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

Christian Clason; Karl Kunisch

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 243-266
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Clason, Christian, and Kunisch, Karl. "A duality-based approach to elliptic control problems in non-reflexive Banach spaces." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 243-266. <http://eudml.org/doc/272853>.

@article{Clason2011,
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.},
author = {Clason, Christian, Kunisch, Karl},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal control; L1; bounded variation (BV); measures; Fenchel duality; semismooth Newton; ; semismooth Newton method},
language = {eng},
number = {1},
pages = {243-266},
publisher = {EDP-Sciences},
title = {A duality-based approach to elliptic control problems in non-reflexive Banach spaces},
url = {http://eudml.org/doc/272853},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Clason, Christian
AU - Kunisch, Karl
TI - A duality-based approach to elliptic control problems in non-reflexive Banach spaces
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 1
SP - 243
EP - 266
AB - 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.
LA - eng
KW - optimal control; L1; bounded variation (BV); measures; Fenchel duality; semismooth Newton; ; semismooth Newton method
UR - http://eudml.org/doc/272853
ER -

References

top
  1. [1] 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). Zbl0957.49001MR1857292
  2. [2] C. Amrouche, P.G. Ciarlet and P. Ciarlet, Jr., Vector and scalar potentials, Poincaré's theorem and Korn's inequality. C. R. Math. Acad. Sci. Paris 345 (2007) 603–608. Zbl1135.35007
  3. [3] 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). Zbl1095.49001MR2192832
  4. [4] H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, France (1983). Zbl0511.46001MR697382
  5. [5] G. Chavent and K. Kunisch, Regularization of linear least squares problems by total bounded variation. ESAIM: COCV 2 (1997) 359–376. Zbl0890.49010MR1483764
  6. [6] I. Ekeland and R. Témam, Convex analysis and variational problems. Society for Industrial and Applied Mathematics, Philadelphia, USA (1999). Zbl0939.49002MR1727362
  7. [7] M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration. SIAM J. Sci. Comput.28 (2006) 1–23. Zbl1136.94302MR2219285
  8. [8] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim.13 (2002) 865–888. Zbl1080.90074MR1972219
  9. [9] 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). Zbl1156.49002MR2441683
  10. [10] W. Ring, Structural properties of solutions to total variation regularization problems. ESAIM: M2AN 34 (2000) 799–810. Zbl1018.49021MR1784486
  11. [11] G. Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comp. Optim. Appl.44 (2009) 159–181. Zbl1185.49031MR2556849
  12. [12] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. Zbl0151.15401MR192177
  13. [13] R. Témam, Navier-Stokes equations. AMS Chelsea Publishing, Providence, USA (2001). Zbl0981.35001
  14. [14] M. Ulbrich, Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim.13 (2002) 805–842. Zbl1033.49039MR1972217
  15. [15] G. Vossen and H. Maurer, On L1-minimization in optimal control and applications to robotics. Optimal Control Appl. Methods27 (2006) 301–321. MR2283487

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.