ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2011 - Volume 17, Issue 03  

Research Article

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

Casas, Eduardoa1 and Tröltzsch, Fredia2

a1 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

a2 Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany. troeltzsch@math.tu-berlin.de

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.

(Received December 1 2008)

(Revised December 20 2009)

(Online publication August 06 2010)

Key Words:

  • Quasilinear elliptic equations;
  • optimal control problems;
  • finite element approximations;
  • convergence of discretized controls

Mathematics Subject Classification:

  • 49M25;
  • 35J60;
  • 35B37;
  • 65N30

Footnotes

*  The first author was supported by Spanish Ministry of Education and Science under projects MTM2008-04206 and “Ingenio Mathematica (i-MATH)” No. CSD2006-00032 (Consolider Ingenio 2010). The second author was also supported by the DFG research center “Mathematics for key technologies” (Matheon) in Berlin.