Stability Analysis of Fractional Differential Systems with Order Lying in (1, 2)
© Fengrong Zhang and Changpin Li. 2011
Received: 6 December 2010
Accepted: 7 March 2011
Published: 15 March 2011
The stability of -dimensional linear fractional differential systems with commensurate order and the corresponding perturbed systems is investigated. By using the Laplace transform, the asymptotic expansion of the Mittag-Leffler function, and the Gronwall inequality, some conditions on stability and asymptotic stability are given.
Fractional calculus has a long history with more than three hundred years [1–3]. Up to now, it has been proved that fractional calculus is very useful. Many mathematical models of real problems arising in various fields of science and engineering were established with the help of fractional calculus, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, and electromagnetic waves [4–7].
Recently, the stability theory of fractional differential equations (FDEs) is of main interest in physical systems. Moreover, some stability results have been found [8–17]. These stability results are almost about the linear fractional differential systems with commensurate order (i.e., the fractional derivative order has to be an integer multiple of minimal fractional order ). For example, a necessary and sufficient condition on asymptotic stability of linear fractional differential system with order was first given in . Then, some literatures on the stability of linear fractional differential systems with order have been appeared [11–15]. However, not all the fractional differential systems have fractional orders in (0, 1). There exist fractional models which have fractional orders lying in (1, 2), for example, super-diffusion . Hence, the stability of linear fractional differential systems with order has also been considered by using the conversion methods and transfer function [8, 10]. Almost all of the above literatures dealt with the fractional differential systems with Caputo derivative. Recently, Qian et al.  have investigated the stability of fractional differential systems with Riemann-Liouville derivative whose order lies in (0, 1) in details. It is worth mentioning that not all of the stability conditions are parallel to the corresponding classical integer-order differential equations because of nonlocality and weak singularities of fractional calculus. For example, the solution to an autonomous fractional differential equation cannot define a dynamical system in the sense of semigroup . Of course, some of the mathematical tools for the integer-order differential equation can be applied to fractional kinetics. In , the authors first define the Lyapunov exponents for fractional differential system then determine their bounds, where the basic ideas and techniques are borrowed from [21, 22].
In this paper, we study the stability of autonomous linear fractional differential systems, nonautonomous linear fractional differential systems, and the corresponding perturbed systems with order by using the properties of Mittag-Leffler functions and the Gronwall inequality.
The paper is organized as follows. In Section 2, we first recall some definitions and lemmas used throughout the paper. In Section 3, the stability analysis is presented for autonomous linear fractional differential systems with order . The stability of nonautonomous linear fractional differential systems and the corresponding perturbed systems are studied in Sections 4 and 5, respectively. Conclusions and comments are included in Section 6.
In this section, we recall the most commonly used definitions and properties of fractional derivatives, Mittag-Leffler functions, and their asymptotic expansions.
The following definitions of stability are introduced.
The zero solution of with order is said to be stable if, for any initial values ( ), there exists such that for all . The zero solution is said to be asymptotically stable if, in addition to being stable, as .
It is useful to recall the following asymptotic formulas for our developments in the sequel.
These results were proved in .
Lemma 2.7 (see ).
Lemma 2.8 (Jordan Decomposition ).
Lemma 2.9 (see ).
3. Stability of Autonomous Linear Fractional Differential Systems
3.1. The Riemann-Liouville Derivative Case
The autonomous fractional differential system (3.1) with Riemann-Liouville derivative and the initial conditions (3.2) is asymptotically stable iff . In this case, the components of the state decay towards 0 like . Moreover, the system (3.1) is stable iff either it is asymptotically stable, or those critical eigenvalues which satisfy have the same algebraic and geometric multiplicities.
That is to say, ( ) from the asymptotic expansion (2.12) and ( ). Moreover, the components of the state decay towards 0 like . Taking into account the entire function , we also get the boundedness of ( ; ).
According to the above discussions, the proof is completed.
3.2. The Caputo Derivative Case
The autonomous fractional differential system (3.17) with Caputo derivative and initial conditions (3.18) is asymptotically stable iff . In this case, the components of the state decay towards 0 like . Moreover, the system (3.17) is stable iff either it is asymptotically stable, or those critical eigenvalues which satisfy have the same algebraic and geometric multiplicities.
