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Erschienen in:
2016 | Supplement | Buchkapitel
HJB-POD Feedback Control for Navier-Stokes Equations
verfasst von : Alessandro Alla, Michael Hinze
Erschienen in: Progress in Industrial Mathematics at ECMI 2014
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Abstract
In this report we present the approximation of an infinite horizon optimal control problem for the evolutive Navier-Stokes system. The method is based on a model reduction technique, using a POD approximation, coupled with a Hamilton-Jacobi-Bellman (HJB) equation which characterizes the value function of the corresponding control problem for the reduced system. Although the approximation schemes available for the HJB are shown to be convergent for any dimension, in practice we need to restrict the dimension to rather small numbers and this limitation affects the accuracy of the POD approximation. We will present numerical tests for the control of the time-dependent Navier-Stokes system in two-dimensional spatial domains to illustrate our approach and to show the effectiveness of the method.