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

top
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

top

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/221895>.

@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; quasilinear elliptic equations; finite element approximations},
language = {eng},
month = {8},
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/221895},
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
DA - 2011/8//
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; quasilinear elliptic equations; finite element approximations
UR - http://eudml.org/doc/221895
ER -

References

top
  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.  
  2. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York-Berlin-Heidelberg (1984).  
  3. E. Casas and V. Dhamo, Error estimates for the numerical approximation of a quasilinear Neumann problem under minimal regularity of the data. (Submitted).  
  4. E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math.21 (2007) 67–100.  
  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.  
  6. P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).  
  7. J. Douglas, Jr. and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem. Math. Comp.29 (1975) 689–696.  
  8. M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl.30 (2005) 45–61.  
  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.  
  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.  
  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.  
  12. R. Rannacher and R. Scott, Some optimal error estimates for piecewise finite element approximations. Math. Comp.38 (1982) 437–445.  
  13. P. Raviart and J. Thomas, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles. Masson, Paris (1983).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.