Skip to main content

Global \(O(t^{-\alpha })\) stabilization of fractional-order memristive neural networks with time delays

Abstract

This article is concerned with the global \(O(t^{-\alpha })\) stabilization for a class of fractional-order memristive neural networks with time delays (FMDNNs). Two kinds of control scheme (i.e., state feedback control law and output feedback control law) are employed to stabilize a class of FMDNNs. Several stabilization conditions in form of algebraic criteria are presented based on a new fractional-order Lyapunov function method and Leibniz rule. Some examples are given to substantiate the effectiveness of the presented theoretical results.

Background

Recently, the fractional calculus serving the fractional-order models develops fast in both theoretical and application. The analysis about fractional-order models has attracted increasing attention cause of its promising applications in various areas of science and engineering (see Chen and Chen 2015; Chen et al. 2014; Liu et al. 2015; Liang et al. 2015; Li et al. 2015; Rakkiyappan et al. 2014, 2015b; Stamova 2014; Velmurugan and Rakkiyappan 2016; Velmurugan et al. 2016; Wang et al. 2014; Wu and Zeng 2016; Wu et al. 2016). Comparing with integer-order systems, fractional-order systems show the superiority of describing and modeling the real world or the practical problems such as anomalous diffusion, signal processing, fractal theory and continuum mechanics. Whereas, it is arduously to promote the development of research about fractional-order models for the absence of efficient mathematical tools. As mentioned by Chen and Chen (2016), Chen et al. (2014), some new and useful methods for the qualitative analysis of fractional-order models are very imperative.

On the other hand, memristor is a circuit element which was proposed by Chua (1971) and has been realized the prototype by Hewlett-Packard laboratory in Strukov et al. (2008) and Tour and He (2008). Different from classical resistors, memristor is a nonlinear resistor which owns non-uniqueness values. In addition, the memristor can manage and store a great quantity of information. For its excellent properties about memory, we can build a new model if the conventional resistors are replaced by the memristors in neural networks, which is called memristive neural networks. Some representative works studied on the properties of the memristive systems display its applicability in several interdisciplinary areas (see Bao and Zeng 2013; Guo et al. 2015; Wang et al. 2003; Wu et al. 2012; Wu and Zeng 2012; Wen and Zeng 2012; Zhao et al. 2015). From the description of memristive neural networks, combining memristors with infinite memory is extremely interesting. An advantage of fractional-order systems in comparison to integer-order systems is that fractional-order systems can generate infinite memory. Therefore, merging the memristors into a class of fractional-order neural networks is pretty anticipated. Although stability analysis of fractional-order memristive or memristor-based neural networks has been gradually carried out (see Chen et al. 2014, 2015; Rakkiyappan et al. 2014, 2015b; Velmurugan and Rakkiyappan 2016; Velmurugan et al. 2016), it is worth noting that fractional-order memristive neural networks can exhibit complicated dynamics or chaotic behaviors if the network’s parameters and time delays are appropriately specified.

Noticed that many static or dynamic control laws have been designed to stabilize nonlinear control systems, for instance, Chandrasekar and Rakkiyappan (2016), Chen et al. (2015), Guo et al. (2013), Huang et al. (2009), Lou et al. (2013), Mathiyalagan et al. (2015), Rakkiyappan et al. (2015a), Wu et al. (2016), Yang and Tong (2016). In allusion to different system structures and actual control requirements, lots of stabilization criteria are established, for example, periodic intermittent stabilization (Huang et al. 2009), robust stabilization (Yang and Tong 2016), finite-time stabilization (Zhang et al. 2016), impulsive stabilization (Chandrasekar and Rakkiyappan 2016; Huang 2010; Lou et al. 2013). Despite these fruitful achievements, some stabilization approaches can hardly be widely applied in practical problems due to high gain. In addition, an undeniable fact is that stabilization control schemes of fractional-order systems is little studied. Hence, it is necessary to investigate some appropriate controllers for stabilization of fractional-order systems.

Inspired by the above discussion, in this article, we will study the global \(O(t^{-\alpha })\) stabilization problem for a class of fractional-order memristive neural networks with time delays. We first introduce the concepts about fractional calculation and global stabilization of fractional-order systems. Secondly, for exploring some simple useful controllers, linear state feedback control law and linear output feedback control law are designed to stabilize the fractional-order systems. In addition, stabilization criteria in form of algebraic inequalities are derived by utilizing a new fractional Lyapunov method instead of classical Gronwall inequality. The conditions can be easily verified.

Fractional calculation and model description

Fractional calculation concepts

First of all, some basics of fractional calculation are given which will be used in the later.

Definition 1

(Chen and Chen 2016) The fractional integral with fractional order \(\alpha >0\) of function f(t) is defined as

$$ {}_{t_0}^{RL}D_{t}^{-\alpha }f(t)=\frac{1}{\Gamma (\alpha )}\int _{t_0}^{t}(t-s)^{\alpha -1}f(s){\mathrm {d}}s, $$

where \(t\ge t_0\), \(\Gamma (\cdot )\) is the Gamma function, that is

$$ \Gamma (\alpha )=\int _0^{\infty }s^{\alpha -1}e^{-s}{\mathrm {d}}s. $$

Definition 2

(Chen and Chen 2016) The Riemann–Liouville derivative with fractional order \(\alpha >0\) of function f(t) is defined as

$$ {}_{t_0}^{RL}D_{t}^{\alpha }f(t)=\frac{1}{\Gamma (n-\alpha )}\frac{{\mathrm {d}}^n}{{\mathrm {d}}t^n}\int _{t_0}^{t}\frac{f(s)}{(t-s)^{\alpha -n+1}}{\mathrm {d}}s, $$

where \(t\ge t_0\), \(n-1<\alpha <n\), n is a positive integer. Moreover, when \(0<\alpha <1\), that is

$$ {}_{t_0}^{RL}D_{t}^{\alpha }f(t)=\frac{1}{\Gamma (1-\alpha )}\frac{{\mathrm {d}}}{{\mathrm {d}}t}\int _{t_0}^{t}\frac{f(s)}{(t-s)^{\alpha }}{\mathrm {d}}s. $$

