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

Abstract

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.

1 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 [1–3] and then by Agrawal [4]. Recently, the q-fractional calculus has been payed more attention [5–8] 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 [11–21] 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.

2 Preliminaries

2.1 Definitions and properties of q-caculus

This section is devoted to recall some essential definitions and properties of q-calculus [1–4, 8].

If $q\in R,0< q<1$, a subset A of R is called q-geometric if $qx\in A$ whenever $x\in A$. If a subset A of R is q-geometric, then it contains all geometric sequences $\{xq^{n}\}_{n=0}^{\infty}$, $x\in A$.

Definition 2.1

([8])

Let $f(x)$ be a real function defined on a q-geometric set A. The q-derivative is defined by

$$\begin{aligned}& D_{q}f(x)=\frac{f(qx)-f(x)}{(q-1)x}, \quad x\in A\setminus\{0\}, \end{aligned}$$

(1)

and

$$\begin{aligned}& D_{q}f(x)\vert_{x=0}=\lim_{n\rightarrow\infty} \frac{f(q^{n})-f(0)}{q ^{n}}. \end{aligned}$$

(2)

Setting $q\rightarrow1$, we have $\lim_{q\rightarrow1}D_{q}f(x)=f ^{\prime}(x)$.

Also, the q-integral is given as

$$\begin{aligned}& \int_{0}^{x}f(t)\,d_{q}t=(1-q)x\sum _{n=0}^{\infty}q^{n}f \bigl(q^{n}x\bigr),\quad x\in A, \end{aligned}$$

(3)

and

$$\begin{aligned}& \int_{a}^{b}f(t)\,d_{q}t= \int_{0}^{b}f(t)\,d_{q}t- \int_{0}^{a}f(t)\,d_{q}t,\quad a,b\in A. \end{aligned}$$

(4)

We present here two basic properties concerning q-derivatives.

Property 1

([7])

$$\begin{aligned}& D_{q}(f \pm g) (x)=D_{q}f(x)\pm D_{q}g(x). \end{aligned}$$

(5)

Property 2

([7])

The q-Leibniz product rule is given by

$$\begin{aligned}& D_{q}\bigl[g(x)f(x)\bigr]=g(qx)D_{q}f(x)+f(x)D_{q}g(x), \end{aligned}$$

(6)

where $D_{q}$ is the q-derivative.

The q-analogue of exponent $(s-t)^{(k)}$ is

$$(s-t)^{(0)}=1,\qquad (s-t)^{(k)}=\prod_{j=0}^{k-1} \bigl(x-yq^{j}\bigr),\quad k\in N,x,y \in R. $$

Definition 2.2

([7])

A q-analogue of the Riemann–Liouville fractional integral is defined as

$$\begin{aligned}& I_{q,a}^{\alpha}f(x)= \int_{0}^{x}\frac{(x-qs)^{(\alpha-1)}}{\Gamma _{q}(\alpha)}f(s)\,d_{q}s,\quad \alpha>0. \end{aligned}$$

(7)

If we let $q\rightarrow1$, then the q-analogue of Riemann–Liouville fractional integral ${}_{q}I_{q,a}^{\alpha}f(x)\rightarrow I_{a}^{ \alpha}f(x)$.

Definition 2.3

([6])

The Riemann–Liouville type fractional q-derivative of a function $f:(0, \infty)\rightarrow R$ is defined by

$$\begin{aligned}& \bigl(D_{q,a}^{\alpha}f\bigr) (x)= \textstyle\begin{cases} (I_{q,a}^{-\alpha}f)(x),& \alpha\leq0, \\ (D_{q,a}^{[\alpha]}I_{q,a}^{[\alpha]-\alpha}f)(x),& \alpha>0, \end{cases}\displaystyle \end{aligned}$$

(8)

where $[\alpha]$ denotes the smallest integer greater than or equal to α.

