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Numerical analysis of some optimal control problemsgoverned by a class of quasilinear elliptic equations*

Published online by Cambridge University Press:  06 August 2010

Eduardo Casas  and
Fredi Tröltzsch
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. eduardo.casas@unican.es
Fredi Tröltzsch
Affiliation:
Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany. troeltzsch@math.tu-berlin.de
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Abstract

In this paper, we carry out the numerical analysis of adistributed optimal control problem governed by a quasilinearelliptic equation of non-monotone type. The goal is to prove thestrong convergence of the discretization of the problem by finiteelements. The main issue is to get error estimates for thediscretization of the state equation. One of the difficulties inthis analysis is that, in spite of the partial differentialequation has a unique solution for any given control, theuniqueness of a solution for the discrete equation is an openproblem.

Keywords

Quasilinearelliptic equationsoptimal control problemsfinite elementapproximationsconvergence of discretized controls
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 771 - 800
Copyright
© EDP Sciences, SMAI, 2010

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References

Arada, N., Casas, E. and Tröltzsch, F., Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23 (2002) 201–229. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York-Berlin-Heidelberg (1984).
E. Casas and V. Dhamo, Error estimates for the numerical approximation of a quasilinear Neumann problem under minimal regularity of the data. (Submitted).
Casas, E. and Mateos, M., Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21 (2007) 67–100.
Casas, E. and Tröltzsch, F., Optimality conditions for a class of optimal control problems with quasilinear elliptic equations. SIAM J. Control Optim. 48 (2009) 688–718. CrossRef
P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
Douglas, J., Jr. and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 29 (1975) 689–696. CrossRef
Hinze, M., A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45–61. CrossRef
Hlaváček, I., Reliable solution of a quasilinear nonpotential elliptic problem of a nonmonotone type with respect to the uncertainty in coefficients. J. Math. Anal. Appl. 212 (1997) 452–466. CrossRef
Hlaváček, I., Křížek, M. and Malý, J., Galerkin, On approximations of quasilinear nonpotential elliptic problem of a nonmonotone type. J. Math. Anal. Appl. 184 (1994) 168–189. CrossRef
Liu, L., Křížek, M. and Neittaanmäki, P., Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type. Appl. Math. 41 (1996) 467–478.
Rannacher, R. and Scott, R., Some optimal error estimates for piecewise finite element approximations. Math. Comp. 38 (1982) 437–445. CrossRef
P. Raviart and J. Thomas, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles. Masson, Paris (1983).