Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 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 : https://doi.org/10.1051/cocv/2010025
Classification:  49M25,  35J60,  35B37,  65N30
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {3},
     year = {2011},
     pages = {771-800},
     doi = {10.1051/cocv/2010025},
     zbl = {1228.49033},
     mrnumber = {2826980},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 17 (2011) no. 3, pp. 771-800. doi : 10.1051/cocv/2010025. http://www.numdam.org/item/COCV_2011__17_3_771_0/

[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 1937089 | Zbl 1033.65044

[2] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York-Berlin-Heidelberg (1984). | Zbl 0804.65101

[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 1209.65129

[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 2009948 | Zbl 1119.49309

[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 2486089 | Zbl 1194.49025

[6] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0511.65078

[7] J. Douglas, Jr. and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 29 (1975) 689-696. | MR 431747 | Zbl 0306.65072

[8] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45-61. | MR 2122182 | Zbl 1074.65069

[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 1464890 | Zbl 0919.35047

[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 1275952 | Zbl 0802.65113

[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. | MR 1415252 | Zbl 0870.65096

[12] R. Rannacher and R. Scott, Some optimal error estimates for piecewise finite element approximations. Math. Comp. 38 (1982) 437-445. | MR 645661 | Zbl 0483.65007

[13] P. Raviart and J. Thomas, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles. Masson, Paris (1983). | MR 773854 | Zbl 0561.65069