Definition 3

(Chen and Chen 2016) The Caputo derivative with fractional order \(\alpha >0\) of function \(f\in C^{n+1}([t_0,+\infty ),{\mathcal {R}})\) is defined as

$$ {}_{t_0}^{C}D_{t}^{\alpha }f(t)=\frac{1}{\Gamma (n-\alpha )}\int _{t_0}^{t}\frac{f^{(n)}(s)}{(t-s)^{\alpha -n+1}}{\mathrm {d}}s,$$

where \(t\ge t_0\), \(n-1<\alpha <n\), n is a positive integer. Moreover, when \(0<\alpha <1\), that is

$$ {}_{t_0}^{C}D_{t}^{\alpha }f(t)=\frac{1}{\Gamma (1-\alpha )}\int _{t_0}^{t}\frac{f^{'}(s)}{(t-s)^{\alpha }}{\mathrm {d}}s.$$

Lemma 1

(Chen and Chen 2016) If \(f\in C^1([0,+\infty ),{\mathcal {R}})\), then the following properties hold:

  1. (1)

    \({}_{t_0}^{C}D_{t}^{\alpha }f(t)={}_{t_0}^{RL}D_{t}^{\alpha }f(t)-\frac{f(t_0)}{\Gamma (1-\alpha )}(t-t_0)^{-\alpha }\).

  2. (2)

    If f(t) and \(\vartheta \) and their all derivatives are continuous in \([t_0,t]\), then Leibniz rule for fractional differentiation can be expressed as follows:

    $$ {}_{t_0}^{RL}D_{t}^{\alpha }(\vartheta (t)f(t))=\sum \limits _{k=0}^{n} \frac{{\mathrm {d}}^{k}\vartheta (t)}{{\mathrm {d}}t^k}{\alpha \atopwithdelims ()k}{}_{t_0}^{RL}D_{t}^{\alpha -k}f(t)-R_{n}^{\alpha }, $$

where \(0<\alpha <1\), \(n\ge \alpha \),

$$\begin{aligned} &R_{n}^{\alpha }(t)= {} \frac{(-1)^{n}(t-\alpha )^{n-\alpha +1}}{n! \Gamma (-\alpha )} \int _{0}^{1} \int _{0}^{1} F_{\alpha }(t,\xi ,\eta ) {\mathrm {d}} \xi {\mathrm {d}} \eta ,\\ &F_{\alpha }(t,\xi ,\eta )= {} f(t_{0}+\eta (t-t_0)) \vartheta ^{n+1} (t_0+(t-t_0)(\xi +\eta -\xi \eta )), \end{aligned}$$

and

$$ {\alpha \atopwithdelims ()k}=\frac{\Gamma (\alpha +1)}{k!\Gamma (\alpha -k+1)}. $$

Model description

Consider the fractional-order memristive neural networks with time delays (FMDNNs) described by the following fractional-order equations: for \(i=1,2,\ldots ,n,\)

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha }x_i(t)&= {} -x_i(t)+\sum \limits _{j = 1}^n a_{ij}(x_j(t))g_j(x_j(t))\nonumber \\&\quad +\sum \limits _{j = 1}^n b_{ij}(x_j(t))f_j(x_j(t-\tau (t))) + u_i(t), \end{aligned}$$
(1)

where \(0<\alpha <1\), n is the number of neurons in the networks, \(x_i(t)\) is the state variable of the ith neuron, \(g_j(\cdot )\), \(f_j(\cdot )\) denotes the output of the jth unit at time t and \(t-\tau (t)\), respectively, and \(g_j(0)=f_j(0)=0\). \(\tau (t)\) corresponds to the transmission delay at time t and \(0\le \tau (t)\le \tau \). \(u_i(t)\) denotes the external input, \(a_{ij}(x_j(t))\) and \(b_{ij}(x_j(t))\) represent memristive weights, which are defined as:

$$\begin{aligned} a_{ij}(x_j(t))=\left\{ \begin{array}{ll} {\hat{a}}_{ij}, &{}\quad \left| x_j(t)\right|>T_j, \\ {\check{a}}_{ij}, &{}\quad \left| x_j(t)\right|<T_j, \end{array}\right. \quad b_{ij}(x_j(t))=\left\{ \begin{array}{ll} {\hat{b}}_{ij}, &{}\quad \left| x_j(t)\right| >T_j, \\ {\check{b}}_{ij}, &{}\quad \left| x_j(t)\right| <T_j, \end{array}\right. \end{aligned}$$
(2)

for \(i,j=1,2,\ldots ,n\), \(a_{ij}\)T \(_{j}\)) = \({\hat{a}}_{ij}\) or \({\check{a}}_{ij},\) \(b_{ij}\)T \(_{j}\)) = \({\hat{b}}_{ij}\) or \({\check{b}}_{ij},\) where the switching jumps \(T_j> 0\), \({\hat{a}}_{ij}\), \({\check{a}}_{ij}\), \({\hat{b}}_{ij}\), and \({\check{b}}_{ij}\) are constants.

Remark 1

Note that \(a_{ij}(x_j(t))\) and \(b_{ij}(x_j(t))\) are discontinuous in system (1), then the classical definition of solution for differential equations cannot be applied to (1). To deal with this issue, we introduce the concept of Filippov solution.

