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Transfer functions are called well-posed if they are bounded and analytic on some right half-plane. Two concepts of L 2-stabilization are analyzed for well-posed transfer functions: the usual one, as in Figure 3, and a more general one called stabilization with internal loop, see Figure 4. In both cases we obtain a complete parametrization of all stabilizing controllers in terms of a doubly coprime factorization of the original transfer function. Moreover, the connection between the two stabilization concepts is clarified. We analyze two special subclasses of stabilizing controllers with internal loop, called canonical and dual canonical controllers. We show that, in a certain sense, every stabilizing controller with internal loop is equivalent to a canonical controller, and also to a dual canonical controller.

We are searching for necessary and/or sufficient conditions for the admissibility of unbounded control operators for semigroups on Hilbert spaces, with respect to input functions of class L 2. Our first conjecture is that admissibility of an unbounded input element 6 for a semigroup with generator A is equivalent to a certain decay rate of ∥(sI - A)-1 b∥ as Re s → ∞. The second conjecture states that a control operator B defined on a Hilbert space U is admissible if and only if, for any v ∈ U, Bv is an admissible input element. It is proved that both conjectures hold in many important particular cases (e.g., the first conjecture is true if the semigroup is normal).