International Journal of Differential Equations

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Fractional Differential Equations 2011

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Volume 2011 |Article ID 635165 | https://doi.org/10.1155/2011/635165

Fengrong Zhang, Changpin Li, YangQuan Chen, "Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative", International Journal of Differential Equations, vol. 2011, Article ID 635165, 12 pages, 2011. https://doi.org/10.1155/2011/635165

Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative

Academic Editor: Fawang Liu
Received18 Apr 2011
Accepted14 Jun 2011
Published04 Aug 2011

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:ğ¶ğ·ğ‘ž0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),(1.1) with initial values 𝑥(0)=𝑥0(0),…,𝑥(𝑚−1)(0)=𝑥0(𝑚−1), where 𝑚−1<ğ‘žâ‰¤ğ‘š. 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: ğ”‡ğ‘ž0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),(1.2) where 0<ğ‘žâ‰¤1, ğ”‡ğ‘ž0,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 ℝ+ the set of nonnegative real numbers, by ℝ the set of real numbers, and by ℤ+ the set of positive integer numbers. Let 0<ğ‘ž<1 and set ğ¶ğ‘ž([𝑡0,𝑇],ℝ)={𝑓∈𝐶((𝑡0,𝑇],ℝ),(𝑡−𝑡0)ğ‘žğ‘“(𝑡)∈𝐶([𝑡0,𝑇],ℝ)}, and ğ¶ğ‘ž([𝑡0,𝑇]×Ω,ℝ)={𝑓(𝑡,𝑥(𝑡))∈𝐶((𝑡0,𝑇]×Ω,ℝ),(𝑡−𝑡0)ğ‘žğ‘“(𝑡,𝑥(𝑡))∈𝐶([𝑡0,𝑇]×Ω,ℝ)}, where 𝐶((𝑡0,𝑡],ℝ) 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 ğ‘žâ‰¥0 of function 𝑥(𝑡) is defined as follows: ğ·ğ‘¡âˆ’ğ‘ž0,𝑡1𝑥(𝑡)=Γ(ğ‘ž)𝑡𝑡0(𝑡−𝜏)ğ‘žâˆ’1𝑥(𝜏)𝑑𝜏,(2.1) where Γ(⋅) is the Gamma function.

Definition 2.2. The Riemann-Liouville derivative with order ğ‘ž of function 𝑥(𝑡) is defined as follows: RLğ·ğ‘žğ‘¡0,𝑡1𝑥(𝑡)=𝑑Γ(ğ‘šâˆ’ğ‘ž)𝑚𝑑𝑡𝑚𝑡𝑡0(𝑡−𝜏)ğ‘šâˆ’ğ‘žâˆ’1𝑥(𝜏)𝑑𝜏,(2.2) where 𝑚−1â‰¤ğ‘ž<𝑚 and 𝑚∈ℤ+.

Definition 2.3. The Caputo derivative with noninteger order ğ‘ž of function 𝑥(𝑡) is defined as follows: ğ¶ğ·ğ‘žğ‘¡0,𝑡1𝑥(𝑡)=Γ(ğ‘šâˆ’ğ‘ž)𝑡𝑡0(𝑡−𝜏)ğ‘šâˆ’ğ‘žâˆ’1𝑥(𝑚)(𝜏)𝑑𝜏,(2.3) where 𝑚−1<ğ‘ž<𝑚 and 𝑚∈ℤ+.

