q-Mittag-Leffler stability and Lyapunov direct method for differential systems with q-fractional order

In this paper, using the theory of q-fractional calculus, we deal with the q-Mittag-Leffler stability of q-fractional differential systems, and based on it, we analyze the direct Lyapunov method of q-fractional differential systems. Several sufficient criteria are established to guarantee the q-Mittag-Leffler stability and asymptotic stability for the differential systems with q-fractional order.


Introduction
The development of the theory of q-calculus can be dated back to the early 20th century in order to look for a better description of the phenomena having both discrete and continuous behaviors. The q-analog of fractional integrals and derivatives were first studied by Al-Salam [1][2][3] and then by Agrawal [4]. Recently, the q-fractional calculus has been payed more attention [5][6][7][8] because it serves as a bridge between fractional calculus and q-calculus.
In nonlinear systems, Lyapunov's direct method provides an effective way to analyze the stability of a system without explicitly solving the differential equations. Motivated by the application of fractional calculus in nonlinear systems Li,Chen, and Podlubny [9,10] proposed the Mittag-Leffler stability and Lyapunov direct method, and a considerable number results of stability analysis for fractional systems have been reported; see [11][12][13][14][15][16][17][18][19][20][21] and the references therein. However, to our knowledge, the q-Mittag-Leffler stability of q-fractional dynamic systems has not been studied. In this paper, we propose the q-Mittag-Leffler stability and the q-fractional Lyapunov direct method with a hope to enrich the knowledge of the theory of q-fractional calculus. We also present a simple Lyapunov function to get the q-Mittag-Leffler stability for many q-fractional-order systems and show that q-fractional-order dynamical systems also do not have to decay exponentially for the system to be stable in the Lyapunov sense.
Definition 2.1 ([8]) Let f (x) be a real function defined on a q-geometric set A. The qderivative is defined by and Setting q → 1, we have lim q→1 D q f (x) = f (x). Also, the q-integral is given as and We present here two basic properties concerning q-derivatives.
The q-Leibniz product rule is given by where D q is the q-derivative.

Definition 2.2 ([7])
A q-analogue of the Riemann-Liouville fractional integral is defined as If we let q → 1, then the q-analogue of Riemann-Liouville fractional integral q I α q,a f (x) → I α a f (x).

Definition 2.3 ([6])
The Riemann-Liouville type fractional q-derivative of a function f : where [α] denotes the smallest integer greater than or equal to α.
The Caputo type fractional q-derivative of a function f : (0, ∞) → R is define by where [α] denotes the smallest integer greater or equal to α.

q-Mittag-Leffler function
Similar to the Mittag-Leffler function frequently used in the solutions of fractional-order equations, the functions frequently used in the solutions of q-fractional-order equations are the q-analogues of Mittag-Leffler functions defined as and where α > 0 and β ∈ C. When β = 1, the functions e α,β (z, q) and E α,β (z, q) are defined by and

q-Laplace transform of fractional q-integrals, q-derivatives, and q-Mittag-Leffler functions
If n -1 < α ≤ n and I n-α q f (x) ∈ C (n) 1 [0, a], then let (s) = q L s f (x). The q-Laplace transform of the Riemann-Liouville fractional and the Caputo fractional q-derivatives are given by and Taking

q-Mittag-Leffler stability and Lyapunov direct method for differential systems with q-fractional order
Consider the Caputo fractional nonautonomous system q-Mittag-Leffler stability of solutions of the following system: Let f (t, 0) = 0, for all t ∈ [t 0 , t] q , so that system (19) admits the trivial solution.
Now we give some definitions that will be used in studying the q-Mittag-Leffler stability of (19).

Definition 3.1
The trivial solution x(t) = 0 of (19) is said to be asymptotically stable if for all > 0 and t 0 ∈ A, there exists δ = δ(t 0 , ) such that if x 0 < δ implies that lim t→∞ x(t) = 0.
Proof It follows from equations (19) and (20) that There exists a nonnegative function M(t) satisfying Taking the q-Laplace transform of (24) gives where V (s) = q L s {V (t, x(t))}. It then follows that (26) It follows from the inverse Laplace transform that the unique solution of (24) is Since 0 < q < 1, M(t) ≥ 0, and e α,α (-β 3 β 2 (tqτ ) α ; q) are nonnegative functions, we get Substitution of (28) into (21) yields where V (0,x(0)) where m = 0 if and only if x(0) = 0. Because V (t, x) is locally Lipschitz with respect to x and V (0, x(0)) = 0 if and only if x(0), it follows that m is also Lipschitz with respect to x(0) and m(0), which implies the q-Mittag-Leffler stability. In [8], an identity relation between the Caputo fractional q-derivative and the Riemann-Liouville fractional q-derivative is introduced: where α > 0 and n = [α] + 1. When 0 < α < 1, we have Proof From (32) we have and since V (0, x(0)) ≥ 0 and t α q q (1-α) ≥ 0, we obtain the result. Furthermore, if we extend the Lyapunov direct method to the case of q-fractional-order systems, then the asymptotic stability of the corresponding systems can be obtained. The following properties of the q-Mittag-Leffler function and the class-K functions are applied to analysis of the q-fractional Lyapunov direct method.

q-Mittag-Leffler stability of linear systems with q-fractional order
In this section, we present a new result that allows us to find Lyapunov candidate functions for demonstrating the q-Mittag-Leffler of many fractional-order systems using the results of the Lyapunov direct method in Theorem 3.3.

] is the space of all continuous functions on the interval [0, a]). Then, for any time t
Proof Proving expression (35) is equivalent to proving that Using Definition 2.2 and Definition 2.4, (x(t) + x(tq)) C 0 D α q x(t) and C 0 D α q x 2 (t) can be written as So, the left side of expression (36) can be written as Now, let us define the axillary variable y(s) = x(t)x(s), which implies that D q y 2 (s) = y(s) + y(sq) D q y(s) In this way, expression (39) can be written as Since x(t) is regular at zero, using the rule of q-integration by parts, expression (41) becomes Since y 2 (t) = (x(t)x(s)) 2 = 0, it follows that This concludes the proof.

Corollary 4.2 For the q-fractional-order system
where α ∈ (0, 1), x = 0 is the equilibrium point, and . If then the origin of system (44) is q-Mittag-Leffler stable.
Proof Let us propose the following Lyapunov candidate function: Applying Theorem 4.1 results in and thus the origin of system (44) is q-Mittag-Leffler stable.

Proposition 4.3 For the system
where 0 < α < 1 and D q x(t) ∈ C q [0, a], the origin of system (44) is q-Mittag-Leffler stable.
So we can conclude that the trivial solution of system (48) is asymptotically stable. Furthermore, from the expression of exact solution for (48) using two q-analogues of the Mittag-Leffler functions defined by (12) and (13), x(t) = c 1 e (α,1) (-x, q) + c 2 E (α,1) (-x, q), and the properties of these two functions the asymptotical stability can also be derived.

Conclusions
In this paper, we studied the stability of systems with q-fractional order. We proposed the definition of q-Mittag-Leffler stability, presented sufficient criteria of q-Mittag-Leffler stability and the q-fractional Lyapunov direct method of nonlinear systems with q-fractional order. Meanwhile, the q-fractional Lyapunov candidate functions for demonstrating the q-Mittag-Leffler stability of many q-fractional-order systems were discussed. With the rapid development of advanced applied science, we believe that many other study subjects of the q-fractional calculus and q-fractional dynamical systems will attract more attention of researchers. In our following study, we will still focus on the stability problem of q-fractional differential equations in a variety of different forms.