Definition 4

(Rakkiyappan et al. 2014) For system \({}_{t_0}^{C}D_{t}^{\alpha }x(t)=g(x)\), \(0<\alpha <1\), \(x\in {\mathcal{R}}^n\), with a discontinuous right-hand side, a set-valued map is defined as

$$ \psi (x)=\bigcap \limits _{\delta >0} \bigcap \limits _{\mu (N)=0} co\left[ g(B(x,\delta ){\backslash}N)\right] , $$

where co[E] is the closure of convex hull of set E, \(B(x,\delta )=\{y:\Vert y-x \Vert \le \delta \}\), and \(\mu (N)\) is a Lebesgue measure of set N. If x(t), \(t\in [t_0,T]\), is called the solution in Filippov sense of the Cauchy problem for system \({}_{t_0}^{C}D_{t}^{\alpha }x(t)=g(x)\), \(0<\alpha <1\), \(x\in {\mathcal{R}}^n\), with initial condition \(x(t_0)=x_0\), when it is absolutely continuous, and satisfies the differential inclusion as follows:

$$ {}_{t_0}^{C}D_{t}^{\alpha }x(t) \in \psi (x),\quad for\quad a.e. \quad t\in [t_0,T]. $$

For FMDNNs (1), define the set-value maps

$$\begin{aligned}&K(a_{ij}(x_j(t)))=\left\{ \begin{array}{ll} \hat{a}_{ij}, &{}\quad \left| x_j(t)\right|>T_j, \\ co\left\{ \hat{a}_{ij},\check{a}_{ij}\right\} , &{}\quad \left| x_j(t)\right| =T_j, \\ {\check{a}}_{ij}, &{}\quad \left| x_j(t)\right|<T_j, \end{array}\right. \\&K(b_{ij}(x_j(t)))=\left\{ \begin{array}{ll} {\hat{b}}_{ij}, &{}\quad \left| x_j(t)\right| >T_j, \\ co\left\{ {\hat{b}}_{ij},{\check{b}}_{ij}\right\} , &{}\quad \left| x_j(t)\right| =T_j, \\ {\check{b}}_{ij}, &{}\quad \left| x_j(t)\right| <T_j, \end{array}\right. \end{aligned}$$

for \(i,j=1,2,\ldots ,n\), where \(co\{{\hat{a}}_{ij},{\check{a}}_{ij}\}\) denotes the closure of convex hull generated by real numbers \({\hat{a}}_{ij}\) and \({\check{a}}_{ij}\), \(co\{{\hat{b}}_{ij},{\check{b}}_{ij}\}\) denotes the closure of convex hull generated by real numbers \({\hat{b}}_{ij}\) and \({\check{b}}_{ij}\).

Throughout this article we denote \(a_{ij}^{m}=\max \nolimits _{1\le i,j \le n}\{|{\hat{a}}_{ij}|,|{\check{a}}_{ij}|\}\), \(b_{ij}^{m}=\max \nolimits _{1\le i,j \le n}\{|{\hat{b}}_{ij}|,|{\check{b}}_{ij}|\}\). For n-dimensional vector \(v=(v_{1},v_{2},\ldots ,v_{n})^{T}\), the norm of vector v is recorded as \(\Vert v\Vert =\sum \nolimits _{i=1}^{n} |v_{i}|\). \(C_{\tau }:=C([-\tau ,0],{\mathcal {R}})\) is a Banach space of all continuous functions \(\varphi :[-\tau ,0]\rightarrow {\mathcal {R}}\). For \(\varphi \in C_{\tau }\), let \(\Vert \varphi \Vert _{C}=\sup \nolimits _{s\in [-\tau ,0]} \Vert \varphi (s)\Vert \).

Throughout this article, let us suppose: the activation functions \(g_i\), \(f_i\), \(i=1,2,\ldots ,n\), are global Lipschitz, that is, for all \(u,v \in {\mathcal {R}},\) there exist positive constants \(G_i\), \(F_i\) such that

$$ \left| g_i(u)-g_{i}(v)\right| \le G_i \left| u-v\right| ,\quad \left| f_i(u)-f_{i}(v)\right| \le F_i \left| u-v\right| . $$

The objective of this article is to investigate the global \(O(t^{-\alpha })\) stabilization problem for system (1). Therefore, the stabilization problem will be converted to find the suit controller \(u_i(t)\) \((i=1,2,\ldots ,n)\) such that zero solution of the closed-loop system of (1) is globally \(O(t^{-\alpha })\) stable.

From the theories of differential inclusions and set-valued maps, the Filippov solution of FMDNNs (1) can be defined in the following form.

Definition 5

A function \(x(t)=(x_1(t), x_2(t), \ldots , x_n(t))^{T}\) is said to be a Filippov solution of FMDNNs (1) on [0, T) with initial conditions \(x(s)=\varphi (s)\), \(s\in [-\tau ,0]\), if x(t) is absolutely continuous on any compact interval of [0, T), and

$$\begin{aligned}&{}_{t_0}^{C}D_{t}^{\alpha }x_i(t)\in -x_i(t)+\sum \limits _{j = 1}^n K(a_{ij}(x_j(t)))g_j(x_j(t))\nonumber \\&\quad \qquad +\,\sum \limits _{j = 1}^n K(b_{ij}(x_j(t)))f_j(x_j(t-\tau (t))) + u_i(t), \end{aligned}$$
(3)

for \(t\ge t_0\), \(i=1,2,\ldots ,n\). Or equivalently, for \(i,j=1,2,\ldots ,n\), there exist \(\gamma _{ij}^{a}(x_j(t))\in K(a_{ij}(x_j(t)))\), \(\gamma _{ij}^{b}(x_j(t))\in K(b_{ij}(x_j(t)))\) such that

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha }x_i(t)&= -x_i(t)+\sum \limits _{j = 1}^n \gamma _{ij}^{a}(x_j(t))g_j(x_j(t))\nonumber \\&\quad +\sum \limits _{j = 1}^n \gamma _{ij}^{b}(x_j(t))f_j(x_j(t-\tau (t))) + u_i(t). \end{aligned}$$
(4)

Remark 2

Based on the definitions of Filippov solution and fractional-order differential inclusion, we know that FMDNNs (1) is equivalent to the fractional-order differential inclusion (3) in the Filippov framework.

Next, definitions of global \(O(t^{-\alpha })\) stability and global \(O(t^{-\alpha })\) stabilization are given.

Definition 6

[Global \(O(t^{-\alpha })\) stability] The zero solution of FMDNNs (1), where \(u_i(t)=0\), is said to be globally \(O(t^{-\alpha })\) stable if there exists a positive constant M such that \(\Vert x(t,t_0,\varphi )\Vert \le M \Vert \varphi \Vert _{C} O(t^{-\alpha })\) for any \(\varphi \in C_{\tau }\) and \(t \ge t_0\).

