Dirichlet control problems in smooth and nonsmooth convex plain domains

• Autorzy: Casas E. , Mateos M.
• Języki publikacji: EN

Abstrakty

EN

In this paper we collect some results about boundary Dirichlet control problems governed by linear partial differential equations. Some differences are found between problems posed on polygonal domains or smooth domains. In polygonal domains some difficulties arise in the corners, where the optimal control is forced to take a value which is independent of the data of the problem. The use of some Sobolev norm of the control in the cost functional, as suggested in the specialized literature as an alternative to the L² norm, allows to avoid this strange behavior. Here, we propose a new method to avoid this undesirable behavior of the optimal control, consisting in considering a discrete perturbation of the cost functional by using a finite number of controls concentrated around the corners. In curved domains, the numerical approximation of the problem requires the approximation of the domain Ω typically by a polygonal domain Ω_h, this introduces some difficulties in comparing the continuous and the discrete controls because of their definition on different domains Γ and Γ_h, respectively. We complete the existing recent analysis of these problems by proving the error estimates for a full discretization of the control problem. Finally, some numerical results are provided to compare the different alternatives and to confirm the theoretical predictions.

Słowa kluczowe

EN

optimal control; boundary control; Dirichlet control

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2011
• Tom: Vol. 40, no 4
• Strony: 931--955
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Casas E.

autor

Mateos M.

• Departamento de Matemática Aplicada y Ciencias de la Computación E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria 39005 Santander, Spain,

Bibliografia

• Apel,T., Pfefferer, J. and Rösch,A. (2011) Finite element error estimates for Neumann boundary control problems on graded meshes. Comput. Optim. Appl. Online First, DOI 10.1007/510589-011-9427x.
• Bergounioux, M., Ito, K. and Kunisch, K. (1999) Primal-dual strategy for constrained optimal control problems. SIAM J. Control and Optim., 37, 1176-1194.
• Bramble, J.H. and King, J.T. (1994) A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries. Math. Comp. 63 (207), 1-17.
• Bramble, J.H., Pasciak, J.E. and Schatz, A.H. (1986) The construction of preconditioners for elliptic problems by substructuring. I. Math. Comp. 47 (175), 103-134.
• Brenner, S.C. and Scott, L.R. (1994) The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 15. Springer-Verlag, New York.
• Casas, E., Günther, A. and Mateos, M. (2011) A Paradox in the Approximation of Dirichlet Control Problems in Curved Domains. SIAM J. Control Optim., 49(5), 1998-2007.
• Casas, E. and Mateos, M. (2002) Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (5), 1431-1454 (electronic).
• Casas, E., Mateos, M. and Raymond, J.P. (2007) Error estimates for the numerical approximation of a distributed control problem for the steadystate Navier-Stokes equations. SIAM J. Control Optim., 46, 952-982
• Casas, E., Mateos, M. and Tröltzsch, F. (2005) Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31, 193-219.
• Casas, E. and Raymond, J.P. (2006) Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim., 45, 1586-1611.
• Casas, E. and Sokolowski, J. (2010) Approximation of boundary control problems on curved domains. SIAM J. Control Optim., 48, 3746-3780.
• Deckelnick, K., Günther, A. and Hinze, M. (2009) Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and Tyree-dimensional curved domains. SIAM J. Control Optim., 48, 2798-2819.
• Grisvard,P. (1985) Elliptic Problems in Nonsmooth Domains. Pitman, Boston.
• Gunzburger, M., Hou, L. and Svobodny, T. (1991) Analysis and finie element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. RAIRO Modél. Math. Anal. Numér., 25, 711-748.
• Kunisch, K. and Vexler, B. (2007) Constrained Dirichlet boundary control in L2 for a class of evolution equations. SIAM J. Control Optim., 46, 1726-1753.
• Mateos, M. and Rösch, A. (2009) On saturation effects in the Neumann boundary control of elliptic optimal control problems. Comput. Optim. Appl. 49, 359-378.
• May, S., Rannacher, R. and Vexler, B. (2008) Error analysis for a finie element approximation of elliptic Dirichlet boundary control problems. Preprint 05/2008, Lehrstuhl für Numerische Mathematik, Universtät Heidelberg, Germany, http://ganymed.iwr.uniheidelberg.de/Paper/may-rannacher-vexler-2008.pdf
• Nečas, J. (1967) Les Méthodes Directes en Théorie des Equations Elliptiques. Editeurs Academia, Prague.
• Vexler, B. (2007) Finite element approximation of elliptic Dirichlet optima control problems. Numer. Funct. Anal. Optim. 28, 957-973.