We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.
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DOI : 10.1051/cocv/2019068
Mots-clés : Diffusion equation, PDE-constrained optimization, control-constraints, finite element method, error bounds
@article{COCV_2020__26_1_A78_0,
author = {Gudi, Thirupathi and Sau, Ramesh Ch.},
title = {Finite element analysis of the constrained {Dirichlet} boundary control problem governed by the diffusion problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019068},
mrnumber = {4156825},
zbl = {1460.65147},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019068/}
}
TY - JOUR AU - Gudi, Thirupathi AU - Sau, Ramesh Ch. TI - Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019068/ DO - 10.1051/cocv/2019068 LA - en ID - COCV_2020__26_1_A78_0 ER -
%0 Journal Article %A Gudi, Thirupathi %A Sau, Ramesh Ch. %T Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019068/ %R 10.1051/cocv/2019068 %G en %F COCV_2020__26_1_A78_0
Gudi, Thirupathi; Sau, Ramesh Ch. Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 78. doi : 10.1051/cocv/2019068. http://www.numdam.org/articles/10.1051/cocv/2019068/
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The authors acknowledge the support from the UGC center for Advanced Study-II. The first author also thank the support from DST MATRICS Grant.
