Robust Stability of C -Semigroups and an Application 0 to Stability of Delay Equations

Let A be a closed linear operator on a complex Banach space X and let l gD A. be a fixed element of the resolvent set of A. Let U and Y be Banach spaces, and let DgL U, X.and EgL X, Y.be bounded linear operators. We define r A; D, E.by l sup rG0: l gD AqD DE. for all DgL Y, U. with 5D5Fr4 and prove that 1 r A; D, E.s . l ER l, A.D We give two applications of this result. The first is an exact formula for the so-called stability radius of the generator of a C0-semigroup of linear operators on a Hilbert space; it is derived from a precise result about robustness under perturbations of uniform boundedness in the right half-plane of the resolvent of an arbitrary semigroup generator. The second application gives sufficient conditions on the norm of the operators BjgL X. such that the classical solutions of the delay equation n u˙ t. sAu t. q Bju tyhj., tG0, js1 are exponentially stable in L p wyh, 0x; X..