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.
Keywords: 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://archive.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://archive.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://archive.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, Volume 15 (2009) no. 4, pp. 782-809. doi : 10.1051/cocv:2008049. http://archive.numdam.org/articles/10.1051/cocv:2008049/
[1] Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 18 (1997) 235-250. | MR | Zbl
and ,[2] Singular perturbation for the Dirichlet boundary control of elliptic problems. ESAIM: M2AN 37 (2003) 833-850. | Numdam | MR | Zbl
, and ,[3] A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions. Asymptot. Anal. 34 (2003) 121-136. | MR | Zbl
, and ,[4] Error estimates for the numerical approximation of Neumann control problems. Comput. Optim. Appl. 39 (2008) 265-295. | MR | Zbl
and ,[5] 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
and ,[6] The stability in spaces of -projections on some convex sets. Numer. Funct. Anal. Optim. 27 (2006) 117-137. | MR | Zbl
and ,[7] Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31 (2005) 193-219. | Numdam | MR | Zbl
, and ,[8] Basic error estimates for elliptic problems, in Handbook of Numerical Analysis II, North-Holland, Amsterdam (1991) 17-351. | MR | Zbl
,[9] A singularly perturbed mixed boundary value problem. Comm. Partial Diff. Eq. 21 (1996) 1919-1949. | MR | Zbl
and ,[10] A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Amer. Math. Soc. 124 (1996) 591-600. | MR | Zbl
,[11] Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR | Zbl
,[12] 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
and ,[13] The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. (N.S.) 4 (1981) 203-207. | MR | Zbl
and ,[14] The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161-219. | MR | Zbl
and ,[15] Nonlinear parabolic and elliptic equations. Plenum Press, New York (1992). | MR | Zbl
,[16] Stokes and Navier-Stokes equations with nonhomogeneous conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 921-951. | Numdam | MR | Zbl
,[17] 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
,Cited by Sources: