Research Article | Open Access
On the Stability of Some Discrete Fractional Nonautonomous Systems
Using the Lyapunov direct method, the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference is studied. The conditions for uniform stability, uniform asymptotic stability, and uniform global stability are discussed.
The fractional calculus which is as old as the usual calculus deals with the generalization of integration and differentiation of integer order to any order. Recently, there has been great interest in this calculus as it turned out that it has many applications in many fields of science and engineering [1–6].
The stability of fractional-order linear and nonlinear dynamic systems was a subject of many reports [13–19], while the stability of discrete dynamic systems was discussed in many books and articles (see, e.g.,  and the references therein). But the stability of discrete fractional dynamic systems, to our knowledge, has not been reported.
The analysis of the dynamics of complex or hypercomplex systems requires new methods and techniques to be developed. Particularly, the systems which have both continuous evolution and discrete evolution represents a new direction in many fields of science and engineering. For these type of systems the classical or the improved techniques and methods (e.g., with or without nonlocal terms described usually by the fractional derivatives and integrals) are not enough to describe accurately their dynamics. In this line of thought the combination of the fractional and discrete operators leads to a better description of the systems mentioned above. However, these new hybrid methods are at the beginning of their evolutions and new efforts are required to be developed and implemented in practical applications.
In , to demonstrate the advantage of using fractional-order derivatives instead of integer-order derivatives, the authors considered 2 systems. The system with integer-order derivative turned to be unstable. But the second system, where the integer-order derivative was replaced by fractional-order derivative, turned out to be stable. The same argument still holds for integer difference and fractional difference systems. To illustrate this, we consider the following two systems: where and the operator is defined by (2.10). The solution of (1.1) is which clearly tends to as for and thus is unstable. The solution of (1.2) which is defined on has the form where the operator is defined by (2.6). The value of at is computed as follows:
The purpose of this paper is to state the stability theorem for discrete nonlinear dynamic systems in the sense of the Caputo fractional differences.
The paper is organized as follows. In Section 2 basic definitions of fractional calculus and discrete fractional calculus are mentioned. Section 3 presents our main results on the stability of discrete fractional nonautonomous systems.
For a function defined on the interval , the left and right Riemann-Liouville integrals of order , are defined, respectively, by [1–3] while the left and right Riemann-Liouville derivatives of order are defined, respectively, by [1–3] where and is the Gamma function.
The left and right Caputo fractional derivatives of order on the interval are defined, respectively, by
Property 1. We have
Property 2. If , then
Below we present basic definitions and properties of fractional sums and differences.
First we define the set and . The (left) fractional sum of , defined on , of order is defined by [7–12] where , , and . The (right) fractional sum of , defined on , of order is defined by where . Analogous of the case of fractional derivatives, the Riemann left and right fractional differences of order are defined, respectively, by where . It can be obviously noticed that maps functions defined on to functions defined on and maps functions defined on to functions defined on , while maps functions defined on to functions defined on and maps functions defined on to functions defined on .
Similar to the case of fractional derivatives, the left and right Caputo fractional differences of order of a function defined on and are defined, respectively, by  where . The following property gives the relation between the Riemann and Caputo fractional differences .
Property 3. For any , we have the following: Particularly, when we have
Property 4. Let , and let be a function defined on suitable subsets of and . Then, Particularly, when we have
3. Lyapunov Stability Theorems for Nonautonomous Systems of Some Fractional Difference Equations
The stability theory of dynamic systems has a crucial role within the system theory and engineering. The stability of equilibrium points is usually done by utilizing the Lyapunov stability. Hypercomplex systems, namely, systems with both discrete and continuous behavior, need new methods and techniques to define their Lyapunov stability.
In this section we extend the method of the Lyapunov functions to study the stability of solutions of the following system: where , , , is continuous, and . We note that a Lyapunov function for the system (3.1) must depend on and . Now let , for all so that the system (3.1) admits the trivial solution. Next we list some definitions that will be used in studying the stability properties of (3.1).
Definition 3.1. The trivial solution of (3.1) is said to be(i)stable if, for each and , there exists a such that for any solution with one has , for all ,(ii)uniformly stable if it is stable and depends solely on ,(iii)asymptotically stable if it is stable and for all there exists if implies that . (iv)uniformly asymptotically stable if it is uniformly stable and, for each , there exists and such that implies for all and for all ,(v)globally asymptotically stable if it is asymptotically stable for all ,(vi)globally uniformly asymptotically stable if it is uniformly asymptotically stable for all .
Definition 3.2. A function is said to belong to the class if and only if , , and is strictly monotonically increasing in . If , , and , then is said to belong to class .
