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

Axel Kröner (1) , Karl Kunisch (1, 2) , Hasnaa Zidani (3, 4)

1 RICAM - Johann Radon Institute for Computational and Applied Mathematics
2 IM - Institut für Mathematik [Graz]
3 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
4 OC - Optimisation et commande

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.

Mots clés: Hamilton-Jacobi appraoch; Feedback control; undamped wave equation

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

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, 2014, 21 (2), pp.442 - 464. ⟨10.1051/cocv/2014033⟩. ⟨hal-00924089⟩