Definition 2.4. The Mittag-Leffler function is defined by 𝐸𝛼(𝑧)=âˆžî“ğ‘˜=0𝑧𝑘,Γ(𝑘𝛼+1)(2.4) where 𝛼>0, 𝑧∈ℝ. The two-parameter Mittag-Leffler function is defined by 𝐸𝛼,𝛽(𝑧)=âˆžî“ğ‘˜=0𝑧𝑘,Γ(𝑘𝛼+𝛽)(2.5) where 𝛼>0 and 𝛽∈ℝ, 𝑧∈ℝ.
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 ğ”‡ğ‘žğ‘¡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥) if and only if 𝑓(𝑡,𝑥eq)=ğ”‡ğ‘žğ‘¡0,𝑡𝑥(𝑡)|𝑥(𝑡)=𝑥eq for all 𝑡>𝑡0, where ğ”‡ğ‘žğ‘¡0,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 ğ”‡ğ‘žğ‘¡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)) with order ğ‘žâˆˆ(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 𝔹⊂ℝ𝑛 be a domain containing the origin. The zero solution of ğ”‡ğ‘žğ‘¡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)) is said to be Mittag-Leffler stable if 𝑚𝑥‖𝑥(𝑡)‖≤0î€¸ğ¸ğ‘žî€·î€·âˆ’ğœ†ğ‘¡âˆ’ğ‘¡0î€¸ğ‘žî€¸î€¾ğ‘,(2.6) where 𝑡0 is the initial time and 𝑥0 is the corresponding initial value, ğ‘žâˆˆ(0,1), 𝜆≥0,𝑏>0, 𝑚(0)=0, 𝑚(𝑥)≥0, and 𝑚(𝑥) is locally Lipschitz on 𝑥∈𝔹⊂ℝ𝑛 with the Lipschitz constant ℒ0.

Definition 2.8. Let 𝔹⊂ℝ𝑛 be a domain containing the origin. The zero solution of ğ”‡ğ‘žğ‘¡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)) is said to be generalized Mittag-Leffler stable if 𝑚𝑥‖𝑥(𝑡)‖≤0𝑡−𝑡0î€¸âˆ’ğ›¾ğ¸ğ‘ž,1−𝛾−𝜆𝑡−𝑡0î€¸ğ‘žî€¸î€¾ğ‘,(2.7) where 𝑡0 is the initial time and 𝑥0 is the corresponding initial value, ğ‘žâˆˆ(0,1), âˆ’ğ‘ž<𝛾≤1âˆ’ğ‘ž, 𝜆≥0,𝑏>0, 𝑚(0)=0,𝑚(𝑥)≥0, and 𝑚(𝑥) is locally Lipschitz on 𝑥∈𝔹⊂ℝ𝑛 with the Lipschitz constant ℒ0.

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∶ℝ+→ℝ+ 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∈ℝ+.

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

Lemma 2.12 (see [20]). Let 𝑣,𝑤∈𝐶1âˆ’ğ‘ž([𝑡0,𝑇],ℝ) be locally Hölder continuous for an exponent 0<ğ‘ž<𝜈≤1, â„Žâˆˆğ¶([𝑡0,𝑇]×ℝ,ℝ) and (i)RLğ·ğ‘žğ‘¡0,𝑡𝑣(𝑡)≤ℎ(𝑡,𝑣(𝑡)),(ii)RLğ·ğ‘žğ‘¡0,𝑡𝑤(𝑡)≥ℎ(𝑡,𝑤(𝑡)), 𝑡0<𝑡≤𝑇,with nonstrict inequalities (i) and (ii), where 𝑣0=Γ(ğ‘ž)𝑣(𝑡)(𝑡−𝑡0)1âˆ’ğ‘ž|𝑡=𝑡0 and 𝑤0=Γ(ğ‘ž)𝑤(𝑡)(𝑡−𝑡0)1âˆ’ğ‘ž|𝑡=𝑡0. Suppose further that ℎ satisfies the standard Lipschitz condition ℎ(𝑡,𝑥)−ℎ(𝑡,𝑦)≤ℒ(𝑥−𝑦),𝑥≥𝑦,ℒ>0.(2.8) Then, 𝑣0≤𝑤0 implies 𝑣(𝑡)≤𝑤(𝑡), 𝑡0<𝑡≤𝑇.

Remark 2.13. In Lemma 2.12, if we replace RLğ·ğ‘žğ‘¡0,𝑡 by ğ¶ğ·ğ‘žğ‘¡0,𝑡, but other conditions remain unchanged, then the same result holds.

