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A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD∗

Published online by Cambridge University Press:  21 January 2013

Eileen Kammann ,
Fredi Tröltzsch  and
Stefan Volkwein
Eileen Kammann
Affiliation:
Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany.. kammann@math.tu-berlin.de; troeltzsch@math.tu-berlin.de
Fredi Tröltzsch
Affiliation:
Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany.. kammann@math.tu-berlin.de; troeltzsch@math.tu-berlin.de
Stefan Volkwein
Affiliation:
Institut für Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany.; Stefan.Volkwein@uni-konstanz.de
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Abstract

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.

Keywords

Optimal controlsemilinear partial differential equationserror estimationproper orthogonal decomposition
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 555 - 581
Copyright
© EDP Sciences, SMAI, 2013

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