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

¹ 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
² Departmento de Matemáticas, E.P.I. Gijón, Universidad de Oviedo, Campus de Gijón, 33203 Gijón, Spain
mmateos@uniovi.es
³ Center for Mathematical Sciences, Technische Universität München, Bolzmannstrasse 3, 85748 Garching b. München, Germany
vexler@ma.tum.de

Received: 31 May 2013
Revised: 25 November 2013

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.