On the regularity and stability of Pritchard–Salamon systems

Abstract

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, S B F ( ⋅ ) is the corresponding admissible perturbation C 0 -semigroup. We show that the C 0 -semigroup S B F ( ⋅ ) persists norm continuity, compactness and analyticity of C 0 -semigroup S ( ⋅ ) on W and V, respectively. We also characterize the compactness and norm continuity of Δ B F ( t ) = S B F ( t ) − S ( t ) for t > 0 . In particular, we unexpectedly find that Δ B F ( 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 S B F ( ⋅ ) and S ( ⋅ ) , so we obtain some new stability results.