Elsevier

Automatica

Volume 45, Issue 8, August 2009, Pages 1965-1969
Automatica

Technical communique
Mittag–Leffler stability of fractional order nonlinear dynamic systems

https://doi.org/10.1016/j.automatica.2009.04.003Get rights and content

Abstract

In this paper, we propose the definition of Mittag–Leffler stability and introduce the fractional Lyapunov direct method. Fractional comparison principle is introduced and the application of Riemann–Liouville fractional order systems is extended by using Caputo fractional order systems. Two illustrative examples are provided to illustrate the proposed stability notion.

Introduction

In nonlinear systems, Lyapunov’s direct method (also called the Lyapunov’s second method) provides a way to analyze the stability of a system without explicitly solving the differential equations. The method generalizes the idea, which shows the system is stable if there exist some Lyapunov function candidates for the system. The Lyapunov direct method is a sufficient condition to show the stability of nonlinear systems, which means the system may still be stable, even if one cannot find a Lyapunov function candidate to conclude the system stability property.

As far as the motivation of this paper is concerned, we note that many systems exhibit the fractional phenomena, such as motions in complex media/environments, random walk of bacteria in fractal substance and the chemotaxi behavior and food seeking of microbes (Cohen, Golding, Ron, & Ben-Jacob, 2001), etc. These phenomena are always related to the complexity and heredity of systems due to the fractional properties of system components, such as the fractional viscoelastic material, the fractional circuit element and the fractal structure, etc (Bagley and Torvik, 1983a, Bagley and Torvik, 1983b). In particular, the memristor (a contraction for memory resistor), which is said to be the missing circuit element (Chua, 1971), shows some hereditary properties. Allowing for the fact that the fractional calculus itself is a kind of convolution, the memristor is naturally likely to be linked to fractional calculus. Finally, it is possible that, in the future, there will be more fractional order dynamic systems in micro/nano scales.

Recently, fractional calculus was introduced to the stability analysis of nonlinear systems, for example Momani and Hadid (2004), Zhang, Li, and Chen (2005), Chen (2006), Tarasov (2007), Sabatier (2008) and Li, Chen, Podlubny, and Cao (2008), where integer-order methods of stability analysis were extended to fractional order dynamic systems. However, as pointed out in Chen (2006), the decay of generalized energy of a dynamic system does not have to be exponential for the system to be stable. The energy decay actually can be of any rate, including power law decay. For extending the application of fractional calculus in nonlinear systems, we propose the Mittag–Leffler stability and the fractional Lyapunov direct method with a view to enrich the knowledge of both system theory and fractional calculus. Meanwhile, the fact that computation becomes faster and memory becomes cheaper makes the application of fractional calculus, in reality, possible and affordable (Chen, 2006).

This work is motivated by the simple fact, as also indicated in Chen (2006), that the generalized energy of a system does not have to decay exponentially for the system to be stable in the sense of Lyapunov.

Our contributions of this paper include:

  • The study of the fractional Lyapunov direct method and the Mittag–Leffler stability of nonautonomous systems.

  • The extension of the application of Riemann–Liouville fractional order systems by using Caputo fractional order systems.

  • The fractional comparison principle and several other fractional inequalities extend the application of fractional calculus.

Section snippets

Caputo and Riemann–Liouville fractional derivatives

Fractional calculus plays an important role in modern science (Chen and Moore, 2002, Podlubny, 1999a, Podlubny, 1999b, Sabatier et al., 2007, Tarasov, 2007, Xu and Tan, 2006). In this paper, we use both Riemann–Liouville and Caputo fractional order operators as our main tools, which are shown in Podlubny (1999b). The uniform formula of a fractional integral with α(0,1) is defined as aDtαf(t)=1Γ(α)atf(τ)(tτ)1αdτ, where f(t) is an arbitrary integrable function, aDtα is the fractional

