INVARIANCE OF ASYMPTOTIC STABILITY OF PERTURBED LINEAR-SYSTEMS ON HILBERT-SPACE

The questions of stabilizability of structurally perturbed or uncertain linear systems in Hilbert space of the form x = (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 greater-than-or-equal-to 0, in a Hilbert space X; B is a bounded linear operator from another Hilbert space U to X; and {P(r), r-epsilon-OMEGA} is a family of bounded or unbounded perturbations of A in X, where OMEGA 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 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.