Definition 7

[Global \(O(t^{-\alpha })\) stabilization] FMDNNs (1) is said to be globally \(O(t^{-\alpha })\) stabilized if there exists an appropriate feedback control law such that the closed-loop system of (1) is globally \(O(t^{-\alpha })\) stable.

Main results

State feedback control law

Two kinds of linear controller about state feedback are given, i.e., the linear controller without or with time delays. Firstly, we propose the following state control rule without time delays:

$$ u_i(t)=\sum \limits _{j = 1}^n p_{ij}x_j(t), $$
(5)

for \(i=1,2,\ldots ,n\).

Theorem 1

FMDNNs (1) with the state feedback control rule (5) can be achieved global \(O(t^{-\alpha })\) stabilization for any \(\varphi \in C_{\tau }\) if there exist a constant \(r>\tau \) and n positive constants \(\beta _i\) \((i=1,2,\ldots ,n)\) such that

$$ \sum \limits _{j = 1}^n \beta _j p_{ij} \le \beta _i\left( 1-\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}\right) -\sum \limits _{j = 1}^n \beta _j \left( a_{ij}^{m}G_j+\left( \frac{r}{r-\tau }\right) ^{\alpha }b_{ij}^{m}F_j\right) , $$
(6)

for all \(i=1,2,\ldots ,n\).

Proof

Define two Lyapunov functions as follows:

$$\begin{aligned} \left\{ \begin{array}{l} W(t)=\max \left\{ \frac{|x_i(t)|}{\beta _i},i=1,2,\ldots ,n \right\} ,\\ V(t)=(t-t_0+r)^\alpha W(t), \end{array} \right. \end{aligned}$$
(7)

and let

$$\begin{aligned} \left\{ \begin{array}{l} {\overline{W}}(t)=\sup \nolimits _{-\tau \le \theta \le t} W(\theta ), \\ {\overline{V}}(t)=\sup \nolimits _{-\tau \le \theta \le t} V(\theta ) , \end{array} \right. \end{aligned}$$
(8)

for \(t\ge t_0\).

From Leibniz rule for fractional differentiation, we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha } V(t)&= {} {}_{t_0}^{RL}D_{t}^{\alpha }V(t)-\frac{V(t_0)}{\Gamma (1-\alpha )}(t-t_0)^{-\alpha }\nonumber \\&= {} {}_{t_0}^{RL}D_{t}^{\alpha }\left( (t-t_{0}+r)^{\alpha }W(t)\right) -\frac{r^{\alpha }W(t_0)}{\Gamma (1-\alpha )}(t-t_0)^{-\alpha }\nonumber \\&= {} (t-t_{0}+r)^{\alpha }{}_{t_0}^{RL}D_{t}^{\alpha }W(t) +\dfrac{\Gamma (\alpha +1)}{\Gamma (\alpha )}\alpha (t-t_{0}+r)^{\alpha -1}{}_{t_0}^{RL}D_{t}^{\alpha -1}W(t)\nonumber \\&\quad +\dfrac{\Gamma (\alpha +1)}{2 \Gamma (\alpha -1)}\alpha (\alpha -1)(t-t_{0}+r)^{\alpha -2}{}_{t_0}^{RL}D_{t}^{\alpha -2}W(t)\nonumber \\&\quad -R_2^{\alpha }(t)-\dfrac{r^{\alpha }W(t_0)}{\Gamma (1-\alpha )}(t-t_0)^{-\alpha }\nonumber \\&\le {} (t-t_{0}+r)^{\alpha }{}_{t_0}^{C}D_{t}^{\alpha }W(t) +(t-t_{0}+r)^{\alpha } \frac{W(t_0)}{\Gamma (1-\alpha )} (t-t_0)^{-\alpha }\nonumber \\&\quad +\alpha ^2(t-t_{0}+r)^{\alpha -1}{}_{t_0}^{RL}D_{t}^{\alpha -1}W(t)\nonumber \\&\quad +\frac{\alpha ^2(\alpha -1)^2}{2}(t-t_{0}+r)^{\alpha -2}{}_{t_0}^{RL}D_{t}^{\alpha -2}W(t) -\frac{r^{\alpha }W(t_0)}{\Gamma (1-\alpha )}(t-t_0)^{-\alpha }\nonumber \\&\le {} (t-t_{0}+r)^{\alpha }{}_{t_0}^{C}D_{t}^{\alpha }W(t) +\left[ \left( \frac{t-t_{0}+r}{t-t_0}\right) ^{\alpha }- \left( \frac{r}{t-t_0}\right) ^{\alpha } \right] \frac{W(t_0)}{\Gamma (1-\alpha )}\nonumber \\&\quad +\alpha ^2(t-t_{0}+r)^{\alpha -1}{}_{t_0}^{RL}D_{t}^{\alpha -1}W(t)\nonumber \\&\quad +\frac{\alpha ^2(\alpha -1)^2}{2}(t-t_{0}+r)^{\alpha -2}{}_{t_0}^{RL}D_{t}^{\alpha -2}W(t)\nonumber \\ &\le {} (t-t_{0}+r)^{\alpha }{}_{t_0}^{C}D_{t}^{\alpha }W(t) +(1+2 \alpha )\frac{{\overline{V}}(t)}{r^{\alpha } \Gamma (1-\alpha )}\nonumber \\&\quad +\alpha ^2(t-t_{0}+r)^{\alpha -1}{}_{t_0}^{RL}D_{t}^{\alpha -1}W(t)\nonumber \\&\quad +\frac{\alpha ^2(\alpha -1)^2}{2}(t-t_{0}+r)^{\alpha -2}{}_{t_0}^{RL}D_{t}^{\alpha -2}W(t), \end{aligned}$$
(9)

for \(t\ge t_{0}\).