Definition 2.4

([6])

The Caputo type fractional q-derivative of a function $f:(0, \infty)\rightarrow R$ is define by

$$\begin{aligned}& \bigl(^{C}D_{q,a}^{\alpha}f\bigr) (x)= \textstyle\begin{cases} (I_{q,a}^{-\alpha}f)(x),& \alpha\leq0, \\ (I_{q,a}^{[\alpha]-\alpha]}D_{q,a}^{[\alpha]}f)(x),& \alpha>0, \end{cases}\displaystyle \end{aligned}$$

(9)

where $[\alpha]$ denotes the smallest integer greater or equal to α.

2.2 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

$$\begin{aligned}& e_{\alpha,\beta}(z,q)=\sum_{n=0}^{\infty} \frac{z^{n\alpha}}{ \Gamma_{q}(n\alpha+\beta)} \quad \bigl(\bigl\vert z(1-q)^{\alpha}\bigr\vert < 1\bigr) \end{aligned}$$

(10)

and

$$\begin{aligned}& E_{\alpha,\beta}(z,q)=\sum_{n=0}^{\infty} \frac{q^{\frac{ \alpha n(n-1)}{2}}z^{n\alpha}}{\Gamma_{q}(n\alpha+\beta)}\quad (z \in C), \end{aligned}$$

(11)

where $\alpha>0$ and $\beta\in\mathcal{C}$. When $\beta=1$, the functions $e_{\alpha,\beta}(z,q)$ and $E_{\alpha,\beta}(z,q)$ are defined by

$$\begin{aligned}& e_{\alpha,1}(z,q)=\sum_{n=0}^{\infty} \frac{z^{n\alpha}}{ \Gamma_{q}(n\alpha+1)} \quad \bigl(\bigl\vert z(1-q)^{\alpha}\bigr\vert < 1\bigr) \end{aligned}$$

(12)

and

$$\begin{aligned}& E_{\alpha,1}(z,q)=\sum_{n=0}^{\infty} \frac{q^{ \frac{\alpha n(n-1)}{2}}z^{n\alpha}}{\Gamma_{q}(n\alpha+1)} \quad (z \in C). \end{aligned}$$

(13)

2.3 q-Laplace transform of fractional q-integrals, q-derivatives, and q-Mittag-Leffler functions

Theorem 2.5

([6])

If $f\in\mathscr{L}_{q}^{1}[0,a]$ and $\Phi(s)=_{q}L_{s}f(x)$, then

$$\begin{aligned}& _{q}L_{s}I_{q}^{\alpha}f(x)= \frac{(1-q)^{\alpha}}{s^{\alpha}} \Phi(s)\quad \textit{for } \alpha>0. \end{aligned}$$

(14)

If $n-1<\alpha\leq n$ and $I_{q}^{n-\alpha}f(x)\in C_{1}^{(n)}[0,a]$, then let $\Phi(s)=_{q}L_{s}f(x)$. The q-Laplace transform of the Riemann–Liouville fractional and the Caputo fractional q-derivatives are given by

$$\begin{aligned}& _{q}L_{s}^{C}D_{q}^{\alpha}f(x)= \frac{s^{\alpha}}{(1-q)^{\alpha}}\Biggl( \Phi(s)-\sum_{r=0}^{n-1}D_{q}^{r}f \bigl(0^{+}\bigr)\frac{(1-q)^{r}}{s ^{r+1}}\Biggr) \end{aligned}$$

(15)

and

$$\begin{aligned}& _{q}L_{s}D_{q}^{\alpha}f(x)= \frac{s^{\alpha}}{(1-q)^{\alpha}} \Phi(s)-\sum_{m=1}^{n}D_{q}^{\alpha-m}f \bigl(0^{+}\bigr) \frac{s^{m-1}}{(1-q)^{m}}. \end{aligned}$$

(16)

Theorem 2.6

([6])

