Mathematical analysis of an observer for solving inverse source wave problem

The objective of this work is to propose a method using observers to estimate a source term of a wave equation from internal measurements in a subdomain. The first part of the work consists of proving an identifiability result from classical observability conditions for wave equations. We show that the source reconstruction is an ill-posed inverse problem of degree 1 or 2 depending on the measurements type. This inverse problem is solved using observers, a sequential strategy, that is proven to be equivalent to a minimization of a cost functional with Tikhonov regularization.