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.

DOI: 10.1051/cocv/2014033

Keywords: Optimal control, wave equation, Hamilton−Jacobi Bellman equation, spectral elements

^{1}; Kunisch, Karl

^{2}; Zidani, Hasnaa

^{3}

@article{COCV_2015__21_2_442_0, author = {Kr\"oner, Axel and Kunisch, Karl and Zidani, Hasnaa}, 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}, number = {2}, year = {2015}, doi = {10.1051/cocv/2014033}, zbl = {1318.49069}, mrnumber = {3348407}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014033/} }

TY - JOUR AU - Kröner, Axel AU - Kunisch, Karl AU - Zidani, Hasnaa TI - Optimal feedback control for undamped wave equations by solving a HJB equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 442 EP - 464 VL - 21 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014033/ DO - 10.1051/cocv/2014033 LA - en ID - COCV_2015__21_2_442_0 ER -

%0 Journal Article %A Kröner, Axel %A Kunisch, Karl %A Zidani, Hasnaa %T Optimal feedback control for undamped wave equations by solving a HJB equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 442-464 %V 21 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014033/ %R 10.1051/cocv/2014033 %G en %F COCV_2015__21_2_442_0

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