If $\vert \frac{s}{1-q}\vert >\vert a\vert ^{\frac {1}{Re(\alpha)}}$, then

$$\begin{aligned}& _{q}L_{s}\bigl(x^{\beta-1}e_{\alpha,\beta}(ax;q) \bigr)=\frac{1}{1-q}\frac{( \frac{s}{1-q})^{\alpha-\beta}}{(\frac{s}{1-q})^{\alpha}-a}. \end{aligned}$$

(17)

Taking $\beta=1$, we have

$$\begin{aligned}& _{q}L_{s}\bigl(e_{\alpha,1}(ax;q)\bigr)= \frac{1}{1-q}\frac{(\frac{s}{1-q})^{ \alpha-1}}{(\frac{s}{1-q})^{\alpha}-a}. \end{aligned}$$

(18)

3 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:

$$\begin{aligned}& \textstyle\begin{cases} ^{C}D_{q}^{\alpha}x(t)=f(t,x(t)), \\ x(t_{0})=x_{0}, \end{cases}\displaystyle \end{aligned}$$

(19)

where $t\geq t_{0},t,t_{0}\in A,A=[t_{0}, t]_{q},0<\alpha<1$, and $f:[t_{0}, t] \times R \rightarrow R$ is a function with $f \in \mathscr{L}_{q,1}[t_{0}, t]$. Let $f(t,0)=0$, for all $t \in[t_{0}, t]_{q}$, 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 $\epsilon>0$ and $t_{0}\in A$, there exists $\delta=\delta(t_{0},\epsilon)$ such that if $\Vert x_{0}\Vert <\delta$ implies that $\lim_{t\rightarrow\infty} \Vert x(t)\Vert =0$.

Definition 3.2

(q-Mittag-Leffler stability)

The solution of (19) is said to be q-Mittag-Leffler stability if

$$\begin{aligned}& \bigl\Vert x(t)\bigr\Vert \leq\bigl\{ m\bigl[x(t_{0}) \bigr]e_{q,\alpha}\bigl(-\lambda(t-t_{0})^{\alpha}\bigr) \bigr\} ^{b}, \end{aligned}$$

(20)

where $t_{q}\in A$ is the initial time, $\alpha\in(0,1),\lambda \geq0,b>0,m(0)=0,m(x)\geq0$, and $m(x)$ is locally Lipschitz on $x\in B\subset R$ with Lipschitz constant $m_{0}$. We further assume that $t_{0}=0$.

Theorem 3.3

Let $x=0$ be an equilibrium point for system (19), and let $D\subset R$ be a domain containing origin. Let $V(t, x(t)):[0,T] \times D\rightarrow R$ be a continuously differentiable function and locally Lipschitz with respect to x such that

$$\begin{aligned}& \beta_{1}\bigl\Vert x(t)\bigr\Vert ^{a}\leq V\bigl(t, x(t)\bigr)\leq\beta_{2}\bigl\Vert x(t)\bigr\Vert ^{ab}, \end{aligned}$$

(21)

$$\begin{aligned}& {}^{C}_{0}D_{q}^{\alpha}V\bigl(t, x(t) \bigr) \leq(-\beta_{3})\bigl\Vert x(t)\bigr\Vert ^{ab}, \end{aligned}$$

(22)

where $t\in[0,T],t>0$, $0<\alpha<1$, and $\beta_{1}$, $\beta_{2}$, $\beta _{3}$, a, and b are arbitrary positive constants. Then $x=0$ is q-Mittag-Leffler stable.

Proof

It follows from equations (19) and (20) that

$$\begin{aligned}& {}^{C}_{0}D_{q}^{\alpha}V\bigl(t, x(t) \bigr)\leq-\frac{\beta_{3}}{\beta_{2}}V\bigl(t, x(t)\bigr). \end{aligned}$$

(23)

There exists a nonnegative function $M(t)$ satisfying

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}V\bigl(t, x(t) \bigr)+M(t)=-\frac{\beta_{3}}{\beta_{2}}V\bigl(t, x(t)\bigr). \end{aligned}$$

(24)

