#### Abstract

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.

#### 1. Introduction

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 [1]. Recent advances in fractional calculus have been reported in [2].

Recently, stability of fractional differential systems has attracted increasing interest. In 1996, Matignon [3] firstly studied the stability of linear fractional differential systems with the Caputo derivative. Since then, many researchers have done further studies on the stability of linear fractional differential systems [4–11]. For the nonlinear fractional differential systems, the stability analysis is much more difficult and only a few are available.

Some authors [12, 13] studied the following nonlinear fractional differential system: with initial values , where . They discussed the continuous dependence of solution on initial conditions and the corresponding structural stability by applying Gronwall's inequality. In [14] the authors dealt with the following fractional differential system: where , denotes either the Caputo, or the Riemann-Liouville fractional derivative operator. They proposed fractional Lyapunov's second method and firstly extended the exponential stability of integer order differential systems to the Mittag-Leffler stability of fractional differential systems. Moreover, the pioneering work on the generalized Mittag-Leffler stability and the generalized fractional Lyapunov direct method was proposed in [15].

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 [16, 17], we also study the stability of nonlinear fractional differential systems by utilizing the comparison method. Our contribution in this paper is that we have relaxed the condition of the Lyapunov-like function and that we have further studied the stability. The present paper is organized as follows. In Section 2, some definitions and lemmas are introduced. In Section 3, sufficient conditions on asymptotical stability and generalized Mittag-Leffler stability are given. The comparison method is applied to the analysis of the stability of fractional differential systems in Section 4. Conclusions are included in the last section.

#### 2. Preliminaries and Notations

Let us denote by the set of nonnegative real numbers, by the set of real numbers, and by the set of positive integer numbers. Let and set , and , where denotes the space of continuous functions on the interval .

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 [18].

*Definition 2.1. *The fractional integral with noninteger order of function is defined as follows:
where is the Gamma function.

*Definition 2.2. *The Riemann-Liouville derivative with order of function is defined as follows:
where and .

*Definition 2.3. *The Caputo derivative with noninteger order of function is defined as follows:
where and .

*Definition 2.4. *The Mittag-Leffler function is defined by
where , . The two-parameter Mittag-Leffler function is defined by
where and , .

Clearly . The following definitions are associated with the stability problem in the paper.

*Definition 2.5. *The constant is an equilibrium of fractional differential system if and only if for all , where means either the Caputo or the Riemann-Liouville fractional derivative operator.

Throughout the paper, we always assume that .

*Definition 2.6 (see [15]). *The zero solution of with order is said to be stable if, for any initial value , there exists an such that for all . The zero solution is said to be asymptotically stable if, in addition to being stable, as .

*Definition 2.7. *Let be a domain containing the origin. The zero solution of is said to be Mittag-Leffler stable if
where is the initial time and is the corresponding initial value, , , , , and is locally Lipschitz on with the Lipschitz constant .

*Definition 2.8. *Let be a domain containing the origin. The zero solution of is said to be generalized Mittag-Leffler stable if
where is the initial time and is the corresponding initial value, , , , , and is locally Lipschitz on with the Lipschitz constant .

*Remark 2.9. *Mittag-Leffler stability and generalized Mittag-Leffler stability both belong to algebraical stability, which also imply asymptotical stability (see [15]).

*Definition 2.10. *A function is said to belong to class- if is continuous function such that and it is strictly increasing.

*Definition 2.11 (see [19]). *The class- functions and are said to be with local growth momentum at the same level if there exist , such that for all . The class- functions and are said to be with global growth momentum at the same level if there exist such that for all .

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

Lemma 2.12 (see [20]). *Let be locally Hölder continuous for an exponent , and *(i)*,*(ii)*, ,**with nonstrict inequalities (i) and (ii), where and . Suppose further that satisfies the standard Lipschitz condition
**
Then, implies , .*

*Remark 2.13. *In Lemma 2.12, if we replace by , but other conditions remain unchanged, then the same result holds.

Lemma 2.14 (see [16]). *Let , and *(i)*,*(ii)*, **where , , , and . Assume that both inequalities are nonstrict and is nondecreasing in for each . Further, suppose that satisfies the standard Lipschitz condition
**
Then, implies , .*

*Remark 2.15. *In Lemmas 2.12 and 2.14, can be .

#### 3. Stability of Nonlinear Fractional Differential Systems

Let us consider the following nonlinear fractional differential system [14, 15]: with the initial condition , where is piecewise continuous in and is a domain that contains the equilibrium point , . Here and throughout the paper, we always assume there exists a unique solution to system (3.1) with the initial condition .

Recently, Li et al. [14, 15] investigated the Mittag-Leffler stability and the generalized Mittag-Leffler stability (the asymptotic stability) of system (3.1) by using the fractional Lyapunov's second method, where the following theorem has been presented.

Theorem 3.1. *Let be an equilibrium point of system (3.1) with , and let be a domain containing the origin. Let be a continuously differentiable function and locally Lipschitz with respect to such that
**
where , , , and , , , , and are arbitrary positive constants. Then is Mittag-Leffler stable (locally asymptotically stable). If the assumptions hold globally on , then is globally Mittag-Leffler stable (globally asymptotically stable).*

In the following, we give a new proof for Theorem 3.1.

*Proof of Theorem 3.1. *From (3.2) and (3.3), we can get
Obviously, for the initial value , the linear fractional differential equation
has a unique solution .

Taking into account Remark 2.13 and the relationship between (3.4) and (3.5), we obtain
where is a nonnegative function [21]. Substituting (3.6) in (3.2) yields
where from the asymptotic expansion of Mittag-Leffler function [22]. Hence the proof is completed.

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

Theorem 3.2. *Let be an equilibrium point of system (3.1), and let be a domain containing the origin. Assume that there exist a continuously differentiable function and class- function satisfying
**
where , . Then is locally stable. If the assumptions hold globally on , then is globally stable.*

*Proof. *Proceeding the same way as that in the proof of Theorem 3.1, it follows from (3.9) that . Again taking into account (3.8), one can get
where . Therefore, the equilibrium point is stable. So the proof is finished.

In the above two theorems, the stronger requirements on function have been assumed to ensure the existence of . This undoubtedly increases the difficulty in choosing the function . In fact, we can weaken the continuously differential function as . Here we give the corresponding results.

Theorem 3.3. *Let be an equilibrium point of system (3.1), and let be a domain containing the origin, . Assume there exists a class- function such that
**
where , , and . Then is locally asymptotically stable. If the assumptions hold globally on , then is globally asymptotically stable.*

*Proof. *Note that the linear fractional differential equation
has a unique solution for initial value .

Taking into account Lemma 2.12 and the relationship between (3.12) and (3.13), we obtain
Substituting (3.14) into (3.11) gives
from the definition of class-. This completes the proof.

Corollary 3.4. *Let be an equilibrium point of system (3.1), let be a domain containing the origin, and let be locally Lipschitz with respect to . Assume ,
**
where , , , and , are arbitrary positive constants. Then is generalized Mittag-Leffler stable. If the assumptions hold globally on , then is globally generalized Mittag-Leffler stable.*

*Proof. *In Theorem 3.3, by replacing by , we can get
so the conclusion holds.

Theorem 3.5. *Let be an equilibrium point of system (3.1), let be a domain containing the origin, and let be locally Lipschitz with respect to . Assume *(i)*there exist class- functions having global growth momentum at the same level and satisfying
*(ii)*there exists such that and have global growth momentum at the same level, **where , , and . Then is locally generalized Mittag-Leffler stable. If the assumptions hold globally on , then is globally generalized Mittag-Leffler stable.*

*Proof. *It follows from condition (i) that there exists such that
On the other hand, the linear fractional differential equation
has a unique solution
for the initial value .

Using (3.19), (3.20), and Lemma 2.12, we obtain
where is a nonnegative function [23, 24].

In addition, using condition (ii), one gets
for all , where .

Substituting (3.23) into (3.22), we finally obtain

Hence, the zero solution of system (3.1) is locally generalized Mittag-Leffler stable. If the assumptions hold globally on , then is globally generalized Mittag-Leffler stable. The proof is completed.

*Remark 3.6. *The nonnegative function tends to zero as approaches infinity from the asymptotic expansion of two-parameter Mittag-Leffler function [22], so the zero solution of system (3.1) satisfying the conditions of Theorem 3.5 is also asymptotically stable.

#### 4. The Comparison Results on the Stability

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 (3.1) with and the scalar fractional differential equation where the initial value , , is Lipschitz in and , . Also, we assume there exists a unique solution for system (4.1) with the initial value .

Theorem 4.1. *For system (3.1), let be an equilibrium point of system (3.1), and let be a domain containing the origin. Assume that there exist a Lyapunov-like function and a class- function such that , , and satisfies the inequality
**
Suppose further that is nondecreasing in for each . *(i)*If the zero solution of (4.1) is stable, then the zero solution of system (3.1) is stable; *(ii)*if the zero solution of (4.1) is asymptotically stable, then the zero solution of system (3.1) is asymptotically stable, too.*

*Proof. *Let denote the solution of system (3.1) with initial value . Along the solution curve , can be written as and
where . Applying the fractional integral operator to both sides of (4.1) leads to
Now, taking and applying Lemma 2.14 to inequalities (4.3) and (4.4), one has , .(i)If the zero solution of (4.1) is stable, then for any initial value , there exists such that for all . Therefore, taking into account , one gets
that is, , and the zero solution of system (3.1) is stable.(ii)One can directly derive
from the same argument in (i). Then, taking the limit to both sides of (4.6) and combining with the definition of class- function, one can obtain .

The proof is thus finished.

*Remark 4.2. *In Theorem 4.1 and system (4.1), if we replace order by , but other conditions remain unchanged, then the result in Theorem 4.1 still holds.

Especially, if the class- function in Theorem 4.1 and have global growth momentum at the same level, then we can have similar comparison result on the generalized Mittag-Leffler stability as follows.

Theorem 4.3. *For system (3.1), let be an equilibrium of system (3.1), and let be a domain containing the origin. Assume that there exists a Lyapunov-like function such that , , and is locally Lipschitz in and satisfies the inequality
**
where , . Suppose further that is nondecreasing in for each . Then the zero solution of system (3.1) is also locally generalized Mittag-Leffler stable if the zero solution of (4.1) is locally generalized Mittag-Leffler stable. In addition, if the assumptions hold globally on , then the globally generalized Mittag-Leffler stability of zero solution of (4.1) implies the globally generalized Mittag-Leffler stability of zero solution of system (3.1).*

*Proof. *First, from Definition 2.8, if the zero solution of (4.1) is generalized Mittag-Leffler stable, then there exist , , such that
where , and is locally Lipschitz in with Lipschitz constant .

Taking and noting that holds from Theorem 4.1, then taking into account (4.8) and , we obtain
Furthermore,
Let . Then it follows that
where due to . It is obvious that is a nonnegative function from and . In addition, is locally Lipschitz in since and are locally Lipschitz in . So, the zero solution of system (3.1) is generalized Mittag-Leffler stable. The proof is completed.

#### 5. Conclusion

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 by using a Lyapunov-like function. Compared to [15], we weaken the continuously differential function as . Sufficient conditions on generalized Mittag-Leffler stability and asymptotical stability are derived. Meanwhile, comparison method is applied to the analysis of the stability of fractional differential systems by fractional differential inequalities.

#### Acknowledgments

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.