Global Exponential Stability of Impulsive Functional Differential Equations via Razumikhin Technique

Shiguo Peng,Liping Yang

doi:10.1155/2010/987372

Shiguo Peng, Liping Yang

Open Access

https://doi.org/10.1155/2010/987372

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Abstract

This paper develops some new Razumikhin-type theorems on global exponential stability of impulsive functional differential equations. Some applications are given to impulsive delay differential equations. Compared with some existing works, a distinctive feature of this paper is to address exponential stability problems for any finite delay. It is shown that the functional differential equations can be globally exponentially stabilized by impulses even if it may be unstable itself. Two examples verify the effectiveness of the proposed results.

Highlights

Functional differential equations FDEs which include delay differential equations DDEs play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, economic, and social sciences 1, 2
We will present some Razumikhin-type theorems on global exponential stability for system 2.2 based on the Lyapunov-Razumikhin method
Our new results are more practically applicable than those in the literature, since the restrictive condition that the supper bound of time delay is less than the length of all the impulsive intervals is removed here

Summary

Introduction

Functional differential equations FDEs which include delay differential equations DDEs play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, economic, and social sciences 1, 2. In 14, 15 , the authors have investigated exponential stability of IFDEs by using the method of Lyapunov functions and Razumikhin techniques. In 16 , the authors have studied exponential stability by using the method of Lyapunov functional. Some results in 15, 16 imposed a restrictive condition on time delays which were less than the length of all the impulsive intervals see, e.g., 15, Theorems 3.1-3.2 and 16, Theorem 3.1. The aim of this paper is to establish global exponential stability criteria for IFDEs by employing the Razumikhin technique which illustrate that impulses do contribute to the stability of some IFDEs and the restrictive condition that the time delays are less than the length of all the impulsive intervals can be removed in this paper

Preliminaries

Razumikhin-Type Theorems

Applications and Examples

Conclusion