Exponential stability of nonlinear delay equation with constant decay rate via perturbed system method

Abstract

This paper studies the exponential stability of nonlinear differential equations with constant decay rate under the assumption that the corresponding crisp equation (without delay, simply, nondelay equation) is exponentially stable. Different from most publications dealing with delay systems by applying Lyapunov-type methods, the perturbed system method is used in this paper. It shall be shown that the considered equations will remain exponentially stable provided the time lag is small enough. Moreover, we formulate and estimate the threshold of delay ensuring exponential stability when a constant decay rate appears explicitly in system model, which is better than the existing results.

A comment on “Exponential stability of nonlinear delay equation with constant decay rate via perturbed system method”

“…We consider the following delay system presented in [5], We agree that the perturbed system method presented in [5] for exponential stability of nonlinear delay system is interesting. It tries to show that the delay system will remain exponential stability, provided the time lag is small enough.…”

“…which gives (7): The previous inequality (3) (or (7) obtained in [5]) is not correct because the use of Gronwall inequality contains an error, there's no version of Gronwall inequality to establish the passage from (2) to (3), and the following counterexample illustrate this. In fact, the inequality used by the authors in [5] can not be considered as a new or other version of Gronwall inequality as explained in the following.…”

“…First, we show that the use of the Gronwall inequality in the proof of Theorem 1 in [5] is not correct. Indeed the authors obtain: …”

“…It tries to show that the delay system will remain exponential stability, provided the time lag is small enough. However, the proof of Theorem 1 in [5] contains a mistake, so that the result presented in this Theorem is not correct. The goal of this paper is to present a new and corrected version of this theorem.…”

“…The goal of this paper is to present a new and corrected version of this theorem. A counterexample is given to prove that the inequality used in [5] is not correct and can not be considered as a new or other version of Gronwall inequality. Furthermore, examples are provided to demonstrate the less conservatism of the obtained results based on Gronwall inequality.…”