New results on the operator Carleson measure criterion

We consider control systems of the form ẋ = Ax + Bu where A is the generator of a diagonal semigroup double-struckTsign on l² and B is an unbounded operator from a Hilbert space U to l². In a previous paper by Hansen & Weiss, a condition called the operator Carleson measure criterion was shown to be necessary for the admissibility of the control operator B. Furthermore this condition was shown to be sufficient if double-struckTsign is either analytic or invertible. In this paper we continue the analysis of admissibility as related to the operator Carleson measure criterion. We show that the operator Carleson measure criterion is satisfied if and only if the input-to-state transfer function has a certain decay rate. We also extend the previous sufficiency results of Hansen & Weiss to a more general class of diagonal semigroups. To achieve our aims, we derive some general results (not confined to diagonal semigroups) concerning Lyapunov equations and feedback-type perturbations, which are of independent interest.