Robust stability of C0-semigroups and an application to stability of delay equations

Let A be a closed linear operator on a complex Banach space X and let lambda is an element of rho(A) be a fixed element of the resolvent set of A. Let U and Y be Banach spaces, and let D is an element of L(U, X) and E is an element of L(X, Y) be bounded linear operators. We define r(lambda)(A; D, E) by sup{r greater than or equal to 0: lambda is an element of rho(A + D Delta E) for all Delta is an element of L(Y, U) with parallel to Delta parallel to less than or equal to r} and prove that r lambda(A; D, E) = 1/parallel to ER(lambda, A)D parallel to We give two applications of this result. The first is an exact formula for the so-called stability radius of the generator of a C-0-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 B-j is an element of L(X) such that the classical solutions of the delay equation (u) over dot(t) = Au(t) + Sigma(j = 1)B(1)u(t - h(j)), t greater than or equal to 0, are exponentially stable in L-p([-h, 0]; X). (C) 1998 Academic Press.