This theorem can be proved in the same manner as that in the proof of Theorem 3.1, so it is omitted here.
4. Stability of Nonautonomous Linear Fractional Differential Systems
4.1. The Riemann-Liouville Derivative Case
Similarly, we can derive the following conclusion.
So, the zero solution of (4.1) is asymptotically stable.
4.2. The Caputo Derivative Case
The main stability results of this subsection are derived as follows.
The proof line is similar to that of Theorem 4.1.
So, the zero solution of (4.10) is asymptotically stable.
5. Stability of the Perturbed Systems
The following theorem can be proved by the same argument used in the proof of Theorem 4.1.
If the matrix such that , , the critical eigenvalues which satisfy have the same algebraic and geometric multiplicities. Moreover, suppose that there exists a positive function which satisfies the following conditions:
then the zero solution of (5.1) is stable.
then the zero solution of (5.1) is asymptotically stable.
It is well know that many physical phenomena having memory and genetic characteristics can be described by using the fractional differential systems. Especially, the fractional differential systems with order have recently gained an increasing attention [23, 28–31]. It should be noted that [29, 30] are earlier and interesting work on fractional interval systems. Motivated by the above research activities, in this paper, we have studied the stability of linear fractional differential systems and the corresponding perturbed systems with Rimann-Liouville derivative and Caputo derivative for the commensurate order . The main analytic tools used in this paper are the Mittag-Leffler function and the Gronwall inequality. For the autonomous linear fractional differential systems with order , the necessary and sufficient conditions on stability and asymptotic stability are given, which are almost the same as those with the fractional derivative order . But the components of the state decay towards 0 like , which is different from the case with Caputo derivative order . For the nonautonomous linear fractional differential systems, we have derived some sufficient conditions on stability and asymptotic stability. We have further given the asymptotic stability results of the perturbed systems with order .
The authors wish to thank Professors D. Baleanu and Juan J. Trujillo for their kind invitation to submit our paper to their special issue on Fractional Models and their Applications. They also thank the anonymous reviewers of this paper for their careful reading and invaluable correction suggestions. The present work was partially supported by the National Natural Science Foundation of China under Grant no. 10872119 and the the Key Disciplines of Shanghai Municipality under Grant no. S30104.
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
- Debnath L: A brief historical introduction to fractional calculus. International Journal of Mathematical Education in Science and Technology 2004,35(4):487–501. 10.1080/00207390410001686571MathSciNetView ArticleGoogle Scholar
- Ross B: A brief history and exposition of the fundamental theory of fractional calculus. Lecture Notes in Mathematics 1975, 457: 1–36. 10.1007/BFb0067096View ArticleMATHGoogle Scholar
- Koeller RC: Applications of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics 1984,51(2):299–307. 10.1115/1.3167616MathSciNetView ArticleMATHGoogle Scholar
- Sun HH, Abdelwahab AA, Onaral B: Linear approximation of transfer function with a pole of fractional power. IEEE Transactions on Automatic Control 1984,29(5):441–444. 10.1109/TAC.1984.1103551View ArticleMATHGoogle Scholar
- Sun HH, Onaral B, Tsao YY: Application of the positive reality principle to metal electrode linear polarization phenomena. IEEE Transactions on Biomedical Engineering 1984,31(10):664–674.View ArticleGoogle Scholar
- Heaviside O: Electromagnetic Theory. Chelsea, New York, NY, USA; 1971.MATHGoogle Scholar
- Tavazoei MS, Haeri M: A note on the stability of fractional order systems. Mathematics and Computers in Simulation 2009,79(5):1566–1576. 10.1016/j.matcom.2008.07.003MathSciNetView ArticleMATHGoogle Scholar