By computing, it follows that

$$\begin{aligned} \alpha ^2(t-t_{0}+r)^{\alpha -1}{}_{t_0}^{RL}D_{t}^{\alpha -1}W(t)&=\alpha ^2(t-t_{0}+r)^{-(1-\alpha )}{}_{t_0}^{RL}D_{t}^{-(1-\alpha )}W(t)\nonumber \\&\le \alpha ^2(t-t_{0}+r)^{-(1-\alpha )}\frac{1}{\Gamma (1-\alpha )}\overline{W}(t)\int _{t_{0}}^t (t-s)^{-\alpha } {\mathrm {d}}s\nonumber \\&\le \alpha ^2(t-t_{0}+r)^{-(1-\alpha )}\frac{(t-t_0)^{1-\alpha }}{(1-\alpha )\Gamma (1-\alpha )}\overline{W}(t)\nonumber \\&\le \alpha ^2\left( \frac{t-t_0}{t-t_{0}+r}\right) ^{1-\alpha }\frac{1}{\Gamma (2-\alpha )}\overline{W}(t)\nonumber \\&\le \frac{\alpha ^2}{\Gamma (2-\alpha )}W({\widetilde{\theta }})\nonumber \\&\le \frac{\alpha ^2}{r^{\alpha }\Gamma (2-\alpha )}{\overline{V}}(t), \end{aligned}$$
(10)

and

$$\begin{aligned}&\frac{\alpha ^2(\alpha -1)^2}{2}(t-t_{0}+r)^{\alpha -2}{}_{t_0}^{RL}D_{t}^{\alpha -2}W(t)\nonumber \\&\quad \le \frac{\alpha ^2(\alpha -1)^2}{2}(t-t_{0}+r)^{\alpha -2}\frac{1}{\Gamma (2-\alpha )}\int _{t_{0}}^t (t-s)^{1-\alpha }W(s){\mathrm {d}}s\nonumber \\&\quad \le \frac{\alpha ^2(\alpha -1)^2}{2}(t-t_{0}+r)^{\alpha -2}\frac{\overline{W}(t)}{\Gamma (2-\alpha )}\int _{t_{0}}^t (t-s)^{1-\alpha }{\mathrm {d}}s\nonumber \\&\quad \le \frac{\alpha ^2(\alpha -1)^2}{2}(t-t_{0}+r)^{\alpha -2}\frac{(t-t_0)^{2-\alpha }\overline{W}(t)}{(2-\alpha )\Gamma (2-\alpha )}\nonumber \\&\quad \le \frac{\alpha ^2(\alpha -1)^2}{2}\left( \frac{t-t_0}{t-t_{0}+r}\right) ^{2-\alpha }\frac{\overline{W}(t)}{(2-\alpha )\Gamma (2-\alpha )}\nonumber \\&\quad \le \frac{\alpha ^2(\alpha -1)^2}{2}\frac{\overline{W}(t)}{(2-\alpha )\Gamma (2-\alpha )}\nonumber \\&\quad \le \frac{\alpha ^2(\alpha -1)^2}{2r^{\alpha }(2-\alpha )\Gamma (2-\alpha )}{\overline{V}}(t)\nonumber \\&\quad \le \frac{\alpha ^2}{r^{\alpha }\Gamma (2-\alpha )}{\overline{V}}(t), \end{aligned}$$
(11)

for \(t\ge t_{0}\), \({\overline{W}}(t)=W({\widetilde{\theta }})\), where \({\widetilde{\theta }}\in [-\tau ,t]\).

From (9) to (11), we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha }V(t)&\le (t-t_{0}+r)^{\alpha } {}_{t_0}^{C}D_t^{\alpha } W(t)+\frac{ 2 \alpha ^2}{r^{\alpha }\Gamma (2-\alpha )}{\overline{V}}(t)+ \frac{(1+2\alpha ) {\overline{V}}(t)}{r^{\alpha } \Gamma (1-\alpha )}\nonumber \\&= (t-t_{0}+r)^{\alpha } {}_{t_0}^{C}D_t^{\alpha } W(t)+\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}{\overline{V}}(t). \end{aligned}$$
(12)

It is obvious that there exists a \(k\in \{1,2,\ldots ,n\}\) such that

$$ W(t)=\frac{|x_k(t)|}{\beta _k}, $$

for given \(t\ge t_0\).

From (7) and (8), we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha } W(t)&=\frac{1}{\beta _k}{}_{t_0}^{C}D_{t}^{\alpha }|x_k(t)|\nonumber \\&\le \frac{1}{\beta _k} sgn(x_k(t)) {}_{t_0}^{C}D_{t}^{\alpha }x_k(t)\nonumber \\&\le \frac{1}{\beta _k} sgn(x_k(t))\left\{ -x_k(t)+\sum \limits _{j=1}^{n}\gamma _{kj}^{a}(x_j(t))g_j(x_j(t))\right. \nonumber \\&\quad +\left. \sum \limits _{j=1}^{n}\gamma _{kj}^{b}(x_j(t))f_j(x_j(t-\tau (t)))+\sum \limits _{j=1}^{n}p_{kj}x_j(t)\right\} \nonumber \\&\le \frac{1}{\beta _k}sgn(x_k(t))\left\{ -x_k(t)+\sum \limits _{j=1}^{n} a_{kj}^{m}G_j |x_j(t)|\right. \nonumber \\&\quad +\left. \sum \limits _{j=1}^{n} b_{kj}^{m}F_j|x_j(t-\tau (t))|+\sum \limits _{j=1}^{n}p_{kj}x_j(t)\right\} \nonumber \\&\le - \frac{|x_k(t)|}{\beta _k}+\frac{1}{\beta _k}\sum \limits _{j=1}^{n} \beta _j a_{kj}^{m} G_j \frac{|x_j(t)|}{\beta _j} +\frac{1}{\beta _k} \sum \limits _{j=1}^{n} \beta _j b_{kj}^{m} F_j \frac{|x_j(t-\tau (t))|}{\beta _j}\nonumber \\&\quad +\frac{1}{\beta _k} \sum \limits _{j=1}^{n} \beta _j p_{kj} \frac{|x_j(t)|}{\beta _j}\nonumber \\&\le -W(t)+\frac{1}{\beta _k}\sum \limits _{j=1}^n \beta _j (a_{kj}^{m}G_j+p_{kj} )W(t) +\frac{1}{\beta _k}\sum \limits _{j=1}^{n} \beta _j b_{kj}^{m} F_j W(t-\tau (t))\nonumber \\&\le -\left[ 1-\frac{1}{\beta _k}\sum \limits _{j=1}^n \beta _j (a_{kj}^{m}G_j+p_{kj})\right] W(t) +\frac{1}{\beta _k}\sum \limits _{j=1}^{n} \beta _j b_{kj}^{m} F_j \overline{W}(t-\tau (t)). \end{aligned}$$
(13)

