Stability conditions for abstract functional differential equations in Hilbert space

Abstract

Stability conditions for functional differential equations of the form: du(t)/dt = Au(t) + bAu(t − h) + (a* Au)(t) are studied, where A is the infinitesimal generator of an analytic semigroup in a Hilbert space, b ≠ 0 and the convolution term contains a square integrable real function a ≠ 0. Norm discontinuity of the solution semigroup of the equation with discrete delay is avoided by studying the inverse of the characteristic operator. Sufficient and necessary conditions for the uniform exponential stability of the solution semigroup are obtained. The results are applied to a retarded partial integrodifferential equation.