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Robustness of stability with respect to small delays, e.g., motivated by feedback systems in control theory, is of great theoretical and practical important, but this property does not hold for many systems. In this paper, we introduce the conception of robustness with respect to small time-varying delays for exponential stability of the non-autonomous linear systems. Sufficient conditions are given for the non-autonomous systems to be robust, and examples are provided to illustrate that the conditions are satisfied for a large class of the non-autonomous parabolic systems.

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In this paper we show the well-posedness of the following constant delay equation:where (A, D(A)) generates a strongly continuous semigroup (T (t)) t 0 on a Banach space X and (B, D(B)) is an unbounded and closed linear operator on X. Our approach is to transform the delay equation into an abstract Cauchy problem in a special phase space and apply semigroup theory. Furthermore, we show that the solution semigroup corresponding to the above delay equation is uniformly exponentially bounded. Finally, we give an example to explain the well-posedness of delay equations.  2004 Elsevier Inc. All rights reserved.