Reconstructing initial data using iterative observers for wave type systems

Abstract

An iterative algorithm for solving initial data inverse problems from partial observations has been proposed in 2010 by Ramdani, Tucsnak and Weiss (Automatica, 2010). In this work, we are concerned with the convergence of this algorithm when the inverse problem is ill-posed, i.e. when the observations are not sufficient to reconstruct any initial data. We prove that the state space can be decomposed as a direct sum, stable by the algorithm, corresponding to the observable and unobservable part of the initial data. We show that this result holds for both locally distributed and boundary observation.