- Matignon D: Stability results for fractional differential equations with applications to control processing. Proceedings of the IMACS-SMC, 1996 2: 963–968.Google Scholar
- Malti R, Cois O, Aoun M, Levron F, Oustaloup A: Computing impulse response energy of fractional transfer function. Proceedings of the 15th IFAC World Congress, 2002, Barcelona, Spain Google Scholar
- Moze M, Sabatier J, Oustaloup A: LMI characterization of fractional systems stability. In Advances in Fractional Calculus. Edited by: Sabatier J, Agrawal OP, Tenreiro Machado JA. Springer, Dordrecht, The Netherlands; 2007:419–434.View ArticleGoogle Scholar
- Deng W, Li C, Lü J: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics 2007,48(4):409–416. 10.1007/s11071-006-9094-0MathSciNetView ArticleMATHGoogle Scholar
- Odibat ZM: Analytic study on linear systems of fractional differential equations. Computers & Mathematics with Applications 2010,59(3):1171–1183.MathSciNetView ArticleMATHGoogle Scholar
- Radwan AG, Soliman AM, Elwakil AS, Sedeek A: On the stability of linear systems with fractional-order elements. Chaos, Solitons and Fractals 2009,40(5):2317–2328. 10.1016/j.chaos.2007.10.033View ArticleMATHGoogle Scholar
- Sabatier J, Moze M, Farges C: LMI stability conditions for fractional order systems. Computers & Mathematics with Applications 2010,59(5):1594–1609. 10.1016/j.camwa.2009.08.003MathSciNetView ArticleMATHGoogle Scholar
- Qian D, Li C, Agarwal RP, Wong PJY: Stability analysis of fractional differential system with Riemann-Liouville derivative. Mathematical and Computer Modelling 2010,52(5–6):862–874. 10.1016/j.mcm.2010.05.016MathSciNetView ArticleMATHGoogle Scholar
- Li CP, Zhang FR: A survey on thestability of fractional differential equations. The European Physical Journal Special Topics 2011, 193: 27–47. 10.1140/epjst/e2011-01379-1View ArticleGoogle Scholar
- Matignon D: Reprsentations en variables d'tat demodles de guides d'ondes avec derivation fractionnaire, Thèse de Doctorat. Université Paris-Sud 11; 1994.Google Scholar
- Li CP, Zhao ZG: Numerical approximation of nonlinear fractional differential equuations with subdiffusion and Ssperdiffusion. Computers and Mathematics with Applications. In pressGoogle Scholar
- Li CP, Gong ZG, Qian DL, Chen YQ: On the bound of the Lyapunov exponents for the fractional differential systems. Chaos 2010,20(1, article 013127):7.MathSciNetView ArticleMATHGoogle Scholar
- Li C, Xia X: On the bound of the Lyapunov exponents for continuous systems. Chaos 2004,14(3):557–561. 10.1063/1.1768911MathSciNetView ArticleMATHGoogle Scholar
- Li CP, Chen G: Estimating the Lyapunov exponents of discrete systems. Chaos 2004,14(2):343–346. 10.1063/1.1741751MathSciNetView ArticleMATHGoogle Scholar
- Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science, Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
- Wen X-J, Wu Z-M, Lu J-G: Stability analysis of a class of nonlinear fractional-order systems. IEEE Transactions on Circuits and Systems II 2008,55(11):1178–1182.View ArticleGoogle Scholar
- Zhang FZ: Matrix Theory, Universitext. Springer, New York, NY, USA; 1999:xiv+277.View ArticleGoogle Scholar
- Corduneanu C: Principles of Differential and Integral Equations. Allyn and Bacon, Boston, Mass, USA; 1971:vi+201.Google Scholar
- Sun Z-Z, Wu XN: A fully discrete difference scheme for a diffusion-wave system. Applied Numerical Mathematics 2006,56(2):193–209. 10.1016/j.apnum.2005.03.003MathSciNetView ArticleMATHGoogle Scholar
- Ahn H-S, Chen YQ, Podlubny I: Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Applied Mathematics and Computation 2007,187(1):27–34. 10.1016/j.amc.2006.08.099MathSciNetView ArticleMATHGoogle Scholar
- Ahn H-S, Chen YQ: Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica 2008,44(11):2985–2988. 10.1016/j.automatica.2008.07.003MathSciNetView ArticleMATHGoogle Scholar
- Luchko YF, Rivero M, Trujillo JJ, Velasco MP: Fractional models, non-locality, and complex systems. Computers & Mathematics with Applications 2010,59(3):1048–1056.MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.