A duality-based approach to elliptic control problems in non-reflexive Banach spaces
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 243-266.

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.

DOI : 10.1051/cocv/2010003
Classification : 49J52, 49J20, 49K20
Mots-clés : optimal control, L1, bounded variation (BV), measures, Fenchel duality, semismooth Newton 
@article{COCV_2011__17_1_243_0,
     author = {Clason, Christian and Kunisch, Karl},
     title = {A duality-based approach to elliptic control problems in non-reflexive {Banach} spaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {243--266},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     doi = {10.1051/cocv/2010003},
     mrnumber = {2775195},
     zbl = {1213.49041},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2010003/}
}
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
SP  - 243
EP  - 266
VL  - 17
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2010003/
DO  - 10.1051/cocv/2010003
LA  - en
ID  - COCV_2011__17_1_243_0
ER  - 
%0 Journal Article
%A Clason, Christian
%A Kunisch, Karl
%T A duality-based approach to elliptic control problems in non-reflexive Banach spaces
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 243-266
%V 17
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2010003/
%R 10.1051/cocv/2010003
%G en
%F COCV_2011__17_1_243_0
Clason, Christian; Kunisch, Karl. A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 243-266. doi : 10.1051/cocv/2010003. http://www.numdam.org/articles/10.1051/cocv/2010003/

[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). | MR | Zbl

[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. | Zbl

[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). | MR | Zbl

[4] H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, France (1983). | MR | Zbl

[5] G. Chavent and K. Kunisch, Regularization of linear least squares problems by total bounded variation. ESAIM: COCV 2 (1997) 359-376. | Numdam | MR | Zbl

[6] I. Ekeland and R. Témam, Convex analysis and variational problems. Society for Industrial and Applied Mathematics, Philadelphia, USA (1999). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[10] W. Ring, Structural properties of solutions to total variation regularization problems. ESAIM: M2AN 34 (2000) 799-810. | EuDML | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[13] R. Témam, Navier-Stokes equations. AMS Chelsea Publishing, Providence, USA (2001). | Zbl

[14] M. Ulbrich, Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13 (2002) 805-842. | MR | Zbl

[15] G. Vossen and H. Maurer, On L1-minimization in optimal control and applications to robotics. Optimal Control Appl. Methods 27 (2006) 301-321. | MR

Cité par Sources :