Uniform convergence of the POD method and applications to optimal control

Karl Kunisch; Markus Müller

doi:10.3934/dcds.2015.35.4477

Article Contents

2015, Volume 35, Issue 9: 4477-4501. Doi: 10.3934/dcds.2015.35.4477

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Uniform convergence of the POD method and applications to optimal control

Karl Kunisch^1, and
Markus Müller^1,

1.
Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz

Received: April 30, 2014

Revised: August 31, 2014

Published: May 2017

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Abstract

We consider proper orthogonal decomposition (POD) based Galerkin approximations to parabolic systems and establish uniform convergence with respect to forcing functions. The result is used to prove convergence of POD approximations to optimal control problems that automatically update the POD basis in order to avoid problems due to unmodeled dynamics in the POD reduced order system. A numerical example illustrates the results.

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Mathematics Subject Classification: 35K20, 49J20, 49K20, 65K10.

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