Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 771-800.

In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness of a solution for the discrete equation is an open problem.

DOI : 10.1051/cocv/2010025
Classification : 49M25, 35J60, 35B37, 65N30
Mots-clés : quasilinear elliptic equations, optimal control problems, finite element approximations, convergence of discretized controls 
@article{COCV_2011__17_3_771_0,
     author = {Casas, Eduardo and Tr\"oltzsch, Fredi},
     title = {Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {771--800},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {3},
     year = {2011},
     doi = {10.1051/cocv/2010025},
     mrnumber = {2826980},
     zbl = {1228.49033},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2010025/}
}
TY  - JOUR
AU  - Casas, Eduardo
AU  - Tröltzsch, Fredi
TI  - Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2011
SP  - 771
EP  - 800
VL  - 17
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2010025/
DO  - 10.1051/cocv/2010025
LA  - en
ID  - COCV_2011__17_3_771_0
ER  - 
%0 Journal Article
%A Casas, Eduardo
%A Tröltzsch, Fredi
%T Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 771-800
%V 17
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2010025/
%R 10.1051/cocv/2010025
%G en
%F COCV_2011__17_3_771_0
Casas, Eduardo; Tröltzsch, Fredi. Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 771-800. doi : 10.1051/cocv/2010025. http://archive.numdam.org/articles/10.1051/cocv/2010025/

[1] N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23 (2002) 201-229. | MR | Zbl

[2] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York-Berlin-Heidelberg (1984). | Zbl

[3] E. Casas and V. Dhamo, Error estimates for the numerical approximation of a quasilinear Neumann problem under minimal regularity of the data. (Submitted). | Zbl

[4] E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21 (2007) 67-100. | MR | Zbl

[5] E. Casas and F. Tröltzsch, Optimality conditions for a class of optimal control problems with quasilinear elliptic equations. SIAM J. Control Optim. 48 (2009) 688-718. | MR | Zbl

[6] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl

[7] J. Douglas, Jr. and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 29 (1975) 689-696. | MR | Zbl

[8] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45-61. | MR | Zbl

[9] I. Hlaváček, 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. | MR | Zbl

[10] I. Hlaváček, M. Křížek and J. Malý, On Galerkin approximations of quasilinear nonpotential elliptic problem of a nonmonotone type. J. Math. Anal. Appl. 184 (1994) 168-189. | MR | Zbl

[11] L. Liu, M. Křížek and P. Neittaanmäki, Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type. Appl. Math. 41 (1996) 467-478. | EuDML | MR | Zbl

[12] R. Rannacher and R. Scott, Some optimal error estimates for piecewise finite element approximations. Math. Comp. 38 (1982) 437-445. | MR | Zbl

[13] P. Raviart and J. Thomas, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles. Masson, Paris (1983). | MR | Zbl

Cité par Sources :