Invariance of asymptotic stability of perturbed linear systems on Hilbert space

Abstract

The questions of stabilizability of structurally perturbed or uncertain linear systems in Hilbert space of the form$\dot x = (A + P(r))x + Bu$ are considered. The operatorA is assumed to be the infinitesimal generator of aC ₀-semigroup of contractionsT(t),t≥0, in a Hilbert spaceX;B is a bounded linear operator from another Hilbert spaceU toX; and {P(r),r ∈ Ω} is a family of bounded or unbounded perturbations ofA inX, 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$\dot x = Ax + Bu$ has similar properties. In particular, it is shown that, for certain classes of perturbations, weak and strong stabilizability properties are preserved for the same state feedback operator.