Taking the q-Laplace transform of (24) gives

$$\begin{aligned}& \frac{s^{\alpha}}{(1-q)^{\alpha}}(V(s)-\frac {1}{s}V\bigl(0,x(0)\bigr)+M(s)=- \frac{ \beta_{3}}{\beta_{2}}V(s), \end{aligned}$$

(25)

where $V(s)=_{q}L_{s}\{V(t,x(t))\}$. It then follows that

$$\begin{aligned} V(s) =&V(0,x(0))\frac{\frac{s^{\alpha-1}}{(1-q)^{\alpha}}}{\frac{s ^{\alpha}}{(1-q)^{\alpha}}+\frac{\beta_{3}}{\beta_{2}}}-\frac {M(s)}{\frac{s ^{\alpha}}{(1-q)^{\alpha}} +\frac{\beta_{3}}{\beta_{2}}} \\ =&V(0,x(0))\frac{1}{1-q}\frac{(\frac{s}{1-q})^{\alpha-1}}{( \frac{s}{1-q})^{\alpha}+\frac{\beta_{3}}{\beta_{2}}}-(1-q)M(s) \frac{1}{1-q} \frac{1}{\frac{s^{\alpha}}{(1-q)^{\alpha}}+\frac{\beta _{3}}{\beta_{2}}}. \end{aligned}$$

(26)

It follows from the inverse Laplace transform that the unique solution of (24) is

$$\begin{aligned}& V(t)=V\bigl(0,x(0)\bigr)e_{\alpha,1}\biggl(-\frac{\beta_{3}}{\beta_{2}}t;q\biggr)- \int_{0} ^{t}M(\tau) (t-q\tau)^{\alpha-1}e_{\alpha,\alpha} \biggl(-\frac{\beta _{3}}{\beta_{2}}(t-q\tau)^{\alpha};q\biggr)\,d\tau. \end{aligned}$$

(27)

Since $0< q<1$, $M(t)\geq0$, and $e_{\alpha,\alpha}(-\frac{\beta _{3}}{\beta_{2}}(t-q\tau)^{\alpha};q)$ are nonnegative functions, we get

$$\begin{aligned}& V(t)\leq V\bigl(0,x(0)\bigr)e_{\alpha,1}\biggl(-\frac{\beta_{3}}{\beta_{2}}t;q \biggr). \end{aligned}$$

(28)

Substitution of (28) into (21) yields

$$\begin{aligned}& \bigl\Vert x(t)\bigr\Vert \leq\biggl[\frac{V(0,x(0))}{\beta_{1}}e_{\alpha,1} \biggl(-\frac{\beta _{3}}{\beta_{2}}t;q\biggr)\biggr]^{\frac{1}{a}}, \end{aligned}$$

(29)

where $\frac{V(0,x(0))}{\beta_{1}}>0$ for $x(0)\neq0$.

Let $m=\frac{V(0,x(0))}{\beta_{1}}\geq0$. Then we have

$$\begin{aligned}& \bigl\Vert x(t)\bigr\Vert \leq\biggl[m e_{\alpha,1}\biggl(- \frac{\beta_{3}}{\beta_{2}}t;q\biggr)\biggr]^{ \frac{1}{a}}, \end{aligned}$$

(30)

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:

$$\begin{aligned}& f(t)=_{t_{0}}D_{q}^{\alpha}f(t)-_{t_{0}}D_{q}^{\alpha} \Biggl(\sum_{k=0} ^{n-1}\frac{D_{q}^{k}f(0^{+})}{\Gamma_{q}(k+1)}x^{k} \Biggr), \end{aligned}$$

(31)

where $\alpha>0$ and $n=[\alpha]+1$. When $0<\alpha<1$, we have

$$\begin{aligned}& ^{C}_{t_{0}}D_{q}^{\alpha}f(t)=_{t_{0}}D_{q}^{\alpha}f(t)- \frac{(t-t _{0})_{q}^{\alpha}}{\Gamma_{q}(1-\alpha)}f(t_{0}). \end{aligned}$$

(32)

 □

Theorem 3.4

If the assumptions in Theorem 3.3 are satisfied except replacing ${}^{C}_{t_{0}}D_{q}^{\alpha}$ by ${t_{0}}D_{q}^{ \alpha}$, then the trivial solution of (19) is q-Mittag-Leffler stable.

