Closed-loop perturbations of well-posed linear systems

We are concerned with the perturbation of a rather general class of linear time-invariant systems, namely well-posed linear system (WPLS), under additive linear perturbations seen as feedback laws. Let Σ be a WPLS with (A, B, C) as generating triple. For all control operator E, and all observation operator F such that (A, E, F ) is the generating triple of a WPLS, we prove that, if (A, B, F ) and (A, E, C) are also the generating triples of some WPLS, for all admissible feedback operator K for (A, E, F ), we can construct a WPLS Σ K whose generating triple is (A K , B K , C K ), where A K is the infinitisemal generator of the closed-loop of(A, E, F ) by the feedback operator K. Furthermore, we give necessary and sufficient condition such that exact controllability persists from Σ to Σ K . In particular, we show that this is the case for all sufficiently small bounded operator K.