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This paper deals with the stability of nonlinear fractional differential systems equipped with the Caputo derivative. At first, a sufficient condition on asymptotical stability is established by using a Lyapunov-like function. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In addition, some other sufficient conditions on stability are also presented.

Fractional calculus is more than 300 years old, but it did not attract enough interest at the early stage of development. In the last three decades, fractional calculus has become popular among scientists in order to model various physical phenomena with anomalous decay, such as dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, and viscoelastic systems [

Recently, stability of fractional differential systems has attracted increasing interest. In 1996, Matignon [

Some authors [

In this paper, we further study the stability of nonlinear fractional differential systems with Caputo derivative by utilizing a Lyapunov-like function. Taking into account the relation between asymptotical stability and generalized Mittag-Leffler stability, we are able to weaken the conditions assumed for the Lyapunov-like function. In addition, based on the comparison principle of fractional differential equations [

Let us denote by

Let us first introduce several definitions, results, and citations needed here with respect to fractional calculus which will be used later. As to fractional integrability and differentiability, the reader may refer to [

The fractional integral with noninteger order

The Riemann-Liouville derivative with order

The Caputo derivative with noninteger order

The Mittag-Leffler function is defined by

Clearly

The constant

Throughout the paper, we always assume that

The zero solution of

Let

Let

Mittag-Leffler stability and generalized Mittag-Leffler stability both belong to algebraical stability, which also imply asymptotical stability (see [

A function

The class-

It is useful to recall the following lemmas for our developments in the sequel.

Let

In Lemma

Let

In Lemmas

Let us consider the following nonlinear fractional differential system [

Recently, Li et al. [

Let

In the following, we give a new proof for Theorem

From (

Taking into account Remark

According to the above results, we have the following theorem.

Let

Proceeding the same way as that in the proof of Theorem

In the above two theorems, the stronger requirements on function

Let

Note that the linear fractional differential equation

Taking into account Lemma

Let

In Theorem

Let

there exist class-

there exists

It follows from condition (i) that there exists

Using (

In addition, using condition (ii), one gets

Substituting (

Hence, the zero solution of system (

The nonnegative function

It is well known that the comparison method is an effective way in judging the stability of ordinary differential systems. In this section, we will discuss similar results on the stability of fractional differential systems by using the comparison method.

In what follows, we consider system (

For system (

If the zero solution of (

if the zero solution of (

Let

If the zero solution of (

One can directly derive

The proof is thus finished.

In Theorem

Especially, if the class-

For system (

First, from Definition

Taking

In this paper, we have studied the stability of the zero solution of nonlinear fractional differential systems with the Caputo derivative and the commensurate order

The present work was partially supported by the National Natural Science Foundation of China under Grant no. 10872119 and the Shanghai Leading Academic Discipline Project under Grant no. S30104.