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New regularity results and improved error estimates for optimal control problems with state constraints∗

Published online by Cambridge University Press:  05 June 2014

Eduardo Casas ,
Mariano Mateos  and
Boris Vexler
Eduardo Casas
Affiliation:
Departmento 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
Mariano Mateos
Affiliation:
Departmento de Matemáticas, E.P.I. Gijón, Universidad de Oviedo, Campus de Gijón, 33203 Gijón, Spain; mmateos@uniovi.es
Boris Vexler
Affiliation:
Center for Mathematical Sciences, Technische Universität München, Bolzmannstrasse 3, 85748 Garching b. München, Germany; vexler@ma.tum.de
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Abstract

In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence for the error in L2(Ω) of the control variable is h | log h | in dimensions 2 and 3.

Keywords

Optimal controlstate constraintselliptic equationsBorel measureserror estimates
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 803 - 822
Copyright
© EDP Sciences, SMAI 2014

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