Recovering the observable part of the initial data of an infinite-dimensional linear system with perturbed skew-adjoint generator using observers

We consider the problem of recovering the initial data (or initial state) of infinite-dimensional linear systems generated by a perturbed skew-adjoint operator. It is well-known that this inverse problem is well posed if the system is exactly observable, but this assumption may be very restrictive in some applications. In this paper we are interested in the slight generalization of the results by Haine [12] in the case where the generator is a simple perturbation of a skew-adjoint operator. The reconstruction algorithm is based on iterative forward and backward observers, using the algorithm of Ramdani, Tucsnak and Weiss [19].