| Issue |
ESAIM: COCV
Volume 21, Number 2, April-June 2015
|
|
|---|---|---|
| Page(s) | 442 - 464 | |
| DOI | https://doi.org/10.1051/cocv/2014033 | |
| Published online | 09 March 2015 | |
Optimal feedback control for undamped wave equations by solving a HJB equation∗,∗∗
1 Johann Radon Institute for
Computational and Applied Mathematics (RICAM), Austrian Academy of
Sciences, Altenberger Straße
69, 4040
Linz,
Austria.
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2 University of Graz, Institute of
Mathematics and Scientific Computing, Heinrichstr. 36, 8010 Graz, Austria and Johann
Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of
Sciences, Altenberger Straße
69, 4040
Linz, Austria.
.
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3 INRIA Saclay & ENSTA
ParisTech, 828 Boulevard des
Marchaux, 91762
Palaiseau cedex,
France.
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Received:
17
December
2013
Abstract
An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton−Jacobi Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional spaces which makes the numerical resolution by the HJB approach infeasible. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach.
Mathematics Subject Classification: 49J20 / 35L05 / 49J15
Key words: Optimal control / wave equation / Hamilton−Jacobi Bellman equation / spectral elements
The work of the second author was supported in part by the Fonds zur Förderung der wissenschaftlichen Forschung under SFB 32, Mathematical Optimization and Applications in the Biomedical Sciences.
The work of the third author was partially supported by the EU under the 7th Framework Program Marie Curie Initial Training Network FP7-PEOPLE-2010-ITN, SADCO project, GA number 264735-SADCO.
© EDP Sciences, SMAI, 2015
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