Recovering the initial state of a Well-Posed Linear System with skew-adjoint generator

We consider the problem of recovering the initial data of a Well-Posed Linear System with unitary semigroups. 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. Thus, we are interested in systems which are not exactly observable, and in particular, where we cannot expect a full reconstruction. We propose to use the algorithm studied by Ramdani et al. in (Automatica 46:1616–1625, 2010) and prove that it always converges towards the observable part of the initial state