Penalization of Dirichlet optimal control problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 782-809.

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.

DOI : https://doi.org/10.1051/cocv:2008049
Classification : 49M30,  35B30,  35B37
Mots clés : Dirichlet optimal control, Robin penalization, regularity of solutions
@article{COCV_2009__15_4_782_0,
author = {Casas, Eduardo and Mateos, Mariano and Raymond, Jean-Pierre},
title = {Penalization of Dirichlet optimal control problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {782--809},
publisher = {EDP-Sciences},
volume = {15},
number = {4},
year = {2009},
doi = {10.1051/cocv:2008049},
zbl = {1175.49027},
mrnumber = {2567245},
language = {en},
url = {http://www.numdam.org/item/COCV_2009__15_4_782_0/}
}
Casas, Eduardo; Mateos, Mariano; Raymond, Jean-Pierre. Penalization of Dirichlet optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 782-809. doi : 10.1051/cocv:2008049. http://www.numdam.org/item/COCV_2009__15_4_782_0/

[1] 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. | MR 1448889 | Zbl 0885.49010

[2] F. Ben Belgacem, H. El Fekih and H. Metoui, Singular perturbation for the Dirichlet boundary control of elliptic problems. ESAIM: M2AN 37 (2003) 833-850. | Numdam | MR 2020866 | Zbl 1051.49012

[3] 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. | MR 1992281 | Zbl 1043.35014

[4] E. Casas and M. Mateos, Error estimates for the numerical approximation of Neumann control problems. Comput. Optim. Appl. 39 (2008) 265-295. | MR 2396868 | Zbl 1191.49030

[5] 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). | MR 2272157 | Zbl 1123.65061

[6] 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. | MR 2212288 | Zbl 1112.65064

[7] 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. | Numdam | MR 2150243 | Zbl 1081.49023

[8] P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis II, North-Holland, Amsterdam (1991) 17-351. | MR 1115237 | Zbl 0875.65086

[9] M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem. Comm. Partial Diff. Eq. 21 (1996) 1919-1949. | MR 1421215 | Zbl 0879.35017

[10] Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Amer. Math. Soc. 124 (1996) 591-600. | MR 1301021 | Zbl 0841.46021

[11] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR 775683 | Zbl 0695.35060

[12] 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). | MR 1632548 | Zbl 0917.49003

[13] D. Jerison and C. Kenig, The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. (N.S.) 4 (1981) 203-207. | MR 598688 | Zbl 0471.35026

[14] D. Jerison and C.E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161-219. | MR 1331981 | Zbl 0832.35034

[15] C.V. Pao, Nonlinear parabolic and elliptic equations. Plenum Press, New York (1992). | MR 1212084 | Zbl 0777.35001

[16] J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 921-951. | Numdam | MR 2371113 | Zbl 1136.35070

[17] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | MR 192177 | Zbl 0151.15401