1 Johann Radon Institute for
Computational and Applied Mathematics (RICAM), Austrian Academy of
Sciences, Altenberger Straße
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.
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.
© EDP Sciences, SMAI, 2015