Closed-loop perturbations of well-posed linear systems

Abstract

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 ΣK whose generating triple is (AK, BK, CK), where AK 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 ΣK. In particular, we show that this is the case for all sufficiently small bounded operator K.