Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay

and Applied Analysis 3 In particular, if x t0 0, we obtain c t0 D q t x t t0D q t x t . 2.4 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


Introduction
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 16 and Krasovski 17 .Whereas Razumikhin 16 used the Lyapunov-type functions V x t depending on the current value x t of the solution, Krasovski 17 proposed to use functionals V x t depending on the whole solution segment x t , that is, the true state of the delay system.The reader can see 18 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 24 .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.

Caputo and Riemann-Liouville Fractional Derivatives
In the fractional calculus the Riemann-Liouville and the Caputo fractional derivatives are defined, respectively 1, 2 , where x t is an arbitrary differentiable function, n ∈ N, and t 0 D q t and c t 0 D q t are the Riemann-Liouville and Caputo fractional derivatives of order q on t 0 , t , respectively, and Γ • denotes the Gamma function.
For 0 < q ≤ 1 we have

2.2
Some properties of the Riemann -Liouville and the Caputo derivatives are recalled below 1, 2 .When 0 < q < 1, we have Abstract and Applied Analysis 3 In particular, if x t 0 0, we obtain

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 α > 0 and z ∈ C. The Mittag-Leffler function with two parameters appears most frequently and has the following form: where α > 0, β > 0, and z ∈ C. For β 1 we obtain E α,1 z E α z .Also we mention that E 1,1 z e z .

Fractional Nonlinear Time-Delay System
Let C a, b , R n be the set of continuous functions mapping the interval a, b to R n .In many situations, one may wish to identify a maximum time delay r of a system.In this case, we are often interested in the set of continuous function mapping −r, 0 to R n , for which we simplify the notation to C C −r, 0 , R n .For any A > 0 and any continuous function of time ψ ∈ C t 0 − r, t 0 A , R n , t 0 ≤ t ≤ t 0 A, let ψ t θ ∈ C be a segment of function ψ defined as ψ t θ ψ t θ , −r ≤ θ ≤ 0. Consider the Caputo fractional nonlinear time-delay system where x t ∈ R n , 0 < α ≤ 1, and f : R × C → R n .Equation 3.1 indicates the Caputo derivatives of the state variable x on t 0 , t and x ξ for t − r ≤ ξ ≤ t.As such, to determine the future evolution of the state, it is necessary to specify the initial state variables x t in a time interval of length r, say, from t 0 − r to t 0 , that is, where φ ∈ C is given.In other words we have x t 0 θ φ θ , −r ≤ θ ≤ 0.
Throughout the paper we will use the Euclidean norm for vectors denoted by • .The space of continuous initial functions C −r, 0 , R n is provided with the supremum norm

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 t required the value of x t in the interval t − r, t , that is, x t , it is natural to expect that for a time-delay system, corresponding the Lyapunov function be a functional V t, x t depending on x t , which also should measure the deviation of x t from the trivial solution 0.
Definition 4.1.Let V t, φ be differentiable, and let x t τ, ϕ be the solution of 3.1 at time t with initial condition x τ ϕ.Then, we calculate the Riemann-Liouville and the Caputo derivatives of V t, x t with respect to t and evaluate it at t τ as follows, respectively, where 0 < q ≤ 1.
Definition 4.2 exponential stability 25 .The solution of 3.1 is said to be exponential stable if there exist b > 0 and a ≥ 0 such that for every solution x t, ϕ , ϕ ∈ C −r, 0 , R n the following exponential estimate holds:

4.2
Definition 4.3 Mittag-Leffler stability .The solution of 3.1 is said to be Mittag-Leffler stable if and m x is locally the Lipschitz on x ∈ B ⊂ R n with the Lipschitz constant m 0 .Theorem 4.4.Suppose that α 1 , α 2 , and β are positive constants and V, w : C −r, 0 , R n → R are continuous functionals.If the following conditions are satisfied for all ϕ ∈ C −r, 0 , R n : That is the solution of 3. where M t ≥ 0 for t ≥ 0. From 3.2 , we have V x t 0 ϕ V φ .Then, the solution of 4.7 with initial condition V x t 0 ϕ V ϕ is given by where * is convolution operator.Since both t γ−1 and E γ,γ −βt γ are nonnegative functions, it follows that Then conditions 1 and 2 yield Comparing the left and the right hand sides, we have Proof.By using 2.3 we have Proof.It follows from Lemma 4.5 and V x t ϕ ≥ 0 that for all t ≥ t 0 .Following the same proof as in Theorem 4.4 yields

Conclusions
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.