A generalization of the concept of similarity of infinite-dimensional linear systems with applications to stability, stabilizability, and detectability

In the present paper a generalization of the concept of similarity called D^∞ similarity for infinite-demensional linear systems is introduced. We show that this D^∞ similarity preserves the spectrum in the standard sense, and the exponential growth bound, stability, stabilizability, and detectability in the extension sense. For the case where one of the semigroups is strongly differentiable at some t0 ≥ 0, it is shown that D^∞ similarity preserves the exponential growth bound and weak and exponential stability in the standard sense. For the case where one of the semigroups is analytic, it is shown that D^∞ similarity preserves stabilizability and detectability in the standard sense.