Lemma 2.14 (see [16]). Let 𝑣,𝑤∈𝐶1âˆ’ğ‘ž([𝑡0,𝑇],ℝ), â„Žâˆˆğ¶([𝑡0,𝑇]×ℝ,ℝ) and (i)𝑣(𝑡)≤(𝑣0/Γ(ğ‘ž))(𝑡−𝑡0)ğ‘žâˆ’1∫+(1/Γ(ğ‘ž))𝑡𝑡0(𝑡−𝑠)ğ‘žâˆ’1ℎ(𝑠,𝑣(𝑠))𝑑𝑠,(ii)𝑤(𝑡)≥(𝑤0/Γ(ğ‘ž))(𝑡−𝑡0)ğ‘žâˆ’1∫+(1/Γ(ğ‘ž))𝑡𝑡0(𝑡−𝑠)ğ‘žâˆ’1ℎ(𝑠,𝑤(𝑠))𝑑𝑠, where 𝑡0<𝑡≤𝑇, 𝑣0=Γ(ğ‘ž)𝑣(𝑡)(𝑡−𝑡0)1âˆ’ğ‘ž|𝑡=𝑡0, 𝑤0=Γ(ğ‘ž)𝑤(𝑡)(𝑡−𝑡0)1âˆ’ğ‘ž|𝑡=𝑡0, and 0<ğ‘ž<1. Assume that both inequalities are nonstrict and ℎ(𝑡,𝑥) is nondecreasing in 𝑥 for each 𝑡. Further, suppose that ℎ satisfies the standard Lipschitz condition ℎ(𝑡,𝑥)−ℎ(𝑡,𝑦)≤ℒ(𝑥−𝑦),𝑥≥𝑦,ℒ>0.(2.9) Then, 𝑣0≤𝑤0 implies 𝑣(𝑡)≤𝑤(𝑡), 𝑡0<𝑡≤𝑇.

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]: ğ¶ğ·ğ‘žğ‘¡0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),(3.1) with the initial condition 𝑥0=𝑥(𝑡0), where 𝑓∶[𝑡0,∞)×Ω→ℝ𝑛 is piecewise continuous in 𝑡 and Ω⊂ℝ𝑛 is a domain that contains the equilibrium point 𝑥eq=0, 0<ğ‘ž<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 ğ‘¥ğ‘’ğ‘ž=0 be an equilibrium point of system (3.1) with 𝑡0=0, and let 𝔻⊂ℝ𝑛 be a domain containing the origin. Let 𝑉(𝑡,𝑥(𝑡))∶[0,∞)×𝔻→ℝ+ be a continuously differentiable function and locally Lipschitz with respect to 𝑥 such that 𝛼1â€–ğ‘¥â€–ğ‘Žâ‰¤ğ‘‰(𝑡,𝑥(𝑡))≤𝛼2â€–ğ‘¥â€–ğ‘Žğ‘,(3.2)𝐶𝐷𝑝0,𝑡𝑉(𝑡,𝑥(𝑡))≤−𝛼3â€–ğ‘¥â€–ğ‘Žğ‘,(3.3) where 𝑡≥0, 𝑥∈𝔻, 𝑝∈(0,1), and 𝛼1, 𝛼2, 𝛼3, ğ‘Ž, and 𝑏 are arbitrary positive constants. Then ğ‘¥ğ‘’ğ‘ž=0 is Mittag-Leffler stable (locally asymptotically stable). If the assumptions hold globally on ℝ𝑛, then ğ‘¥ğ‘’ğ‘ž=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))𝐸𝑝−𝛼3𝛼2𝑡𝑝,(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))𝛼1𝐸𝑝−𝛼3𝛼2𝑡𝑝1/ğ‘Ž,(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 ğ‘¥ğ‘’ğ‘ž=0 be an equilibrium point of system (3.1), and let 𝔻⊂ℝ𝑛 be a domain containing the origin. Assume that there exist a continuously differentiable function 𝑉(𝑡,𝑥(𝑡))∶[𝑡0,∞)×𝔻→ℝ+ and class-𝒦 function 𝛼 satisfying 𝑉(𝑡,𝑥(𝑡))≥𝛼(‖𝑥‖),(3.8)𝐶𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡))≤0,(3.9) where 𝑥∈𝔻, 𝑝∈(0,1). Then ğ‘¥ğ‘’ğ‘ž=0 is locally stable. If the assumptions hold globally on ℝ𝑛, 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)‖≤𝛼−1𝑉𝑡0𝑡,𝑥0,(3.10) where 𝑡≥𝑡0. 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))∈𝐶1−𝑝([𝑡0,∞)×𝔻,ℝ+). Here we give the corresponding results.