Fractional nonautonomous systems

Consider the Caputo or Riemann–Liouville fractional nonautonomous system (Podlubny, 1999a, Tarasov, 2007) t0Dtαx(t)=f(t,x) with initial condition x(t0), where D denotes either Caputo or Riemann–Liouville fractional operator, α(0,1), f:[t0,]×ΩRn is piecewise continuous in t and locally Lipschitz in x on [t0,]×Ω, and ΩRn is a domain that contains the origin x=0. The equilibrium point of (7) is defined as follows:

Definition 1

The constant x0 is an equilibrium point of fractional dynamic system (7), if

Mittag–Leffler stability

Lyapunov stability provides an important tool for stability analysis in nonlinear systems. In fact, stability issues have been extensively covered by Lyapunov and there are several tests associated with this name. We primarily consider what is often called Lyapunov’s direct method, which involves finding a Lyapunov function candidate for a given nonlinear system. If such a function exists, the system is stable. Applying Lyapunov’s direct method, one can search for an appropriate function. Note

Fractional order extension of Lyapunov direct method

By using the Lyapunov direct method, we can get the asymptotic stability of the corresponding systems. In this section, we extend the Lyapunov direct method to the case of fractional order systems, which leads to the Mittag–Leffler stability.

Theorem 5

Letx=0be an equilibrium point for the system(7)andDRnbe a domain containing the origin. LetV(t,x(t)):[0,)×DRbe a continuously differentiable function and locally Lipschitz with respect toxsuch thatα1xaV(t,x(t))α2xab,0CDtβV(t,x(t))α3xab,wheret0,

Fractional Lyapunov direct method by using the class-K functions

In this section, the class-K functions are applied to the analysis of fractional Lyapunov direct method.

Definition 9

A continuous function α:[0,t)[0,) is said to belong to class-K if it is strictly increasing and α(0)=0 (Khalil, 2002).

Lemma 10 Fractional Comparison Principle

Letx(0)=y(0)and0CDtβx(t)0CDtβy(t), whereβ(0,1). Thenx(t)y(t).

Proof

It follows from 0CDtβx(t)0CDtβy(t) that there exists a nonnegative function m(t) satisfying 0CDtβx(t)=m(t)+0CDtβy(t). Taking the Laplace transform of Eq. (18) yields sβX(s)sβ1x(0)=M(s)+sβY(s)sβ1y(0), where M

Two illustrative examples

The following illustrative examples are provided to show the usefulness of the Mittag–Leffler stability notion.

Example 13

Consider the following fractional order dynamic system 0CDtαx(t)=0tg(τ,x)dτ+h(t,x), where g(t,x)0, x(0)>0, α(0,1) and h(t,x) equal 0 or 0tg(τ,x)dτ for x0 or x=0, respectively. The equilibrium point x=0 is asymptotically stable.

Proof

It follows from 0tg(τ,0)dτ+h(t,0)=0 and Definition 1 that x=0 is the equilibrium point of system (23). Let the Lyapunov candidate be V(x)=x2, it follows

Conclusion and future work

In this paper, we studied the stabilization of nonlinear fractional order dynamic systems. We discussed fractional nonautonomous systems and the application of the Lipschitz condition to fractional order systems. We proposed the definition of Mittag–Leffler stability and the fractional Lyapunov direct method, which enriched the knowledge of both the system theory and the fractional calculus. We introduced the fractional comparison principle. We partly extended the application of

Acknowledgments

We would like to thank Mr. Y. Cao and Dr. C. P. Li for discussions, and the reviewers and the Associate Editor for their useful comments on our paper. Y. Li is supported by the State Scholarship Fund of China (Grant No.LiuJinChu[2007]3020-2007102037). I. Podlubny was supported by grants APVV LPP-0283-06 and VEGA 1/4058/07. Y. Chen and I. Podlubny were funded in part by the National Academy of Sciences under the Collaboration in Basic Science and Engineering Program/Twinning Program supported by

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This work was completed while Y. Li visited the Center for Self-Organizing and Intelligent Systems (CSOIS), Utah State University from August 2007 to August 2008. Part of this paper was presented at the Third IFAC Workshop on Fractional Derivative and Applications (FDA08), November 5-7, 2008, Ankara, Turkey. This paper was recommended for publication in revised form by Associate Editor Wei Kang under the direction of Editor Andre L. Tits.

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