Research Article | Open Access
S. J. Sadati, D. Baleanu, A. Ranjbar, R. Ghaderi, T. Abdeljawad (Maraaba), "Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay", Abstract and Applied Analysis, vol. 2010, Article ID 108651, 7 pages, 2010. https://doi.org/10.1155/2010/108651
Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay
Fractional calculus started to play an important role for analysis of the evolution of the nonlinear dynamical systems which are important in various branches of science and engineering. In this line of taught in this paper we studied the stability of fractional order nonlinear time-delay systems for Caputo's derivative, and we proved two theorems for Mittag-Leffler stability of the fractional nonlinear time delay systems.
During the last decade the fractional calculus [1–5] has gained importance in both theoretical and applied aspects of several branches of science and engineering [6–15]. However there are several open problems in this area. One of them is related to the stability of the fractional systems in the presence of the delay. Both delay and fractional systems are describing separately the evolution of the dynamical systems involving memory effect. Particularly, for the time delay case we mention the seminal works on the applicability of Lyapunov’s second method [16, 17].
Since 1950s different types of the Lyapunov functions have been proposed for the stability analysis of delay systems, see the pioneering works of Razumikhin  and Krasovskiǐ . Whereas Razumikhin  used the Lyapunov-type functions depending on the current value (t) of the solution, Krasovskiǐ  proposed to use functionals depending on the whole solution segment , that is, the true state of the delay system. The reader can see  for more details.
Some literatures published about stability of fractional-order linear time delay systems can be found in [19, 20]. In the base of Lyapunov’s second method, some work has been done in the field of stability of fractional order nonlinear systems without delay [21–23]. Razumikhin theorem for the fractional nonlinear time-delay systems was extended recently in . However few attempts were done in order to combine these two powerful concepts and to observe what the benefits of this combination are.
The main aim of this paper is to establish the Mittag-Leffler stability theorem for fractional order nonlinear time-delay systems.
The organization of the manuscript is given belo.
In Section 2 some basic definitions of fractional calculus used in this paper are mentioned. Section 3 introduces briefly the fractional nonlinear time-delay systems. Section 4 deals with the Mittag-Leffler stability theorem when both fractional derivatives and delay are taken into account.
Finally, Section 5 is devoted to our conclusions.
2. Fractional Calculus
2.1. Caputo and Riemann-Liouville Fractional Derivatives
In the fractional calculus the Riemann-Liouville and the Caputo fractional derivatives are defined, respectively [1, 2], where is an arbitrary differentiable function, , and and are the Riemann-Liouville and Caputo fractional derivatives of order on , respectively, and denotes the Gamma function.
When , we have In particular, if , we obtain
2.2. Mittag-Leffler Function
Similar to the exponential function frequently used in the solutions of integer-order systems, a function frequently used in the solutions of fractional-order systems is the Mittag-Leffler function defined as where and . The Mittag-Leffler function with two parameters appears most frequently and has the following form: where , , and . For we obtain . Also we mention that .
3. Fractional Nonlinear Time-Delay System
Let be the set of continuous functions mapping the interval to . In many situations, one may wish to identify a maximum time delay of a system. In this case, we are often interested in the set of continuous function mapping to , for which we simplify the notation to . For any and any continuous function of time , , let be a segment of function defined as , .
Consider the Caputo fractional nonlinear time-delay system where , , and . Equation (3.1) indicates the Caputo derivatives of the state variable on and for . As such, to determine the future evolution of the state, it is necessary to specify the initial state variables in a time interval of length , say, from to , that is, where is given. In other words we have .
Throughout the paper we will use the Euclidean norm for vectors denoted by . The space of continuous initial functions is provided with the supremum norm
4. Mittag-Leffler Stability of Fractional Nonlinear Time Delay Systems
As in the study of systems without delay, an effective method for determining the stability of a time-delay system is the Lyapunov method. Since in a time-delay system the “state” at time required the value of in the interval , that is, , it is natural to expect that for a time-delay system, corresponding the Lyapunov function be a functional depending on , which also should measure the deviation of from the trivial solution .
Definition 4.1. Let be differentiable, and let be the solution of (3.1) at time with initial condition . Then, we calculate the Riemann-Liouville and the Caputo derivatives of with respect to and evaluate it at as follows, respectively, where .
Definition 4.3 (Mittag-Leffler stability). The solution of (3.1) is said to be Mittag-Leffler stable if where , , , , , and is locally the Lipschitz on with the Lipschitz constant .
Theorem 4.4. Suppose that , and are positive constants and are continuous functionals. If the following conditions are satisfied for all :(1)(2),(3) has fractional derivative of order for all ,(4) for all and then That is the solution of (3.1) is Mittag-Leffler stable.
Proof. Given any , condition () implies that
From (4.5) and condition (), we have
where for .
From (3.2), we have . Then, the solution of (4.7) with initial condition is given by where * is convolution operator. Since both and are nonnegative functions, it follows that Then conditions () and () yield Comparing the left and the right hand sides, we have
Lemma 4.5. Let and , then
Proof. By using (2.3) we have . Because and we get .
Theorem 4.6. Assume that the assumptions in Theorem 4.4 are satisfied except replacing by then one has
Corollary 4.7. For one has exponential stability 
Some complex systems which appear in several areas of science and engineering involve delay and they have memory effect. The combined use of the fractional derivative and delay may lead to a better description of such systems. From these reasons in this paper we obtained the Mittag-Leffler stability theorem in the presence of both the Riemann-Liouville or the Caputo fractional derivatives and delay. The obtained theorems contain particular cases of the fractional calculus versions as well as the time-delay ones.
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