Technical communiqueMittag–Leffler stability of fractional order nonlinear dynamic systems☆
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:
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The study of the fractional Lyapunov direct method and the Mittag–Leffler stability of nonautonomous systems.
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The extension of the application of Riemann–Liouville fractional order systems by using Caputo fractional order systems.
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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 is defined as where is an arbitrary integrable function, is the fractional
Fractional nonautonomous systems
Consider the Caputo or Riemann–Liouville fractional nonautonomous system (Podlubny, 1999a, Tarasov, 2007) with initial condition , where denotes either Caputo or Riemann–Liouville fractional operator, , is piecewise continuous in and locally Lipschitz in on , and is a domain that contains the origin . The equilibrium point of (7) is defined as follows: Definition 1 The constant 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 Letbe an equilibrium point for the system(7)andbe a domain containing the origin. Letbe a continuously differentiable function and locally Lipschitz with respect tosuch thatwhere,
Fractional Lyapunov direct method by using the class- functions
In this section, the class- functions are applied to the analysis of fractional Lyapunov direct method.
Definition 9 A continuous function is said to belong to class- if it is strictly increasing and (Khalil, 2002).
Lemma 10 Fractional Comparison Principle Letand, where. Then.
Proof It follows from that there exists a nonnegative function satisfying Taking the Laplace transform of Eq. (18) yields , where
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 where , , and equal 0 or for or , respectively. The equilibrium point is asymptotically stable.
Proof It follows from and Definition 1 that is the equilibrium point of system (23). Let the Lyapunov candidate be , 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.