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.