Proof

From (32) we have

$$\begin{aligned}& {}^{C}_{0}D_{q}^{\alpha}V\bigl(t, x(t) \bigr)=_{0}D_{q}^{\alpha}V\bigl(t,x(t)\bigr)- \frac{t _{q}^{\alpha}}{\Gamma_{q}(1-\alpha)}V\bigl(0,x(0)\bigr)\quad \mbox{for } t \in[0,T], \end{aligned}$$

(33)

and since $V(0,x(0))\geq0$ and $\frac{t_{q}^{\alpha}}{\Gamma_{q}(1- \alpha)}\geq0$, 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. □

Remark 3.5

Since

$$\begin{aligned}& D_{q}e_{{\alpha,1}}\bigl((-\lambda t;q)\bigr)=-\lambda t^{\alpha-1}e_{\alpha,\alpha-1}(-\lambda t;q), \end{aligned}$$

(34)

where $t>0$, $0<\alpha<1$, $\lambda>0$, the q-Mittag-Leffler function $e_{{\alpha,1}}(((-\lambda t)^{\alpha};q))$ is decreasing, so the q-Mittag-Leffler stability implies the asymptotic stability.

4 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.

Theorem 4.1

Let $x(t) \in R$ be defined in a suitable q-geometric set $A=[0,a]_{q}$, $D_{q}x(t)\in C_{q}[0,q]$ (where $C_{q}[0,a]$ is the space of all continuous functions on the interval $[0, a]$). Then, for any time $t>0$, $t\in A$,

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}x^{2}(t) \leq\bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{\alpha }x(t),\quad 0< \alpha< 1. \end{aligned}$$

(35)

Proof

Proving expression (35) is equivalent to proving that

$$\begin{aligned}& \bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{\alpha}x(t)-^{C}_{0}D_{q}^{\alpha}x^{2}(t) \geq0. \end{aligned}$$

(36)

Using Definition 2.2 and Definition 2.4, $(x(t)+x(tq))^{C}_{0}D_{q}^{ \alpha}x(t)$ and ${}^{C}_{0}D_{q}^{\alpha}x^{2}(t)$ can be written as

$$\begin{aligned}& \bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{\alpha}x(t)= \bigl(x(t)+x(tq)\bigr)\frac{1}{\Gamma(1- \alpha)} \int_{0}^{t}(t-qs)^{-\alpha}D_{q}x(s)\,d_{q}s \end{aligned}$$

(37)

and

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}x^{2}(t)= \frac{1}{\Gamma(1-\alpha)} \int_{0} ^{t}(t-qs)^{-\alpha}\bigl(x(s)+x(qs) \bigr)D_{q}x(s)\,d_{q}s. \end{aligned}$$

(38)

So, the left side of expression (36) can be written as

$$\begin{aligned}& \frac{1}{\Gamma(1-\alpha)} \int_{0}^{t}(t-qs)^{-\alpha }\bigl[ \bigl(x(t)-x(s)\bigr)+\bigl(x(tq)-x(sq)\bigr)\bigr]D _{q}x(s)\,d_{q}s. \end{aligned}$$

(39)

Now, let us define the axillary variable $y(s)=x(t)-x(s)$, which implies that

$$\begin{aligned} D_{q}y^{2}(s) =&\bigl(y(s)+y(sq)\bigr)D_{q}y(s) \\ =&-\bigl[\bigl(x(t)-x(s)\bigr)+\bigl(x(tq)-x(sq)\bigr)\bigr]D_{q}x(s). \end{aligned}$$

(40)

