We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to confirm the validity of our approach.

Keywords: optimal control, reduced basis method, a posteriori error estimation, model order reduction, parameter-dependent systems, partial differential equations, parabolic problems

@article{M2AN_2014__48_6_1615_0, author = {K\"archer, Mark and Grepl, Martin A.}, title = {\protect\emph{A {Posteriori} {}Error} {Estimation} for {Reduced} {Order} {Solutions} of {Parametrized} {Parabolic} {Optimal} {Control} {Problems}}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1615--1638}, publisher = {EDP-Sciences}, volume = {48}, number = {6}, year = {2014}, doi = {10.1051/m2an/2014012}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2014012/} }

TY - JOUR AU - Kärcher, Mark AU - Grepl, Martin A. TI - A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 1615 EP - 1638 VL - 48 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2014012/ DO - 10.1051/m2an/2014012 LA - en ID - M2AN_2014__48_6_1615_0 ER -

%0 Journal Article %A Kärcher, Mark %A Grepl, Martin A. %T A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 1615-1638 %V 48 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2014012/ %R 10.1051/m2an/2014012 %G en %F M2AN_2014__48_6_1615_0

Kärcher, Mark; Grepl, Martin A.A PosterioriError Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 (2014) no. 6, pp. 1615-1638. doi : 10.1051/m2an/2014012. http://www.numdam.org/articles/10.1051/m2an/2014012/

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