Stabilizability of perturbed linear systems on Hilbert space

The questions of controllability and stabilizability of structurally perturbed (or uncertain) linear systems in Hilbert space of the form dx/dt=(A+P(r))x+ Bu are considered. The operator A is assumed to be the infinitesimal generator of a C₀-semigroup of contractions T(t), t⩾0, in a Hilbert space X. B is a bounded linear operator from another Hilbert space U to X, and {P(r), r εΩ} is a family of bounded or unbounded perturbations of A in X where Ω is an arbitrary set not necessarily carrying any topology. Sufficient conditions are presented that guarantee controllability and stabilizability of the perturbed system given that the unperturbed system dx/dt =Ax+Bu, has similar properties. In particular it is shown that for certain class of perturbations weak and strong stabilizability properties are preserved for the same state feedback operator