On the regularity and stability of Pritchard–Salamon systems

Let Σ(S(·),B,−) be a Pritchard–Salamon system for (W,V ), where W and V are Hilbert spaces. Suppose U is a Hilbert space and F ∈ L(W,U) is an admissible output operator, SBF (·) is the corresponding admissible perturbation C0-semigroup. We show that the C0-semigroup SBF (·) persists norm continuity, compactness and analyticity of C0-semigroup S(·) on W and V , respectively. We also characterize the compactness and norm continuity of ΔBF (t) = SBF (t) − S(t) for t > 0. In particular, we unexpectedly find that ΔBF (t) is norm continuous for t > 0 on W and V if the embedding from W into V is compact. Moreover, from this we give some relations between the spectral bounds and growth bounds of SBF (·) and S(·), so we obtain some new stability results.  2006 Elsevier Inc. All rights reserved