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 : 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},
     mrnumber = {2567245},
     zbl = {1175.49027},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008049/}
}
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
PY  - 2009
SP  - 782
EP  - 809
VL  - 15
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008049/
DO  - 10.1051/cocv:2008049
LA  - en
ID  - COCV_2009__15_4_782_0
ER  - 
%0 Journal Article
%A Casas, Eduardo
%A Mateos, Mariano
%A Raymond, Jean-Pierre
%T Penalization of Dirichlet optimal control problems
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2009
%P 782-809
%V 15
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2008049/
%R 10.1051/cocv:2008049
%G en
%F 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/articles/10.1051/cocv:2008049/

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[6] E. Casas and J.-P. Raymond, The stability in W s,p (Γ) spaces of L 2 -projections on some convex sets. Numer. Funct. Anal. Optim. 27 (2006) 117-137. | MR | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

Cité par Sources :