In this way, expression (39) can be written as

$$\begin{aligned}& \frac{1}{\Gamma(1-\alpha)} \int_{0}^{t}(t-qs)^{-\alpha}\,d_{q}y^{2}(s)= -\frac{1}{\Gamma(1-\alpha)} \int_{0}^{t}(t-qs)^{-\alpha}\bigl[y(s)+y(sq) \bigr]D _{q}y(s)\,d_{q}s. \end{aligned}$$

(41)

Since $x(t)$ is regular at zero, using the rule of q-integration by parts, expression (41) becomes

$$\begin{aligned} \int_{t_{0}}^{t}(t-qs)^{-\alpha}\,d_{q}y^{2}(s) =&y^{2}(t)(t-qt)^{- \alpha}-\Gamma(1-\alpha)y^{2}(0)t^{-\alpha} \\ &{}-\alpha q \int_{0}(t-qs)^{-\alpha-1}y^{2}(qs)\,d_{q}(s). \end{aligned}$$

(42)

Since $y^{2}(t)=(x(t)-x(s))^{2}=0$, it follows that

$$\begin{aligned}& {}^{C}_{0}D_{q}^{\alpha}x(t)-^{C}_{0}D_{q}^{\alpha}x^{2}(t) \\& \quad =\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-qs)^{-\alpha }[(x(t)-x(s))+(x(tq)-x(sq))]D _{q}x(s)\,d_{q}s \\& \quad =-\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-qs)^{-\alpha}\,d_{q}y ^{2}(s) \\& \quad =\frac{1}{\Gamma(1-\alpha)}y^{2}(0)t^{-\alpha}+\frac{\alpha q}{ \Gamma(1-\alpha)}\int_{0}^{t}(t-qs)^{-\alpha-1}y^{2}(s)\,d_{q}(s) \\& \quad \geq0. \end{aligned}$$

(43)

This concludes the proof. □

Corollary 4.2

For the q-fractional-order system

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}x(t)=f\bigl(t,x(t) \bigr), \end{aligned}$$

(44)

where $\alpha\in(0,1)$, $x=0$ is the equilibrium point, and $D_{q}x(t)\in C_{q}[0,a]$, $f(t,x(t))\in\pounds_{q}^{1}[0,a]$. If

$$\begin{aligned}& \bigl(x(t)+x(tq)\bigr)f\bigl(t,x(t)\bigr)\leq0, \quad \forall x\in A, \end{aligned}$$

(45)

then the origin of system (44) is q-Mittag-Leffler stable.

Proof

Let us propose the following Lyapunov candidate function:

$$\begin{aligned}& V\bigl(t,x(t)\bigr)=x^{2}. \end{aligned}$$

(46)

Applying Theorem 4.1 results in

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}V\bigl(t,x(t)\bigr) \leq\bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{ \alpha}x(t) \leq\bigl(x(t)+x(tq)\bigr)f\bigl(t,x(t)\bigr)\leq0, \end{aligned}$$

(47)

and thus the origin of system (44) is q-Mittag-Leffler stable. □

Proposition 4.3

For the system

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}x(t)=-x(t)-x(tq), \end{aligned}$$

(48)

where $0<\alpha<1$ and $D_{q}x(t)\in C_{q}[0,a]$, the origin of system (44) is q-Mittag-Leffler stable.

Proof

Let $V(x(t))=x^{2}(t)$. Then

$$\begin{aligned} {}^{C}_{0}D_{q}^{\alpha}x^{2}(t) \leq& \bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{ \alpha}x(t) \\ =& -\bigl(x(t)+x(tq)\bigr)^{2}\leq-\big\Vert x(t)\big\Vert ^{2}. \end{aligned}$$

(49)

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),

$$\begin{aligned}& x(t)=c_{1}e_{(\alpha,1)}(-x,q)+c_{2}E_{(\alpha,1)}(-x,q), \end{aligned}$$

(50)

and the properties of these two functions the asymptotical stability can also be derived. □

5 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.