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

Sebastien Imperiale (1, 2) , Philippe Moireau (1, 2) , A Tonnoir (1, 3)

1 M3DISIM - Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine
2 Université Paris-Saclay
3 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie

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.

Domaines: Equations aux dérivées partielles [math.AP]

Sebastien Imperiale, Philippe Moireau, A Tonnoir. Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains. 2018. ⟨hal-01898440v1⟩