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

Advances in Difference Equations20182018:78

https://doi.org/10.1186/s13662-018-1502-5

  • Received: 5 April 2017
  • Accepted: 23 January 2018
  • Published:

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.

Keywords

  • q-fractional calculus
  • q-Mittag-Leffler functions
  • q-Mittag-Leffler stability
  • Lyapunov method

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 [13] and then by Agrawal [4]. Recently, the q-fractional calculus has been payed more attention [58] 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 [1121] 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 [14, 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.

Declarations

Acknowledgements

The authors would like to express our deep gratitude to the referees for their valuable suggestions and their comments. The work is financially supported by the National Nature Science Foundation of China (11601003, 11371027), Natural Science Research Project of Colleges of Anhui Province (KJ2016A023), and Natural Science Foundation of Anhui Province (1508085MA01).

Authors’ contributions

All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Science, Anhui University, Hefei, P.R. China

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