Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Eduardo Casas; Fredi Tröltzsch

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 3, page 771-800
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Casas, Eduardo, and Tröltzsch, Fredi. "Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 771-800. <http://eudml.org/doc/272858>.

@article{Casas2011,
abstract = {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.},
author = {Casas, Eduardo, Tröltzsch, Fredi},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {quasilinear elliptic equations; optimal control problems; finite element approximations; convergence of discretized controls},
language = {eng},
number = {3},
pages = {771-800},
publisher = {EDP-Sciences},
title = {Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations},
url = {http://eudml.org/doc/272858},
volume = {17},
year = {2011},
}

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
PB - EDP-Sciences
VL - 17
IS - 3
SP - 771
EP - 800
AB - 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.
LA - eng
KW - quasilinear elliptic equations; optimal control problems; finite element approximations; convergence of discretized controls
UR - http://eudml.org/doc/272858
ER -

References

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  1. [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. Zbl1033.65044MR1937089
  2. [2] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York-Berlin-Heidelberg (1984). Zbl0804.65101
  3. [3] E. Casas and V. Dhamo, Error estimates for the numerical approximation of a quasilinear Neumann problem under minimal regularity of the data. (Submitted). Zbl1209.65129
  4. [4] E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21 (2007) 67–100. Zbl1119.49309MR2009948
  5. [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. Zbl1194.49025MR2486089
  6. [6] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). Zbl0511.65078MR520174
  7. [7] J. Douglas, Jr. and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem. Math. Comp.29 (1975) 689–696. Zbl0306.65072MR431747
  8. [8] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl.30 (2005) 45–61. Zbl1074.65069MR2122182
  9. [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. Zbl0919.35047MR1464890
  10. [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. Zbl0802.65113
  11. [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. Zbl0870.65096
  12. [12] R. Rannacher and R. Scott, Some optimal error estimates for piecewise finite element approximations. Math. Comp.38 (1982) 437–445. Zbl0483.65007MR645661
  13. [13] P. Raviart and J. Thomas, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles. Masson, Paris (1983). Zbl0561.65069

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