Lyapunov’s direct method for linear systems of functional-differential equations in Sobolev space

Abstract

We establish a criterion for exponential stability in the H1-topology in terms of operator inequalities for a linear FDE system of retarded type by Lyapunov’s direct method. As a corollary, some sufficient condition of exponential stability in terms of the matrix specifying the Stieltjes integral is obtained in the autonomous case. A few examples illustrating the results are exhibited.