Geodesic
Optimal feedback control for undamped wave equations by solving a HJB equation
Kröner, Axel1 ; Kunisch, Karl2  ; Zidani, Hasnaa3 
1 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria.
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.
3 INRIA Saclay & ENSTA ParisTech, 828 Boulevard des Marchaux, 91762 Palaiseau cedex, France.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 442-464

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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.

Reçu le :
DOI : 10.1051/cocv/2014033
Classification : 49J20, 35L05, 49J15
Keywords: Optimal control, wave equation, Hamilton−Jacobi Bellman equation, spectral elements

Kröner, Axel 1 ; Kunisch, Karl 2 ; Zidani, Hasnaa 3

1 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria.
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.
3 INRIA Saclay & ENSTA ParisTech, 828 Boulevard des Marchaux, 91762 Palaiseau cedex, France. 
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     title = {Optimal feedback control for undamped wave equations by solving a {HJB} equation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {442--464},
     publisher = {EDP-Sciences},
     volume = {21},
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     year = {2015},
     doi = {10.1051/cocv/2014033},
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     mrnumber = {3348407},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2014033/}
}
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Kröner, Axel; Kunisch, Karl; Zidani, Hasnaa. Optimal feedback control for undamped wave equations by solving a HJB equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 442-464. doi: 10.1051/cocv/2014033

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