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.
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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
and
Setting \(q\rightarrow1\), we have \(\lim_{q\rightarrow1}D_{q}f(x)=f ^{\prime}(x)\).
Also, the q-integral is given as
and
We present here two basic properties concerning q-derivatives.
Property 1
([7])
Property 2
([7])
The q-Leibniz product rule is given by
where \(D_{q}\) is the q-derivative.
The q-analogue of exponent \((s-t)^{(k)}\) is
Definition 2.2
([7])
A q-analogue of the Riemann–Liouville fractional integral is defined as
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
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
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
and
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
and
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
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
and
Theorem 2.6
([6])
If \(\vert \frac{s}{1-q}\vert >\vert a\vert ^{\frac {1}{Re(\alpha)}}\), then
Taking \(\beta=1\), we have
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:
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
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
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
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
It follows from the inverse Laplace transform that the unique solution of (24) is
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
Substitution of (28) into (21) yields
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
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 \(\alpha>0\) and \(n=[\alpha]+1\). When \(0<\alpha<1\), we have
□
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
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
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 \(t>0\), \(t\in A\),
Proof
Proving expression (35) is equivalent to proving that
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
and
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
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 \(\alpha\in(0,1)\), \(x=0\) 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
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 \(0<\alpha<1\) 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
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),
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.
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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).
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Li, X., Liu, S. & Jiang, W. q-Mittag-Leffler stability and Lyapunov direct method for differential systems with q-fractional order. Adv Differ Equ 2018, 78 (2018). https://doi.org/10.1186/s13662-018-1502-5
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DOI: https://doi.org/10.1186/s13662-018-1502-5