Invariance of asymptotic stability of perturbed linear systems on Hilbert space (Q912058)

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 operator A is assumed to be the infinitesimal generator of a \(C_ 0\)-semigroup of contractions T(t), \(t\geq 0\), in a Hilbert space X; B is a bounded linear operator from another Hilbert space U to X; and \(\{\) P(r), \(r\in \Omega \}\) 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 \(\dot x=Ax+Bu\) 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.