Singular perturbation for the Dirichlet boundary control of elliptic problems

Ben Belgacem, Faker; El Fekih, Henda; Metoui, Hejer

ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 5, pp. 833-850.

A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small penalization parameter. Some numerical results are reported on to highlight the reliability of such an approach.

[1] D.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

[2] N. Arada, H. El Fekih and J.-P. Raymond, Asymptotic analysis of some control problems. Asymptot. Anal. 24 (2000) 343-366. | Zbl

[3] I. Babuška, The finite element method with penalty. Math. Comp. 27 (1973) 221-228. | Zbl

[4] 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. | Zbl

[5] M. Bergounioux and K. Kunisch, Augmented Lagrangian techniques for elliptic state constrained optimal control problems. SIAM J. Control Optim. 35 (1997) 1524-1543. | Zbl

[6] A. Bossavit, Approximation régularisée d'un problème aux limites non homogène. Séminaire J.-L. Lions 12 (Avril 1969). | Zbl

[7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR | Zbl

[8] P. Colli Franzoni, Approssimazione mediante il metodo de penalizazione de problemi misti di Dirichlet-Neumann per operatori lineari ellittici del secondo ordine. Boll. Un. Mat. Ital. A (7) 4 (1973) 229-250. | Zbl

[9] P. Colli Franzoni, Approximation of optimal control problems of systems described by boundary value mixed problems of Dirichlet-Neumann type, in 5th IFIP Conference on Optimization Techniques. Springer, Berlin, Lecture Notes in Computer Science 3 (1973) 152-162. | Zbl

[10] M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem. Commun. Partial Differential Equations 21 1919-1949 (1996). | Zbl

[11] M. Dauge, Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. Springer-Verlag, Lecture Notes in Math. 1341 (1988). | MR | Zbl

[12] P. Grisvard, Singularities in boundary value problems. Masson (1992). | MR | Zbl

[13] 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. Control Optim. 20 (1998) 1795-1814. | Zbl

[14] L.S. Hou and S.S. Ravindran, Numerical approximation of optimal flow control problems by a penalty method: error estimates and numerical results. SIAM J. Sci. Comput. 20 (1999) 1753-1777. | Zbl

[15] A. Kirsch, The Robin problem for the Helmholtz equation as a singular perturbation problem. Numer. Funct. Anal. Optim. 8 (1985) 1-20. | Zbl

[16] I. Lasiecka and J. Sokolowski, Semidiscrete approximation of hyperbolic boundary value problem with nonhomogeneous Dirichlet boundary conditions. SIAM J. Math. Anal. 20 (1989) 1366-1387. | Zbl

[17] J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod (1968). | MR | Zbl

[18] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vols. 1 and 2. Dunod, Paris (1968). | MR | Zbl

[19] T. Masrour, Contrôlabilité et observabilité des sytèmes distribués, problèmes et méthodes. Thesis, École Nationale des Ponts et Chaussées. Paris (1995).