Definition 3.3. A real valued function defined on , where , is said to be positive definite if and only if for all and there exists such that .
Definition 3.4. A real valued function defined on , where , is said to be decrescent if and only if for all and there exists such that .
Now, we can state the theorems regarding the stability of solutions of the system (3.1).
Theorem 3.5. If there exists a positive definite and decrescent scalar function such that and , then the trivial solution of (3.1) is uniformly stable.
Proof. Let be a solution of system (3.1). Since is positive definite and decrescent, there exist such that for all . For each , , we choose a such that . For any solution of (3.1) we have with . Since , by using (2.14) in Property 4 we have . Consequently, and thus .
Theorem 3.6. If there exists a positive definite and decrescent scalar function such that where , then the trivial solution of (3.1) is uniformly asymptotically stable.
Proof. Since all the conditions of Theorem 3.5 are satisfied, the trivial solution of the system (3.1) is uniformly stable. Let and correspond to uniform stability. Choose a fixed and . Now, choose and large enough such that . Such a large can be chosen since . Now, we claim that for all . If this is not true, due to (3.3) and Property 4, we get Substituting , we get which is a contradiction. Thus, there exists a such that . But in this case, since the trivial solution is uniformly stable and is arbitrary, for all whenever .
Theorem 3.7. If there exists a function such that where hold for all , then the trivial solution of (3.1) is globally uniformly asymptotically stable.
Proof. Since the conditions of Theorem 3.6 are satisfied, the trivial solution of (3.1) is uniformly asymptotically stable. It remains to show that the domain of attraction of is all of . Since , in the proof of Theorem 3.6 may be chosen arbitrary large and can be chosen such that it satisfies . Thus, the globally uniformly asymptotic stability of is concluded.
Lemma 3.8. If , then, for , one has
Proof. From Property 3, we have Since and , we obtain
(i) If the assumptions in Theorem 3.5 are satisfied except replacing by , then the trivial solution of (3.1) is uniformly stable.
(ii) If the assumptions in Theorem 3.6 are satisfied except replacing by , then the trivial solution of (3.1) is uniformly asymptotically stable.
(iii) If the assumptions in Theorem 3.7 are satisfied except replacing by , then the trivial solution of (3.1) is globally uniformly asymptotically stable.
In this paper using the Lyapunov direct method, we studied the stability of discrete nonautonomous systems in the sense of the Caputo fractional difference. We listed the conditions for uniform stability, uniformly asymptotic stability, and globally uniformly asymptotic stability for such systems. We will consider the Mittag-Leffler stability for such systems in future works.
This work is partially supported by the Scientific and Technical Research Council of Turkey. The authors would like to thank to the anonymous referee for her/his valuable comments.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Dractional Differential Equations, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
- G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
- B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer-Verlag, New York, NY, USA, 2003.
- R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, Conn, USA, 2006.
- R. L. Bagley and P. J. Torvik, “On the appearance of the fractional derivative in the behavior of real materials,” Journal of Applied Mechanics, Transactions ASME, vol. 51, no. 2, pp. 294–298, 1984.
- K. S. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, pp. 139–152, Nihon University, Koriyama, Japan, 1989.
- F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165–176, 2007.
- F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981–989, 2009.
- F. M. Atıcı and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–12, 2009.
- T. Abdeljawad and D. Baleanu, “Fractional differences and integration by parts,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 574–582, 2011.
- T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011.
- J. Chen, D. M. Xu, and B. Shafai, “On sufficient conditions for stability independent of delay,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 40, no. 9, pp. 1675–1680, 1995.
- M. P. Lazarević, “Finite time stability analysis of fractional control of robotic time-delay systems,” Mechanics Research Communications, vol. 33, no. 2, pp. 269–279, 2006.
- W. Deng, C. Li, and J. Lü, “Stability analysis of linear fractional differential system with multiple time delays,” Nonlinear Dynamics, vol. 48, no. 4, pp. 409–416, 2007.
- F. Merrikh-Bayat and M. Karimi-Ghartemani, “An efficient numerical algorithm for stability testing of fractional-delay systems,” ISA Transactions, vol. 48, no. 1, pp. 32–37, 2009.
- X. Zhang, “Some results of linear fractional order time-delay system,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 407–411, 2008.
- S. Momani and S. Hadid, “Lyapunov stability solutions of fractional integrodifferential equations,” International Journal of Mathematics and Mathematical Sciences, no. 45 48, pp. 2503–2507, 2004.
- 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.
- R. P. Agarwal, Difference Equations and Inequalities, Theory, Methods, and Application, Marcel Dekker, New York, NY, USA, 2000.
Copyright © 2012 Fahd Jarad 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.