Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-18T09:44:54.334Z Has data issue: false hasContentIssue false
  • English
  • Français

Penalization of Dirichlet optimal control problems

Published online by Cambridge University Press:  20 August 2008

Eduardo Casas ,
Mariano Mateos  and
Jean-Pierre Raymond
Eduardo Casas
Affiliation:
Dpto. de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain. [email protected]
Mariano Mateos
Affiliation:
Dpto. de Matemáticas, Universidad de Oviedo, Campus de Viesques, 33203 Gijón, Spain. [email protected]
Jean-Pierre Raymond
Affiliation:
Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier, 31062 Toulouse Cedex 9, France. [email protected]
Get access

Abstract

We apply Robin penalization to Dirichlet optimal control problemsgoverned 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 controlRobin penalizationregularity of solutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 4 , October 2009 , pp. 782 - 809
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alibert, J.-J. and Raymond, J.-P., Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 18 (1997) 235–250. CrossRef
Ben Belgacem, F., El Fekih, H. and Metoui, H., Singular perturbation for the Dirichlet boundary control of elliptic problems. ESAIM: M2AN 37 (2003) 833–850. CrossRef
Ben Belgacem, F., El Fekih, H. and Raymond, J.-P., A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions. Asymptot. Anal. 34 (2003) 121–136.
Casas, E. and Mateos, M., Error estimates for the numerical approximation of Neumann control problems. Comput. Optim. Appl. 39 (2008) 265–295. CrossRef
Casas, E. and Raymond, J.-P., Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Contr. Opt. 45 (2006) 1586–1611 (electronic). CrossRef
Casas, E. and Raymond, J.-P., The stability in $W\sp {s,p}(\Gamma)$ spaces of $L\sp 2$ -projections on some convex sets. Numer. Funct. Anal. Optim. 27 (2006) 117–137. CrossRef
Casas, E., Mateos, M. and Tröltzsch, F., Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31 (2005) 193–219. CrossRef
P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis II, North-Holland, Amsterdam (1991) 17–351.
Costabel, M. and Dauge, M., A singularly perturbed mixed boundary value problem. Comm. Partial Diff. Eq. 21 (1996) 1919–1949.
Ding, Z., A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Amer. Math. Soc. 124 (1996) 591–600. CrossRef
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985).
Hou, L.S. and Ravindran, S.S., 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). CrossRef
Jerison, D. and Kenig, C., The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. (N.S.) 4 (1981) 203–207. CrossRef
Jerison, D. and Kenig, C.E., The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161–219. CrossRef
C.V. Pao, Nonlinear parabolic and elliptic equations. Plenum Press, New York (1992).
Raymond, J.-P., Stokes and Navier-Stokes equations with nonhomogeneous conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 921–951. CrossRef
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. CrossRef