Stability of Nonlinear Fractional Neutral Differential Difference Systems

Wei Jiang,Kewei Liu

doi:10.1155/2014/514631

Abstract

We study the stability of a class of nonlinear fractional neutral differential difference systems equipped with the Caputo derivative. We extend Lyapunov-Krasovskii theorem for the nonlinear fractional neutral systems. Conditions of stability and instability are obtained for the nonlinear fractional neutral systems.

Highlights

Fractional differential equations have attracted great attention
We study the stability of a class of nonlinear fractional neutral differential difference systems equipped with the Caputo derivative
We consider the stability of the following nonlinear fractional neutral differential difference system: cDtα0 [D (t, xt)] = f (t, xt), t > t0, (17)

Summary

Introduction

Fractional differential equations have attracted great attention. It has been proved that fractional differential equations are valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics. For more details about the stability results and the methods available to analyze the stability of fractional differential systems, the reader may refer to the recent survey papers [19, 20] and the references therein. It is necessary to extend Lyapunov’s second method to fractional systems. Lyapunov’s second method was proposed, and the authors extended the exponential stability of integer order differential system to the Mittag-Leffler stability of fractional differential system. In [16,17,18], Baleanu et al extended Lyapunov’s method to fractional functional differential systems and developed the Lyapunov-Krasovskii stability theorem, Lyapunov-Razumikhin stability theorem, and Mittag-Leffler stability theorem for fractional functional differential systems. We consider the stability of a class of nonlinear fractional neutral differential difference equations with the Caputo derivative. We extend the Lyapunov-Krasovskii method for the nonlinear fractional neutral differential difference systems.

Preliminaries

Main Results

An Illustrative Example

Conclusions