We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated.
Mots clés : optimal control, semilinear partial differential equations, error estimation, proper orthogonal decomposition
@article{M2AN_2013__47_2_555_0,
author = {Kammann, Eileen and Tr\"oltzsch, Fredi and Volkwein, Stefan},
title = {\protect\emph{A posteriori }error estimation for semilinear parabolic optimal control problems with application to model reduction by {POD}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {555--581},
publisher = {EDP-Sciences},
volume = {47},
number = {2},
year = {2013},
doi = {10.1051/m2an/2012037},
zbl = {1282.49021},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2012037/}
}
TY - JOUR AU - Kammann, Eileen AU - Tröltzsch, Fredi AU - Volkwein, Stefan TI - A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 555 EP - 581 VL - 47 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2012037/ DO - 10.1051/m2an/2012037 LA - en ID - M2AN_2013__47_2_555_0 ER -
%0 Journal Article %A Kammann, Eileen %A Tröltzsch, Fredi %A Volkwein, Stefan %T A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 555-581 %V 47 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2012037/ %R 10.1051/m2an/2012037 %G en %F M2AN_2013__47_2_555_0
Kammann, Eileen; Tröltzsch, Fredi; Volkwein, Stefan. A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 2, pp. 555-581. doi : 10.1051/m2an/2012037. http://archive.numdam.org/articles/10.1051/m2an/2012037/
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