. CrossRef
R. Témam, Navier-Stokes equations. AMS Chelsea Publishing, Providence, USA (2001).
Ulbrich, M., Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13 (2002) 805842. CrossRef
Vossen, G. and Maurer, H., On L1-minimization in optimal control and applications to robotics. Optimal Control Appl. Methods 27 (2006) 301321. CrossRef

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A duality-based approach to elliptic control problems in non-reflexive Banach spaces*
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

A duality-based approach to elliptic control problems in non-reflexive Banach spaces*
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

A duality-based approach to elliptic control problems in non-reflexive Banach spaces*
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *