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International Journal of Differential Equations
Volume 2011 (2011), Article ID 635165, 12 pages
http://dx.doi.org/10.1155/2011/635165
Research Article

Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative

1Department of Mathematics, Shanghai University, Shanghai 200444, China
2School of Mathematics and Computational Science, China University of Petroleum (East China), Qingdao 266555, China
3Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322-4160, USA

Received 18 April 2011; Accepted 14 June 2011

Academic Editor: Fawang Liu

Copyright © 2011 Fengrong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [411]. 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:𝐶𝐷𝑞0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),(1.1) with initial values 𝑥(0)=𝑥0(0),,𝑥(𝑚1)(0)=𝑥0(𝑚1), where 𝑚1<𝑞𝑚. 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: 𝔇𝑞0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),(1.2) where 0<𝑞1, 𝔇𝑞0,𝑡 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 0<𝑞<1 and set 𝐶𝑞([𝑡0,𝑇],)={𝑓𝐶((𝑡0,𝑇],),(𝑡𝑡0)𝑞𝑓(𝑡)𝐶([𝑡0,𝑇],)}, and 𝐶𝑞([𝑡0,𝑇]×Ω,)={𝑓(𝑡,𝑥(𝑡))𝐶((𝑡0,𝑇]×Ω,),(𝑡𝑡0)𝑞𝑓(𝑡,𝑥(𝑡))𝐶([𝑡0,𝑇]×Ω,)}, where 𝐶((𝑡0,𝑡],) denotes the space of continuous functions on the interval (𝑡0,𝑡].

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 𝑞0 of function 𝑥(𝑡) is defined as follows: 𝐷𝑡𝑞0,𝑡1𝑥(𝑡)=Γ(𝑞)𝑡𝑡0(𝑡𝜏)𝑞1𝑥(𝜏)𝑑𝜏,(2.1) where Γ() is the Gamma function.

Definition 2.2. The Riemann-Liouville derivative with order 𝑞 of function 𝑥(𝑡) is defined as follows: RL𝐷𝑞𝑡0,𝑡1𝑥(𝑡)=𝑑Γ(𝑚𝑞)𝑚𝑑𝑡𝑚𝑡𝑡0(𝑡𝜏)𝑚𝑞1𝑥(𝜏)𝑑𝜏,(2.2) where 𝑚1𝑞<𝑚 and 𝑚+.

Definition 2.3. The Caputo derivative with noninteger order 𝑞 of function 𝑥(𝑡) is defined as follows: 𝐶𝐷𝑞𝑡0,𝑡1𝑥(𝑡)=Γ(𝑚𝑞)𝑡𝑡0(𝑡𝜏)𝑚𝑞1𝑥(𝑚)(𝜏)𝑑𝜏,(2.3) where 𝑚1<𝑞<𝑚 and 𝑚+.

Definition 2.4. The Mittag-Leffler function is defined by 𝐸𝛼(𝑧)=𝑘=0𝑧𝑘,Γ(𝑘𝛼+1)(2.4) where 𝛼>0, 𝑧. The two-parameter Mittag-Leffler function is defined by 𝐸𝛼,𝛽(𝑧)=𝑘=0𝑧𝑘,Γ(𝑘𝛼+𝛽)(2.5) where 𝛼>0 and 𝛽, 𝑧.
Clearly 𝐸𝛼(𝑧)=𝐸𝛼,1(𝑧). The following definitions are associated with the stability problem in the paper.

Definition 2.5. The constant 𝑥eq is an equilibrium of fractional differential system 𝔇𝑞𝑡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥) if and only if 𝑓(𝑡,𝑥eq)=𝔇𝑞𝑡0,𝑡𝑥(𝑡)|𝑥(𝑡)=𝑥eq for all 𝑡>𝑡0, where 𝔇𝑞𝑡0,𝑡 means either the Caputo or the Riemann-Liouville fractional derivative operator.

Throughout the paper, we always assume that 𝑥eq=0.

Definition 2.6 (see [15]). The zero solution of 𝔇𝑞𝑡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)) with order 𝑞(0,1) is said to be stable if, for any initial value 𝑥0, there exists an 𝜀>0 such that 𝑥(𝑡)𝜀 for all 𝑡>𝑡0. The zero solution is said to be asymptotically stable if, in addition to being stable, 𝑥(𝑡)0 as 𝑡+.

Definition 2.7. Let 𝔹𝑛 be a domain containing the origin. The zero solution of 𝔇𝑞𝑡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)) is said to be Mittag-Leffler stable if 𝑚𝑥𝑥(𝑡)0𝐸𝑞𝜆𝑡𝑡0𝑞𝑏,(2.6) where 𝑡0 is the initial time and 𝑥0 is the corresponding initial value, 𝑞(0,1), 𝜆0,𝑏>0, 𝑚(0)=0, 𝑚(𝑥)0, and 𝑚(𝑥) is locally Lipschitz on 𝑥𝔹𝑛 with the Lipschitz constant 0.

Definition 2.8. Let 𝔹𝑛 be a domain containing the origin. The zero solution of 𝔇𝑞𝑡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)) is said to be generalized Mittag-Leffler stable if 𝑚𝑥𝑥(𝑡)0𝑡𝑡0𝛾𝐸𝑞,1𝛾𝜆𝑡𝑡0𝑞𝑏,(2.7) where 𝑡0 is the initial time and 𝑥0 is the corresponding initial value, 𝑞(0,1), 𝑞<𝛾1𝑞, 𝜆0,𝑏>0, 𝑚(0)=0,𝑚(𝑥)0, and 𝑚(𝑥) is locally Lipschitz on 𝑥𝔹𝑛 with the Lipschitz constant 0.

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 𝛼(𝑟) is said to belong to class-𝒦 if 𝛼++ is continuous function such that 𝛼(0)=0 and it is strictly increasing.

Definition 2.11 (see [19]). The class-𝒦 functions 𝛼(𝑟) and 𝛽(𝑟) are said to be with local growth momentum at the same level if there exist 𝑟1>0, 𝑘𝑖>0(𝑖=1,2) such that 𝑘1𝛼(𝑟)𝛽(𝑟)𝑘2𝛼(𝑟) for all 𝑟[0,𝑟1]. The class-𝒦 functions 𝛼(𝑟) and 𝛽(𝑟) are said to be with global growth momentum at the same level if there exist 𝑘𝑖>0(𝑖=1,2) such that 𝑘1𝛼(𝑟)𝛽(𝑟)𝑘2𝛼(𝑟) for all 𝑟+.

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

Lemma 2.12 (see [20]). Let 𝑣,𝑤𝐶1𝑞([𝑡0,𝑇],) be locally Hölder continuous for an exponent 0<𝑞<𝜈1, 𝐶([𝑡0,𝑇]×,) and (i)RL𝐷𝑞𝑡0,𝑡𝑣(𝑡)(𝑡,𝑣(𝑡)),(ii)RL𝐷𝑞𝑡0,𝑡𝑤(𝑡)(𝑡,𝑤(𝑡)), 𝑡0<𝑡𝑇,with nonstrict inequalities (i) and (ii), where 𝑣0=Γ(𝑞)𝑣(𝑡)(𝑡𝑡0)1𝑞|𝑡=𝑡0 and 𝑤0=Γ(𝑞)𝑤(𝑡)(𝑡𝑡0)1𝑞|𝑡=𝑡0. Suppose further that satisfies the standard Lipschitz condition (𝑡,𝑥)(𝑡,𝑦)(𝑥𝑦),𝑥𝑦,>0.(2.8) Then, 𝑣0𝑤0 implies 𝑣(𝑡)𝑤(𝑡), 𝑡0<𝑡𝑇.

Remark 2.13. In Lemma 2.12, if we replace RL𝐷𝑞𝑡0,𝑡 by 𝐶𝐷𝑞𝑡0,𝑡, but other conditions remain unchanged, then the same result holds.

Lemma 2.14 (see [16]). Let 𝑣,𝑤𝐶1𝑞([𝑡0,𝑇],), 𝐶([𝑡0,𝑇]×,) and (i)𝑣(𝑡)(𝑣0/Γ(𝑞))(𝑡𝑡0)𝑞1+(1/Γ(𝑞))𝑡𝑡0(𝑡𝑠)𝑞1(𝑠,𝑣(𝑠))𝑑𝑠,(ii)𝑤(𝑡)(𝑤0/Γ(𝑞))(𝑡𝑡0)𝑞1+(1/Γ(𝑞))𝑡𝑡0(𝑡𝑠)𝑞1(𝑠,𝑤(𝑠))𝑑𝑠, where 𝑡0<𝑡𝑇, 𝑣0=Γ(𝑞)𝑣(𝑡)(𝑡𝑡0)1𝑞|𝑡=𝑡0, 𝑤0=Γ(𝑞)𝑤(𝑡)(𝑡𝑡0)1𝑞|𝑡=𝑡0, and 0<𝑞<1. Assume that both inequalities are nonstrict and (𝑡,𝑥) is nondecreasing in 𝑥 for each 𝑡. Further, suppose that satisfies the standard Lipschitz condition (𝑡,𝑥)(𝑡,𝑦)(𝑥𝑦),𝑥𝑦,>0.(2.9) Then, 𝑣0𝑤0 implies 𝑣(𝑡)𝑤(𝑡), 𝑡0<𝑡𝑇.

Remark 2.15. In Lemmas 2.12 and 2.14, 𝑇 can be +.

3. Stability of Nonlinear Fractional Differential Systems

Let us consider the following nonlinear fractional differential system [14, 15]: 𝐶𝐷𝑞𝑡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),(3.1) with the initial condition 𝑥0=𝑥(𝑡0), where 𝑓[𝑡0,)×Ω𝑛 is piecewise continuous in 𝑡 and Ω𝑛 is a domain that contains the equilibrium point 𝑥eq=0, 0<𝑞<1. Here and throughout the paper, we always assume there exists a unique solution 𝑥(𝑡)𝐶1[𝑡0,) to system (3.1) with the initial condition 𝑥(𝑡0).

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 𝑥𝑒𝑞=0 be an equilibrium point of system (3.1) with 𝑡0=0, and let 𝔻𝑛 be a domain containing the origin. Let 𝑉(𝑡,𝑥(𝑡))[0,)×𝔻+ be a continuously differentiable function and locally Lipschitz with respect to 𝑥 such that 𝛼1𝑥𝑎𝑉(𝑡,𝑥(𝑡))𝛼2𝑥𝑎𝑏,(3.2)𝐶𝐷𝑝0,𝑡𝑉(𝑡,𝑥(𝑡))𝛼3𝑥𝑎𝑏,(3.3) where 𝑡0, 𝑥𝔻, 𝑝(0,1), and 𝛼1, 𝛼2, 𝛼3, 𝑎, and 𝑏 are arbitrary positive constants. Then 𝑥𝑒𝑞=0 is Mittag-Leffler stable (locally asymptotically stable). If the assumptions hold globally on 𝑛, then 𝑥𝑒𝑞=0 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 𝐶𝐷𝑝0,𝑡𝑉𝛼(𝑡,𝑥(𝑡))3𝛼2𝑉(𝑡,𝑥(𝑡)).(3.4) Obviously, for the initial value 𝑉(0,𝑥(0)), the linear fractional differential equation 𝐶𝐷𝑝0,𝑡𝑉𝛼(𝑡,𝑥(𝑡))=3𝛼2𝑉(𝑡,𝑥(𝑡))(3.5) has a unique solution 𝑉(𝑡,𝑥(𝑡))=𝑉(0,𝑥(0))𝐸𝑝((𝛼3/𝛼2)𝑡𝑝).
Taking into account Remark 2.13 and the relationship between (3.4) and (3.5), we obtain 𝑉(𝑡,𝑥(𝑡))𝑉(0,𝑥(0))𝐸𝑝𝛼3𝛼2𝑡𝑝,(3.6) where 𝐸𝑝((𝛼3/𝛼2)𝑡𝑝) is a nonnegative function [21]. Substituting (3.6) in (3.2) yields 𝑥(𝑡)𝑉(0,𝑥(0))𝛼1𝐸𝑝𝛼3𝛼2𝑡𝑝1/𝑎,(3.7) where 𝐸𝑝((𝛼3/𝛼2)𝑡𝑝)0(𝑡+) 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 𝑥𝑒𝑞=0 be an equilibrium point of system (3.1), and let 𝔻𝑛 be a domain containing the origin. Assume that there exist a continuously differentiable function 𝑉(𝑡,𝑥(𝑡))[𝑡0,)×𝔻+ and class-𝒦 function 𝛼 satisfying 𝑉(𝑡,𝑥(𝑡))𝛼(𝑥),(3.8)𝐶𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡))0,(3.9) where 𝑥𝔻, 𝑝(0,1). Then 𝑥𝑒𝑞=0 is locally stable. If the assumptions hold globally on 𝑛, then 𝑥eq=0 is globally stable.

Proof. Proceeding the same way as that in the proof of Theorem 3.1, it follows from (3.9) that 𝑉(𝑡,𝑥(𝑡))𝑉(𝑡0,𝑥(𝑡0)). Again taking into account (3.8), one can get 𝑥(𝑡)𝛼1𝑉𝑡0𝑡,𝑥0,(3.10) where 𝑡𝑡0. Therefore, the equilibrium point 𝑥eq=0 is stable. So the proof is finished.

In the above two theorems, the stronger requirements on function 𝑉 have been assumed to ensure the existence of 𝐶𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡)). This undoubtedly increases the difficulty in choosing the function 𝑉(𝑡,𝑥(𝑡)). In fact, we can weaken the continuously differential function 𝑉(𝑡,𝑥(𝑡)) as 𝑉(𝑡,𝑥(𝑡))𝐶1𝑝([𝑡0,)×𝔻,+). Here we give the corresponding results.

Theorem 3.3. Let 𝑥𝑒𝑞=0 be an equilibrium point of system (3.1), and let 𝔻𝑛 be a domain containing the origin, 𝑉(𝑡,𝑥(𝑡))𝐶1𝑝([𝑡0,)×𝔻,+). Assume there exists a class-𝒦 function 𝛼 such that 𝑉(𝑡,𝑥(𝑡))𝛼(𝑥),(3.11)RL𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡))0,(3.12) where 𝑡>𝑡00, 𝑥𝔻, and 𝑝(0,1). Then 𝑥eq=0 is locally asymptotically stable. If the assumptions hold globally on 𝑛, then 𝑥eq=0 is globally asymptotically stable.

Proof. Note that the linear fractional differential equation RL𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡))=0(3.13) has a unique solution 𝑉(𝑡,𝑥(𝑡))=(𝑉0/Γ(𝑝))(𝑡𝑡0)𝑝1 for initial value 𝑉0=Γ(𝑝)𝑉(𝑡,𝑥(𝑡))(𝑡𝑡0)1𝑝|𝑡=𝑡0.
Taking into account Lemma 2.12 and the relationship between (3.12) and (3.13), we obtain 𝑉𝑉(𝑡,𝑥(𝑡))0Γ(𝑝)𝑡𝑡0𝑝1.(3.14) Substituting (3.14) into (3.11) gives 𝑥(𝑡)𝛼1𝑉0Γ(𝑝)𝑡𝑡0𝑝10(𝑡+),(3.15) from the definition of class-𝒦. This completes the proof.

Corollary 3.4. Let 𝑥eq=0 be an equilibrium point of system (3.1), let 𝔻𝑛 be a domain containing the origin, and let 𝑉(𝑡,𝑥(𝑡))𝐶1𝑝([𝑡0,)×𝔻,+) be locally Lipschitz with respect to 𝑥. Assume 𝑉(𝑡,0)=0, 𝑉(𝑡,𝑥(𝑡))𝑎𝑥𝑏,RL𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡))0,(3.16) where 𝑡>𝑡00, 𝑥𝔻, 𝑝(0,1), and 𝑎, 𝑏 are arbitrary positive constants. Then 𝑥eq=0 is generalized Mittag-Leffler stable. If the assumptions hold globally on 𝑛, then 𝑥eq=0 is globally generalized Mittag-Leffler stable.

Proof. In Theorem 3.3, by replacing 𝛼(𝑥) by 𝑎𝑥𝑏, we can get 𝑉𝑥(𝑡)0𝑎𝑡𝑡0𝑝1𝐸𝑝,𝑝0𝑡𝑡0𝑝1/𝑏,(3.17) so the conclusion holds.

Theorem 3.5. Let 𝑥𝑒𝑞=0 be an equilibrium point of system (3.1), let 𝔻𝑛 be a domain containing the origin, and let 𝑉(𝑡,𝑥(𝑡))𝐶1𝑝([𝑡0,)×𝔻,+) be locally Lipschitz with respect to 𝑥. Assume (i)there exist class-𝒦 functions 𝛼𝑖(𝑖=1,2,3) having global growth momentum at the same level and satisfying 𝛼1(𝑥)𝑉(𝑡,𝑥(𝑡))𝛼2(𝑥),RL𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡))𝛼3(𝑥),(3.18)(ii)there exists 𝑎>0 such that 𝛼1(𝑟) and 𝑟𝑎 have global growth momentum at the same level, where 𝑡>𝑡00, 𝑥𝔻, and 𝑝(0,1). Then 𝑥eq=0 is locally generalized Mittag-Leffler stable. If the assumptions hold globally on 𝑛, then 𝑥eq=0 is globally generalized Mittag-Leffler stable.

Proof. It follows from condition (i) that there exists 𝑘1>0 such that RL𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡))𝛼3(𝑥)𝑘1𝛼2(𝑥)𝑘1𝑉(𝑡,𝑥(𝑡)).(3.19) On the other hand, the linear fractional differential equation RL𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡))=𝑘1𝑉(𝑡,𝑥(𝑡))(3.20) has a unique solution 𝑉𝑉(𝑡,𝑥(𝑡))=0Γ(𝑝)𝑡𝑡0𝑝1𝐸𝑝,𝑝𝑘1𝑡𝑡0𝑝,(3.21) for the initial value 𝑉0=Γ(𝑝)𝑉(𝑡,𝑥(𝑡))(𝑡𝑡0)1𝑝|𝑡=𝑡0.
Using (3.19), (3.20), and Lemma 2.12, we obtain 𝛼1(𝑉𝑥)𝑉(𝑡,𝑥(𝑡))0Γ(𝑝)𝑡𝑡0𝑝1𝐸𝑝,𝑝𝑘1𝑡𝑡0𝑝,(3.22) where 𝐸𝑝,𝑝(𝑘1(𝑡𝑡0)𝑝) is a nonnegative function [23, 24].
In addition, using condition (ii), one gets 𝑘2𝑥𝑎𝛼1(𝑥),(3.23) for all 𝑥𝔻, where 𝑘2>0.
Substituting (3.23) into (3.22), we finally obtain 𝑉𝑥(𝑡)0𝑘𝑎2Γ(𝑝)𝑡𝑡0𝑝1𝐸𝑝,𝑝𝑘1𝑡𝑡0𝑝1/𝑎.(3.24)
Hence, the zero solution of system (3.1) is locally generalized Mittag-Leffler stable. If the assumptions hold globally on 𝑛, then 𝑥eq=0 is globally generalized Mittag-Leffler stable. The proof is completed.

Remark 3.6. The nonnegative function 𝐸𝑝,𝑝(𝑘1(𝑡𝑡0)𝑝) tends to zero as 𝑡 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 𝑓(𝑡,0)=0 and the scalar fractional differential equation RL𝐷𝑞𝑡0,𝑡𝑢(𝑡)=𝑔(𝑡,𝑢),𝑢0=Γ(𝑞)𝑢(𝑡)𝑡𝑡01𝑞|||𝑡=𝑡0,(4.1) where the initial value 𝑢0+, 𝑢(𝑡)𝐶1𝑞([𝑡0,),), 𝑔𝐶([𝑡0,)×,) is Lipschitz in 𝑢 and 𝑔(𝑡,0)=0, 0<𝑞<1. Also, we assume there exists a unique solution 𝑢(𝑡)(𝑡𝑡0) for system (4.1) with the initial value 𝑢0.

Theorem 4.1. For system (3.1), let 𝑥eq=0 be an equilibrium point of system (3.1), and let Ω𝑛 be a domain containing the origin. Assume that there exist a Lyapunov-like function 𝑉𝐶1𝑞([𝑡0,)×Ω,+) and a class-𝒦 function 𝛼 such that 𝑉(𝑡,0)=0, 𝑉(𝑡,𝑥)𝛼(𝑥), and 𝑉(𝑡,𝑥) satisfies the inequality RL𝐷𝑞𝑡0,𝑡𝑉𝑡(𝑡,𝑥)𝑔(𝑡,𝑉(𝑡,𝑥)),(𝑡,𝑥)0,×Ω.(4.2) Suppose further that 𝑔(𝑡,𝑥) is nondecreasing in 𝑥 for each 𝑡. (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 𝑥(𝑡)=𝑥(𝑡,𝑡0,𝑥0) denote the solution of system (3.1) with initial value 𝑥0Ω. Along the solution curve 𝑥(𝑡), 𝑉(𝑡,𝑥(𝑡) can be written as 𝑉(𝑡) and 𝑉𝑉(𝑡)0Γ(𝑞)𝑡𝑡0𝑞1+1Γ(𝑞)𝑡𝑡0(𝑡𝑠)𝑞1𝑔(𝑠,𝑉(𝑠))𝑑𝑠,(4.3) where 𝑉0=Γ(𝑞)𝑉(𝑡)(𝑡𝑡0)1𝑞|𝑡=𝑡0. Applying the fractional integral operator 𝐷𝑡𝑞0,𝑡 to both sides of (4.1) leads to 𝑢𝑢(𝑡)=0Γ(𝑞)𝑡𝑡0𝑞1+1Γ(𝑞)𝑡𝑡0(𝑡𝑠)𝑞1𝑔(𝑠,𝑢(𝑠))𝑑𝑠.(4.4) Now, taking 𝑢0=𝑉0 and applying Lemma 2.14 to inequalities (4.3) and (4.4), one has 𝑉(𝑡)𝑢(𝑡), 𝑡>𝑡0.(i)If the zero solution of (4.1) is stable, then for any initial value 𝑢00, there exists 𝜖>0 such that |𝑢(𝑡)|<𝜖 for all 𝑡>𝑡0. Therefore, taking into account 𝑉(𝑡,𝑥(𝑡))𝛼(𝑥), one gets 𝛼(𝑥(𝑡))𝑉(𝑡,𝑥)𝑢(𝑡)<𝜖,(4.5) that is, 𝑥(𝑡)<𝛼1(𝜖), and the zero solution of system (3.1) is stable.(ii)One can directly derive 𝛼(𝑥(𝑡))𝑉(𝑡,𝑥)𝑢(𝑡)<𝜖(4.6) from the same argument in (i). Then, taking the limit to both sides of (4.6) and combining with the definition of class-𝒦 function, one can obtain lim𝑡+𝑥(𝑡)=0.
The proof is thus finished.

Remark 4.2. In Theorem 4.1 and system (4.1), if we replace order 𝑞 by 𝑝(0,1), but other conditions remain unchanged, then the result in Theorem 4.1 still holds.

Especially, if the class-𝒦 function 𝛼(𝑥) in Theorem 4.1 and 𝑥𝑎 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 𝑥eq=0 be an equilibrium of system (3.1), and let Ω𝑛 be a domain containing the origin. Assume that there exists a Lyapunov-like function 𝑉𝐶1𝑞([𝑡0,)×Ω,+) such that 𝑉(𝑡,0)=0, 𝑉(𝑡,𝑥)𝑘𝑥𝑎, and 𝑉(𝑡,𝑥) is locally Lipschitz in 𝑥 and satisfies the inequality RL𝐷𝑞𝑡0,𝑡𝑉𝑡(𝑡,𝑥)𝑔(𝑡,𝑉(𝑡,𝑥)),(𝑡,𝑥)0,×Ω,(4.7) where 𝑘>0, 𝑎>0. Suppose further that 𝑔(𝑡,𝑥) is nondecreasing in 𝑥 for each 𝑡. 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 𝑛, 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 𝜆0, 𝑏>0, 𝑞<𝛾1𝑞 such that ||||𝑚𝑢𝑢(𝑡)0𝑡𝑡0𝛾𝐸𝑞,1𝛾𝜆𝑡𝑡0𝑞𝑏,(4.8) where 𝑚(0)=0, 𝑚(𝑥)0 and 𝑚(𝑥) is locally Lipschitz in 𝑥 with Lipschitz constant 0.
Taking 𝑢0=𝑉0=Γ(𝑞)𝑉(𝑡,𝑥)(𝑡𝑡0)1𝑞|𝑡=𝑡0 and noting that 𝑉(𝑡,𝑥)𝑢(𝑡) holds from Theorem 4.1, then taking into account (4.8) and 𝑉(𝑡,𝑥)𝑘𝑥𝑎, we obtain 𝑘𝑥(𝑡)𝑎𝑚𝑢𝑉(𝑡,𝑥)0𝑡𝑡0𝛾𝐸𝑞,1𝛾𝜆𝑡𝑡0𝑞𝑏.(4.9) Furthermore, 𝑚𝑡𝑥(𝑡)Γ(𝑞)𝑉𝑡,𝑥0𝑡𝑡01𝑞|||𝑡=𝑡0𝑘1/𝑏𝑡𝑡0𝛾𝐸𝑞,1𝛾𝜆𝑡𝑡0𝑞𝑏/𝑎.(4.10) Let 𝑀(𝑥)=𝑚(Γ(𝑞)𝑉(𝑡,𝑥)(𝑡𝑡0)1𝑞|𝑡=𝑡0)/𝑘1/𝑏. Then it follows that 𝑀𝑥𝑡𝑥(𝑡)0𝑡𝑡0𝛾𝐸𝑞,1𝛾𝜆𝑡𝑡0𝑞𝑏/𝑎,(4.11) where 𝑀(0)=𝑚(Γ(𝑞)𝑉(𝑡,0)(𝑡𝑡0)1𝑞|𝑡=𝑡0)/𝑘1/𝑏=0 due to 𝑉(𝑡,0)=0. It is obvious that 𝑀(𝑥) is a nonnegative function from 𝑚(𝑥),𝑉(𝑡,𝑥)0 and 𝑘>0. In addition, 𝑀(𝑥) is locally Lipschitz in 𝑥 since 𝑚(𝑥) and 𝑉(𝑡,𝑥) are locally Lipschitz in 𝑥. 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 0<𝑞<1 by using a Lyapunov-like function. Compared to [15], we weaken the continuously differential function 𝑉(𝑡,𝑥) as 𝑉(𝑡,𝑥)𝐶1𝑝([𝑡0,)×𝔻,+). 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.

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