Theorem 3.3. Let ğ‘¥ğ‘’ğ‘ž=0 be an equilibrium point of system (3.1), and let 𝔻⊂ℝ𝑛 be a domain containing the origin, 𝑉(𝑡,𝑥(𝑡))∈𝐶1−𝑝([𝑡0,∞)×𝔻,ℝ+). 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>𝑡0≥0, 𝑥∈𝔻, and 𝑝∈(0,1). Then 𝑥eq=0 is locally asymptotically stable. If the assumptions hold globally on ℝ𝑛, 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))(𝑡−𝑡0)𝑝−1 for initial value 𝑉0=Γ(𝑝)𝑉(𝑡,𝑥(𝑡))(𝑡−𝑡0)1−𝑝|𝑡=𝑡0.
Taking into account Lemma 2.12 and the relationship between (3.12) and (3.13), we obtain 𝑉𝑉(𝑡,𝑥(𝑡))≤0Γ(𝑝)𝑡−𝑡0𝑝−1.(3.14) Substituting (3.14) into (3.11) gives ‖𝑥(𝑡)‖≤𝛼−1𝑉0Γ(𝑝)𝑡−𝑡0𝑝−1⟶0(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 𝔻⊂ℝ𝑛 be a domain containing the origin, and let 𝑉(𝑡,𝑥(𝑡))∈𝐶1−𝑝([𝑡0,∞)×𝔻,ℝ+) be locally Lipschitz with respect to 𝑥. Assume 𝑉(𝑡,0)=0, 𝑉(𝑡,𝑥(𝑡))â‰¥ğ‘Žâ€–ğ‘¥â€–ğ‘,RL𝐷𝑝𝑡0,𝑡𝑉(𝑡,𝑥(𝑡))≤0,(3.16) where 𝑡>𝑡0≥0, 𝑥∈𝔻, 𝑝∈(0,1), and ğ‘Ž, 𝑏 are arbitrary positive constants. Then 𝑥eq=0 is generalized Mittag-Leffler stable. If the assumptions hold globally on ℝ𝑛, then 𝑥eq=0 is globally generalized Mittag-Leffler stable.

Proof. In Theorem 3.3, by replacing 𝛼(‖𝑥‖) by ğ‘Žâ€–ğ‘¥â€–ğ‘, we can get ‖𝑉𝑥(𝑡)‖≤0ğ‘Žî€·ğ‘¡âˆ’ğ‘¡0𝑝−1𝐸𝑝,𝑝0⋅𝑡−𝑡0𝑝1/𝑏,(3.17) so the conclusion holds.