And hence

$$\begin{aligned} {}&{}(t-t_0+r)^{\alpha }{}_{t_0}^{C}D_{t}^{\alpha } W(t)\nonumber \\&\quad \le -(t-t_0+r)^{\alpha }\left[ 1-\frac{1}{\beta _k}\sum \limits _{j=1}^n \beta _j \left( a_{kj}^{m}G_j+p_{kj}\right) \right] W(t)\nonumber \\&\quad \qquad +(t-t_0+r)^{\alpha }\frac{1}{\beta _k}\sum \limits _{j=1}^{n} \beta _j b_{kj}^{m} F_j {\overline{W}}(t-\tau (t))\nonumber \\&\quad \le -\left[ 1-\frac{1}{\beta _k}\sum \limits _{j=1}^n \beta _j \left( a_{kj}^{m}G_j+p_{kj}\right) \right] V(t) +\frac{(t-t_0+r)^{\alpha }}{\beta _k\left( t-t_0+r+{\overline{\theta }}\right) ^{\alpha }}\sum \limits _{j=1}^{n} \beta _j b_{kj}^{m} F_j {\overline{V}}(t)\nonumber \\&\quad \le -\left[ 1-\sum \limits _{j=1}^n \frac{\beta _j}{\beta _k} (a_{kj}^{m}G_j+p_{kj})-\left( \frac{r}{r+\overline{\theta }}\right) ^{\alpha }\sum \limits _{j=1}^{n}\frac{\beta _j}{\beta _k} b_{kj}^{m} F_j\right] V(t)\nonumber \\&\quad \le -\left[ 1-\sum \limits _{j=1}^n \frac{\beta _j}{\beta _k} \left( a_{kj}^{m}G_j+p_{kj}\right) -\left( \frac{r}{r-\tau }\right) ^{\alpha }\sum \limits _{j=1}^{n}\frac{\beta _j}{\beta _k} b_{kj}^{m} F_j\right] V(t), \end{aligned}$$
(14)

where \({\overline{\theta}}\in [-\tau ,t]\) such that \({\overline{V}}(t)=(t-t_0+{\overline{\theta }}+r)^{\alpha } {\overline{W}}(t)\), when \({\overline{V}}(t)=V(t)\).

From (12) and (14), we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha }V(t)&\le -\left[ 1-\sum \limits _{j=1}^n \frac{\beta _j}{\beta _k} (a_{kj}^{m}G_j+p_{kj})-\left( \frac{r}{r-\tau }\right) ^{\alpha }\sum \limits _{j=1}^{n}\frac{\beta _j}{\beta _k} b_{kj}^{m} F_j\right] V(t) +\frac{1+ \alpha }{r^{\alpha } \Gamma (2-\alpha )}{\overline{V}}(t)\nonumber \\&\le \left\{ -\left[ 1-\sum \limits _{j=1}^n \frac{\beta _j}{\beta _k} \left( a_{kj}^{m}G_j+p_{kj}\right) -\left( \frac{r}{r-\tau }\right) ^{\alpha }\sum \limits _{j=1}^{n}\frac{\beta _j}{\beta _k} b_{kj}^{m} F_j\right] +\frac{1+ \alpha }{r^{\alpha } \Gamma (2-\alpha )}\right\} V(t), \end{aligned}$$
(15)

when \(\overline{V}(t)=V(t)\).

From (6), it follows that

$$ {}_{t_0}^{C}D_{t}^{\alpha } {\overline{V}}(t)\le 0, $$
(16)

for all \(t\ge t_0\).

On the basis of Definition 2 and Lemma 1, the following inequality holds

$$ \frac{1}{\Gamma (1-\alpha )}\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int _{t_0}^{t} \frac{{\overline{V}}(s)}{(t-s)^{\alpha }} {\mathrm {d}}s \le \frac{{\overline{V}}(t_0)}{\Gamma (1-\alpha )}(t-t_0)^{-\alpha }. $$

It yields

$$ {\overline{V}}(t)\le {\overline{V}}(t_0), $$
(17)

for \(t\ge t_0\). Hence for \(i=1,2,\ldots ,n,\)

$$\begin{aligned} |x_i(t)|&\le \beta _i W(t)\nonumber \\&=\beta _i \frac{V(t)}{(t-t_0+r)^{\alpha }}\le \beta _i \frac{{\overline{V}}(t)}{(t-t_0+r)^{\alpha }}\nonumber \\&\le \beta _i \frac{V(t_0)}{(t-t_0+r)^{\alpha }}=\beta _i \frac{r^{\alpha } {\overline{W}}(t_0)}{(t-t_0+r)^{\alpha }}\nonumber \\&\le \frac{\beta _i r^{\alpha } \Vert \varphi \Vert _C}{\beta _{\min }(t-t_0+r)^{\alpha }}, \end{aligned}$$
(18)

where \(\beta _{\min }=\min \{\beta _i, i=1,2,\ldots ,n\}\), for \(t\ge t_0\), which implies

$$ \Vert x(t)\Vert \le \frac{\Theta r^{\alpha }\Vert \varphi \Vert _C}{(t-t_0+r)^{\alpha }}, $$

where \(\Theta =\frac{1}{\beta _{\min }}\sum \nolimits _{i=1}^{n}\beta _i\). Therefore, FMDNNs (1) can be achieved global \(O(t^{-\alpha })\) stabilization under the designed control law (5). \(\square \)

In the following, we propose the following state control rule with time delays:

$$ u_i(t)=\sum \limits _{j=1}^{n}p_{ij}x_j(t)+\sum \limits _{j=1}^{n}q_{ij}x_{j}(t-\tau (t)),$$
(19)

for \(i=1,2,\ldots ,n\).

