Perturbation of strongly and polynomially stable Riesz-spectral operators

In this paper we consider bounded and relatively bounded finite rank perturbations of a Riesz-spectral operator generating a polynomially stable semigroup of linear operators on a Hilbert space. We concentrate on a commonly encountered situation where the spectrum of the unperturbed operator is contained in the open left half-plane of the complex plane and approaches the imaginary axis asymptotically. We present conditions on the perturbing operator such that the spectrum of the perturbed operator is contained in the open left half-plane of the complex plane and additional conditions for the strong and polynomial stabilities of the perturbed semigroup. We consider two applications of the perturbation results. In the first example we apply the results to the perturbation of a polynomially stabilized one-dimensional wave equation. In the second example we consider the perturbation of a closed-loop system consisting of a distributed parameter system and an observer-based feedback controller solving the robust output regulation problem related to an infinite-dimensional signal generator.