Theorem 3.5. Let ğ‘¥ğ‘’ğ‘ž=0 be an equilibrium point of system (3.1), let 𝔻⊂ℝ𝑛 be a domain containing the origin, and let 𝑉(𝑡,𝑥(𝑡))∈𝐶1−𝑝([𝑡0,∞)×𝔻,ℝ+) 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 ğ‘Ž>0 such that 𝛼1(𝑟) and ğ‘Ÿğ‘Ž have global growth momentum at the same level, where 𝑡>𝑡0≥0, 𝑥∈𝔻, and 𝑝∈(0,1). Then 𝑥eq=0 is locally generalized Mittag-Leffler stable. If the assumptions hold globally on ℝ𝑛, 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))=0Γ(𝑝)𝑡−𝑡0𝑝−1⋅𝐸𝑝,𝑝−𝑘1𝑡−𝑡0𝑝,(3.21) for the initial value 𝑉0=Γ(𝑝)𝑉(𝑡,𝑥(𝑡))(𝑡−𝑡0)1−𝑝|𝑡=𝑡0.
Using (3.19), (3.20), and Lemma 2.12, we obtain 𝛼1(𝑉‖𝑥‖)≤𝑉(𝑡,𝑥(𝑡))≤0Γ(𝑝)𝑡−𝑡0𝑝−1⋅𝐸𝑝,𝑝−𝑘1𝑡−𝑡0𝑝,(3.22) where 𝐸𝑝,𝑝(−𝑘1(𝑡−𝑡0)𝑝) is a nonnegative function [23, 24].
In addition, using condition (ii), one gets 𝑘2î€¸â€–ğ‘¥â€–ğ‘Žâ‰¤ğ›¼1(‖𝑥‖),(3.23) for all 𝑥∈𝔻, where 𝑘2>0.
Substituting (3.23) into (3.22), we finally obtain ‖𝑉𝑥(𝑡)‖≤0ğ‘˜ğ‘Ž2Γ(𝑝)𝑡−𝑡0𝑝−1𝐸𝑝,𝑝−𝑘1𝑡−𝑡0𝑝1/ğ‘Ž.(3.24)
Hence, the zero solution of system (3.1) is locally generalized Mittag-Leffler stable. If the assumptions hold globally on ℝ𝑛, then 𝑥eq=0 is globally generalized Mittag-Leffler stable. The proof is completed.

Remark 3.6. The nonnegative function 𝐸𝑝,𝑝(−𝑘1(𝑡−𝑡0)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 RLğ·ğ‘žğ‘¡0,𝑡𝑢(𝑡)=𝑔(𝑡,𝑢),𝑢0=Γ(ğ‘ž)𝑢(𝑡)𝑡−𝑡01âˆ’ğ‘ž|||𝑡=𝑡0,(4.1) where the initial value 𝑢0∈ℝ+, 𝑢(𝑡)∈𝐶1âˆ’ğ‘ž([𝑡0,∞),ℝ), 𝑔∈𝐶([𝑡0,∞)×ℝ,ℝ) is Lipschitz in 𝑢 and 𝑔(𝑡,0)=0, 0<ğ‘ž<1. Also, we assume there exists a unique solution 𝑢(𝑡)(𝑡≥𝑡0) 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 Ω⊂ℝ𝑛 be a domain containing the origin. Assume that there exist a Lyapunov-like function 𝑉∈𝐶1âˆ’ğ‘ž([𝑡0,∞)×Ω,ℝ+) and a class-𝒦 function 𝛼 such that 𝑉(𝑡,0)=0, 𝑉(𝑡,𝑥)≥𝛼(‖𝑥‖), and 𝑉(𝑡,𝑥) satisfies the inequality RLğ·ğ‘žğ‘¡0,𝑡𝑉𝑡(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 𝑥0∈Ω. Along the solution curve 𝑥(𝑡), 𝑉(𝑡,𝑥(𝑡) can be written as 𝑉(𝑡) and 𝑉𝑉(𝑡)≤0Γ(ğ‘ž)𝑡−𝑡0î€¸ğ‘žâˆ’1+1Γ(ğ‘ž)𝑡𝑡0(𝑡−𝑠)ğ‘žâˆ’1𝑔(𝑠,𝑉(𝑠))𝑑𝑠,(4.3) where 𝑉0=Γ(ğ‘ž)𝑉(𝑡)(𝑡−𝑡0)1âˆ’ğ‘ž|𝑡=𝑡0. Applying the fractional integral operator ğ·ğ‘¡âˆ’ğ‘ž0,𝑡 to both sides of (4.1) leads to 𝑢𝑢(𝑡)=0Γ(ğ‘ž)𝑡−𝑡0î€¸ğ‘žâˆ’1+1Γ(ğ‘ž)𝑡𝑡0(𝑡−𝑠)ğ‘žâˆ’1𝑔(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 𝑢0≥0, 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)‖<𝛼−1(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→+âˆžâ€–ğ‘¥(𝑡)‖=0.
The proof is thus finished.

