A CONSTRUCTIVE PROOF OF GIBSON'S STABILITY THEOREM

A useful stability result due to Gibson [SIAM J. Control Optim., 18 (1980), 311-316] ensures that, perturbing the generator of an exponentially stable semigroup by a compact operator, one obtains an exponentially stable semigroup again, provided the perturbed semigroup is strongly stable. In this paper we give a new proof of Gibson's theorem based on constructive reasoning, extend the analysis to Banach spaces, and relax the above compactness assumption. Moreover, we discuss some applications of such an abstract result to equations and systems of evolution.