# Penalization of Dirichlet optimal control problems

Eduardo Casas; Mariano Mateos; Jean-Pierre Raymond

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 15, Issue: 4, page 782-809
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topCasas, Eduardo, Mateos, Mariano, and Raymond, Jean-Pierre. "Penalization of Dirichlet optimal control problems." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2008): 782-809. <http://eudml.org/doc/90937>.

@article{Casas2008,

abstract = {
We apply Robin penalization to Dirichlet optimal control problems
governed by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.
},

author = {Casas, Eduardo, Mateos, Mariano, Raymond, Jean-Pierre},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Dirichlet optimal control; Robin penalization; regularity of solutions},

language = {eng},

month = {8},

number = {4},

pages = {782-809},

publisher = {EDP Sciences},

title = {Penalization of Dirichlet optimal control problems},

url = {http://eudml.org/doc/90937},

volume = {15},

year = {2008},

}

TY - JOUR

AU - Casas, Eduardo

AU - Mateos, Mariano

AU - Raymond, Jean-Pierre

TI - Penalization of Dirichlet optimal control problems

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/8//

PB - EDP Sciences

VL - 15

IS - 4

SP - 782

EP - 809

AB -
We apply Robin penalization to Dirichlet optimal control problems
governed by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.

LA - eng

KW - Dirichlet optimal control; Robin penalization; regularity of solutions

UR - http://eudml.org/doc/90937

ER -

## References

top- J.-J. Alibert and J.-P. Raymond, Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim.18 (1997) 235–250.
- F. Ben Belgacem, H. El Fekih and H. Metoui, Singular perturbation for the Dirichlet boundary control of elliptic problems. ESAIM: M2AN37 (2003) 833–850.
- F. Ben Belgacem, H. El Fekih and J.-P. Raymond, A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions. Asymptot. Anal.34 (2003) 121–136.
- E. Casas and M. Mateos, Error estimates for the numerical approximation of Neumann control problems. Comput. Optim. Appl.39 (2008) 265–295.
- E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Contr. Opt.45 (2006) 1586–1611 (electronic).
- E. Casas and J.-P. Raymond, The stability in ${W}^{s,p}\left(\Gamma \right)$ spaces of ${L}^{2}$-projections on some convex sets. Numer. Funct. Anal. Optim.27 (2006) 117–137.
- E. Casas, M. Mateos and F. Tröltzsch, Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl.31 (2005) 193–219.
- P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical AnalysisII, North-Holland, Amsterdam (1991) 17–351.
- M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem. Comm. Partial Diff. Eq.21 (1996) 1919–1949.
- Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Amer. Math. Soc.124 (1996) 591–600.
- P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985).
- L.S. Hou and S.S. Ravindran, A penalized Neumann control approach for solving an optimal Dirichlet control problem for the Navier-Stokes equations. SIAM J. Contr. Opt.36 (1998) 1795–1814 (electronic).
- D. Jerison and C. Kenig, The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. (N.S.)4 (1981) 203–207.
- D. Jerison and C.E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal.130 (1995) 161–219.
- C.V. Pao, Nonlinear parabolic and elliptic equations. Plenum Press, New York (1992).
- J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire24 (2007) 921–951.
- G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble)15 (1965) 189–258.

## NotesEmbed ?

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