Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains

¹ Inria – LMS, Ecole Polytechnique, CNRS, Université Paris-Saclay. Inria Saclay-Ile-de-France, 1 rue Honoré d’Estienne d’Orves, 91120 Palaiseau France.
² Normandie Université, INSA de Rouen Normandie, LMI (EA 3226 – FR CNRS 3335), 76000 Rouen, France, 685 Avenue de l’Université, 76801 St Etienne du Rouvray, Cedex France.

^* Corresponding author: philippe.moireau@inria.fr

Received: 18 June 2018
Accepted: 23 April 2019

Abstract

We are interested in reconstructing the initial condition of a wave equation in an unbounded domain configuration from measurements available in time on a subdomain. To solve this problem, we adopt an iterative strategy of reconstruction based on observers and time reversal adjoint formulations. We prove the convergence of our reconstruction algorithm with perfect measurements and its robustness to noise. Moreover, we develop a complete strategy to practically solve this problem on a bounded domain using artificial transparent boundary conditions to account for the exterior domain. Our work then demonstrates that the consistency error introduced by the use of approximate transparent boundary conditions is compensated by the stabilization properties obtained from the use of the available measurements, hence allowing to still be able to reconstruct the unknown initial condition.