Optimal feedback control of undamped wave equations by solving a HJB equation

Axel Kröner 1 Karl Kunisch 1, 2 Hasnaa Zidani 3, 4
4 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, Polytechnique - X, CNRS - Centre National de la Recherche Scientifique : UMR7641
Abstract : An optimal fi nite-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 nite elements lead to approximated problems governed by ODEs in high dimensional space which makes infeasible the numerical resolution by HJB approach. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The e ffect of noise is considered and numerical simulations are presented to show the relevance of the approach.
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ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 21 (2), pp.442 - 464. 〈10.1051/cocv/2014033〉
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Axel Kröner, Karl Kunisch, Hasnaa Zidani. Optimal feedback control of undamped wave equations by solving a HJB equation. ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 21 (2), pp.442 - 464. 〈10.1051/cocv/2014033〉. 〈hal-00924089〉

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