Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T21:56:49.438Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  24 March 2010

Christian Clason  and
Karl Kunisch
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.

Keywords

Optimal controlL1bounded variation (BV)measuresFenchel dualitysemismooth Newton
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 243 - 266
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) 603–608. 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) 359–376. 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) 1–23. CrossRef
Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2002) 865–888. 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) 799–810. CrossRef
Stadler, G., Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comp. Optim. Appl. 44 (2009) 159–181. CrossRef
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. 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) 805–842. CrossRef
Vossen, G. and Maurer, H., On L1-minimization in optimal control and applications to robotics. Optimal Control Appl. Methods 27 (2006) 301–321. CrossRef