Theorem 2

FMDNNs (1) with the state feedback control rule (19) can be achieved global \(O(t^{-\alpha })\) stabilization for any \(\varphi \in C_{\tau }\) if there exist a constant \(r>\tau \) and n positive constants \(\beta _i\) \((i=1,2,\ldots ,n)\) such that

$$ \sum \limits _{j=1}^{n} \beta _j \left( p_{ij}+\left( \frac{r}{r-\tau }\right) ^{\alpha }q_{ij}\right)\le {} \beta _i \left( 1-\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}\right) -\sum \limits _{j=1}^{n} \beta _j \left( a_{ij}^{m}G_j+\left( \frac{r}{r-\tau }\right) ^{\alpha }b_{ij}^{m}F_j\right) ,$$
(20)

for all \(i=1,2,\ldots ,n\).

Proof

Define two Lyapunov functions as follows:

$$\begin{aligned} \left\{ \begin{array}{l} W(t)=\max \left\{ \frac{|x_i(t)|}{\beta _i},i=1,2,\ldots ,n \right\} ,\\ V(t)=(t-t_0+r)^\alpha W(t), \end{array} \right. \end{aligned}$$
(21)

and let

$$\begin{aligned} \left\{ \begin{array}{l} {\overline{W}}(t)=\sup \nolimits _{-\tau \le \theta \le t} W(\theta ), \\ {\overline{V}}(t)=\sup \nolimits _{-\tau \le \theta \le t} V(\theta ), \end{array} \right. \end{aligned}$$
(22)

for \(t\ge t_0\).

Through Theorem 1, we have

$$ {}_{t_0}^{C}D_{t}^{\alpha } V(t) \le (t-t_0+r)^{\alpha }{}_{t_0}^{C}D_{t}^{\alpha }W(t)+\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}{\overline{V}}(t). $$
(23)

It is obvious that there exists a \(k\in \{1,2,\ldots ,n\}\) such that

$$ W(t)=\frac{|x_k(t)|}{\beta _k}, $$

for given \(t\ge t_0\).

From (21) and (22), we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha }W(t)&\le \frac{1}{\beta _k}sgn(x_k(t)){}_{t_0}^{C}D_{t}^{\alpha }x_k(t)\nonumber \\&\le \frac{1}{\beta _k}sgn(x_k(t))\left\{ -x_k(t)+\sum \limits _{j=1}^{n} \gamma _{kj}^{a}(x_j(t))g_j(x_j(t)) + \sum \limits _{j=1}^{n} \gamma _{kj}^{b}(x_j(t))f_j(x_j(t-\tau (t)))\right.\nonumber \\&\quad \left. + \sum \limits _{j=1}^{n}p_{kj}x_j(t)+\sum \limits _{j=1}^{n}q_{kj}x_j(t-\tau (t))\right\} \nonumber \\&\le \frac{1}{\beta _k}sgn(x_k(t))\left\{ -x_k(t)+\sum \limits _{j=1}^{n} a_{kj}^{m} G_j|x_j(t)| +\sum \limits _{j=1}^{n} b_{kj}^{m} F_j|x_j(t-\tau (t))|\right.\nonumber \\&\quad + \left. \sum \limits _{j=1}^{n}p_{kj}x_j(t)+\sum \limits _{j=1}^{n}q_{kj}x_j(t-\tau (t))\right\} \nonumber \\&\le -\left[ 1-\frac{1}{\beta _k}\sum \limits _{j=1}^{n}\beta _j (a_{kj}^{m}G_j+p_{kj})\right] W(t) +\frac{1}{\beta _k}\sum \limits _{j=1}^{n}\beta _j\left(b_{kj}^{m}F_j+q_{kj}\right) {\overline{W}}(t). \end{aligned}$$
(24)

Hence

$$ (t-t_0+r)^{\alpha }{}_{t_0}^{C}D_{t}^{\alpha }W(t)\le {} -\left[ 1-\sum \limits _{j=1}^{n} \frac{\beta _j}{\beta _k}\left( a_{kj}^{m}G_j+p_{kj} +\left( \frac{r}{r-\tau }\right) ^{\alpha }\left( b_{kj}^{m}F_j+q_{kj}\right) \right) \right] V(t), $$
(25)

where \({\overline{\theta}}\in [-\tau ,t]\) such that \({\overline{V}}(t)=(t+{\overline{\theta}}-t_0+r)^{\alpha } {\overline{W}}(t)\), when \({\overline{V}}(t)=V(t)\).

From (23) and (25), we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha } V(t)&\le -\left[ 1-\sum \limits _{j=1}^{n}\frac{\beta _j}{\beta _k}\left( a_{kj}^{m}G_j+p_{kj}+ \left( \frac{r}{r-\tau }\right) ^{\alpha }\left( b_{kj}^{m}F_j+q_{kj}\right) \right) \right] V(t) +\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )} {\overline{V}}(t) \nonumber \\&\le \left\{ -\left[ 1-\sum \limits _{j=1}^{n}\frac{\beta _j}{\beta _k}\left( a_{kj}^{m}G_j+p_{kj}+ \left( \frac{r}{r-\tau }\right) ^{\alpha }\left( b_{kj}^{m}F_j+q_{kj}\right) \right) \right] +\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}\right\} V(t), \end{aligned}$$
(26)

when \({\overline{V}}(t)=V(t)\).

From (20), it follows that

$$ {}_{t_0}^{C}D_{t}^{\alpha } {\overline{V}}(t)\le 0, $$
(27)

for all \(t \ge t_0\).

