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.

Classification: 49K20, 49M05, 49M25, 65N30, 65N15

Keywords: optimal control, state constraints, elliptic equations, Borel measures, error estimates

@article{COCV_2014__20_3_803_0, author = {Casas, Eduardo and Mateos, Mariano and Vexler, Boris}, title = {New regularity results and improved error estimates for optimal control problems with state constraints}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {20}, number = {3}, year = {2014}, pages = {803-822}, doi = {10.1051/cocv/2013084}, zbl = {1293.49044}, mrnumber = {3264224}, language = {en}, url = {http://www.numdam.org/item/COCV_2014__20_3_803_0} }

Casas, Eduardo; Mateos, Mariano; Vexler, Boris. New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, pp. 803-822. doi : 10.1051/cocv/2013084. http://www.numdam.org/item/COCV_2014__20_3_803_0/

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