Remark 4.2. In Theorem 4.1 and system (4.1), if we replace order ğ‘ž 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 â€–ğ‘¥â€–ğ‘Ž 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 Ω⊂ℝ𝑛 be a domain containing the origin. Assume that there exists a Lyapunov-like function 𝑉∈𝐶1âˆ’ğ‘ž([𝑡0,∞)×Ω,ℝ+) such that 𝑉(𝑡,0)=0, 𝑉(𝑡,𝑥)â‰¥ğ‘˜â€–ğ‘¥â€–ğ‘Ž, and 𝑉(𝑡,𝑥) is locally Lipschitz in 𝑥 and satisfies the inequality RLğ·ğ‘žğ‘¡0,𝑡𝑉𝑡(𝑡,𝑥)≤𝑔(𝑡,𝑉(𝑡,𝑥)),(𝑡,𝑥)∈0,∞×Ω,(4.7) where 𝑘>0, ğ‘Ž>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 ℝ𝑛, 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 𝜆≥0, 𝑏>0, âˆ’ğ‘ž<𝛾≤1âˆ’ğ‘ž such that ||||≤𝑚𝑢𝑢(𝑡)0𝑡−𝑡0î€¸âˆ’ğ›¾ğ¸ğ‘ž,1−𝛾−𝜆𝑡−𝑡0î€¸ğ‘žî€¸î€¾ğ‘,(4.8) where 𝑚(0)=0, 𝑚(𝑥)≥0 and 𝑚(𝑥) is locally Lipschitz in 𝑥 with Lipschitz constant ℒ0.
Taking 𝑢0=𝑉0=Γ(ğ‘ž)𝑉(𝑡,𝑥)(𝑡−𝑡0)1âˆ’ğ‘ž|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)â‰¥ğ‘˜â€–ğ‘¥â€–ğ‘Ž, we obtain 𝑘‖𝑥(𝑡)â€–ğ‘Žî€½ğ‘šî€·ğ‘¢â‰¤ğ‘‰(𝑡,𝑥)≤0𝑡−𝑡0î€¸âˆ’ğ›¾ğ¸ğ‘ž,1−𝛾−𝜆𝑡−𝑡0î€¸ğ‘žî€¸î€¾ğ‘.(4.9) Furthermore, âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©ğ‘šî‚µî€·î€·ğ‘¡â€–ğ‘¥(𝑡)‖≤Γ(ğ‘ž)𝑉𝑡,𝑥0𝑡−𝑡01âˆ’ğ‘ž|||𝑡=𝑡0𝑘1/𝑏⋅𝑡−𝑡0î€¸âˆ’ğ›¾ğ¸ğ‘ž,1−𝛾−𝜆𝑡−𝑡0î€¸ğ‘žî€¸âŽ«âŽªâŽªâŽ¬âŽªâŽªâŽ­ğ‘/ğ‘Ž.(4.10) Let 𝑀(𝑥)=𝑚(Γ(ğ‘ž)𝑉(𝑡,𝑥)(𝑡−𝑡0)1âˆ’ğ‘ž|𝑡=𝑡0)/𝑘1/𝑏. Then it follows that 𝑀𝑥𝑡‖𝑥(𝑡)‖≤0𝑡−𝑡0î€¸âˆ’ğ›¾ğ¸ğ‘ž,1−𝛾−𝜆𝑡−𝑡0î€¸ğ‘žî€¸î€¾ğ‘/ğ‘Ž,(4.11) where 𝑀(0)=𝑚(Γ(ğ‘ž)𝑉(𝑡,0)(𝑡−𝑡0)1âˆ’ğ‘ž|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<ğ‘ž<1 by using a Lyapunov-like function. Compared to [15], we weaken the continuously differential function 𝑉(𝑡,𝑥) as 𝑉(𝑡,𝑥)∈𝐶1−𝑝([𝑡0,∞)×𝔻,ℝ+). 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.

References

  1. R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 299–307, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011. View at: Publisher Site | Google Scholar
  3. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proceedings of the IMACS-SMC, vol. 2, pp. 963–968, 1996. View at: Google Scholar
  4. W. H. Deng, C. P. Li, and Q. Guo, “Analysis of fractional differential equations with multi-orders,” Fractals, vol. 15, no. 2, pp. 173–182, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. W. H. Deng, C. P. Li, and J. H. Lü, “Stability analysis of linear fractional differential system with multiple time delays,” Nonlinear Dynamics, vol. 48, no. 4, pp. 409–416, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. M. Moze, J. Sabatier, and A. Oustaloup, “LMI characterization of fractional systems stability,” in Advances in Fractional Calculus, pp. 419–434, Springer, Dordrecht, The Netherlands, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. D. L. Qian and C. P. Li, “Stability analysis of the fractional differential systems with Miller-Ross sequential derivative,” in Proceedings of the 8th World Congress on Intelligent Control and Automation, pp. 213–219, Jinan, China, 2010. View at: Google Scholar
  8. D. L. Qian, C. P. Li, R. P. Agarwal, and P. J. Y. Wong, “Stability analysis of fractional differential system with Riemann-Liouville derivative,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 862–874, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. A. G. Radwan, A. M. Soliman, A. S. Elwakil, and A. Sedeek, “On the stability of linear systems with fractional-order elements,” Chaos, Solitons & Fractals, vol. 40, no. 5, pp. 2317–2328, 2009. View at: Google Scholar
  10. J. Sabatier, M. Moze, and C. Farges, “LMI stability conditions for fractional order systems,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1594–1609, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. M. S. Tavazoei and M. Haeri, “A note on the stability of fractional order systems,” Mathematics and Computers in Simulation, vol. 79, no. 5, pp. 1566–1576, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511–522, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. Y. Li, Y. Q. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. Y. Li, Y. Q. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1810–1821, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. Z. Denton and A. S. Vatsala, “Fractional integral inequalities and applications,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1087–1094, 2010. View at: Google Scholar | Zentralblatt MATH
  17. T. C. Hu, D. L. Qian, and C. P. Li, “Comparison theorems for fractional differential equations,” Communication on Applied Mathematics and Computation, vol. 23, no. 1, pp. 97–103, 2009 (Chinese). View at: Google Scholar
  18. C. P. Li and Z. G. Zhao, “Introduction to fractional integrability and differentiability,” European Physical Journal —Special Topics, vol. 193, no. 1, pp. 5–26, 2011. View at: Publisher Site | Google Scholar
  19. L. Huang, Stability Theory, Peking University Press, Beijing, China, 1992.
  20. V. Lakshmikantham, S. Leela, and D. J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge UK, 2009.
  21. H. Pollard, “The completely monotonic character of the Mittag-Leffler function Eα(−x),” Bulletin of the American Mathematical Society, vol. 54, pp. 1115–1116, 1948. View at: Publisher Site | Google Scholar
  22. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  23. K. S. Miller and S. G. Samko, “A note on the complete monotonicity of the generalized Mittag-Leffler function,” Real Analysis Exchange, vol. 23, no. 2, pp. 753–755, 1997. View at: Google Scholar
  24. W. R. Schneider, “Completely monotone generalized Mittag-Leffler functions,” Expositiones Mathematicae, vol. 14, no. 1, pp. 3–16, 1996. View at: Google Scholar | Zentralblatt MATH

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