On the basis of Definition 3, we get

$$ {\overline{V}}(t)\le {\overline{V}}(t_0), $$
(28)

for \(t\ge t_0\). Hence for \(i=1,2,\ldots ,n,\)

$$ |x_i(t)|\le \beta _i W(t)\le \frac{\beta _i r^{\alpha }\Vert \varphi \Vert _C}{\beta _{\min }(t-t_0+r)^{\alpha }}, $$
(29)

where \(\beta _{\min }=\min \{\beta _i, i=1,2,\ldots ,n\}\), for \(t\ge t_0,\) it follows

$$ \Vert x(t)\Vert \le \frac{\Theta r^{\alpha }\Vert \varphi \Vert _C}{(t-t_0+r)^{\alpha }}, $$

where \(\Theta =\frac{1}{\beta _{\min }}\sum \nolimits _{i=1}^{n}\beta _i\). Therefore, FMDNNs can be achieved global \(O(t^{-\alpha })\) stabilization under the designed control law (19). \(\square \)

Output feedback control law

Two kinds of linear controller about output feedback are given, i.e., the linear output feedback controller without or with time delays. Firstly, we propose the following output feedback control rule without time delays:

$$ u_i(t)=\sum \limits _{j=1}^{n}\omega _{ij}g_j(x_j(t)), $$
(30)

for \(i=1,2,\ldots ,n\).

Theorem 3

FMDNNs (1) with the output feedback control rule (30) can be achieved global \(O(t^{-\alpha })\) stabilization for any \(\varphi \in C_{\tau }\) if there exist a constant \(r>\tau \) and n positive constants \(\beta _i\) \((i=1,2,\ldots ,n)\) such that

$$ \sum \limits _{j = 1}^n \beta _j G_j \omega _{ij}\le {} \beta _i \left( 1-\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}\right) -\sum \limits _{j = 1}^n \beta _j \left( a_{ij}^{m}G_j+\left( \frac{r}{r-{\tau }}\right) ^{\alpha }b_{ij}^{m}F_j\right) , $$
(31)

for all \(i=1,2,\ldots ,n\).

Proof

Define two Lyapunov functions as follows:

$$\begin{aligned} \left\{ \begin{array}{l} W(t)=\max \left\{ \frac{|x_i(t)|}{\beta _i},i=1,2,\ldots ,n \right\} ,\\ V(t)=(t-t_0+r)^\alpha W(t), \end{array} \right. \end{aligned}$$
(32)

and let

$$\begin{aligned} \left\{ \begin{array}{l} {\overline{W}}(t)=\sup \nolimits _{-\tau \le \theta \le t} W(\theta ), \\ {\overline{V}}(t)=\sup \nolimits _{-\tau \le \theta \le t} V(\theta ), \end{array} \right. \end{aligned}$$
(33)

for \(t\ge t_0\).

Through Theorem 1, we have

$$ {}_{t_0}^{C}D_{t}^{\alpha }V(t) \le (t-t_{0}+r)^{\alpha } {}_{t_0}^{C}D_t^{\alpha } W(t)+\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}{\overline{V}}(t). $$
(34)

It is obvious that there exists a \(k\in \{1,2,\ldots ,n\}\) such that

$$ W(t)=\frac{|x_k(t)|}{\beta _k}, $$

for given \(t\ge t_0\).

From (32) and (33), we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha } W(t)&=\frac{1}{\beta _k}{}_{t_0}^{C}D_{t}^{\alpha }|x_k(t)|\nonumber \\&\le \frac{1}{\beta _k} sgn(x_k(t)) {}_{t_0}^{C}D_{t}^{\alpha }x_k(t)\nonumber \\&\le \frac{1}{\beta _k} sgn(x_k(t))\left\{ -x_k(t)+\sum \limits _{j=1}^{n}\gamma _{kj}^{a}(x_j(t))g_j(x_j(t))+\sum \limits _{j=1}^{n}\gamma _{kj}^{b}(x_j(t))f_j(x_j(t-\tau (t)))\right. \nonumber \\&\quad +\left. \sum \limits _{j=1}^{n}\omega _{kj} g_j(x_j(t))\right\} \nonumber \\&\le -\frac{|x_k(t)|}{\beta _k}+\frac{1}{\beta _k}\sum \limits _{j=1}^{n}\beta _{j} a_{kj}^{m}G_j \frac{|x_j(t)|}{\beta _j}+\frac{1}{\beta _k}\sum \limits _{j=1}^{n} \beta _{j}b_{kj}^{m}F_j\frac{|x_j(t-\tau (t))|}{\beta _j}\nonumber \\&\quad +\frac{1}{\beta _k}\sum \limits _{j=1}^{n}\beta _{j} \omega _{kj}G_j \frac{|x_j(t)|}{\beta _j}\nonumber \\&\le -\left[ 1-\frac{1}{\beta _k}\sum \limits _{j=1}^n \beta _{j} G_j\left( a_{kj}^{m}+\omega _{kj}\right) \right] W(t) +\frac{1}{\beta _k}\sum \limits _{j=1}^{n} \beta _j b_{kj}^{m} F_j {\overline{W}}(t-\tau (t)). \end{aligned}$$
(35)

And hence

$$ (t-t_0+r)^{\alpha }{}_{t_0}^{C}D_{t}^{\alpha } W(t)\le {} -\left[ 1-\sum \limits _{j=1}^{n} \frac{\beta _{j}}{\beta _k} \left( G_j\left( a_{kj}^{m}+\omega _{kj}\right) +\left( \frac{r}{r-\tau }\right) ^{\alpha } b_{kj}^{m} F_j\right) \right] V(t), $$
(36)

where \({\overline{\theta }}\in [-\tau ,t]\) such that \({\overline{V}}(t)=(t-t_0+{\overline{\theta }}+r)^{\alpha } {\overline{W}}(t)\), when \({\overline{V}}(t)=V(t)\).

From (34) and (36), we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha }V(t)&\le -\left[ 1-\sum \limits _{j=1}^{n} \frac{\beta _j}{\beta _k}\left( G_j\left( a_{kj}^{m} +\omega _{kj}\right) +\left( \frac{r}{r-\tau }\right) ^{\alpha }b_{kj}^{m}F_j\right) \right] V(t) +\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}{\overline{V}}(t)\nonumber \\&\le \left\{ -\left[ 1-\sum \limits _{j=1}^{n}\frac{\beta _j}{\beta _k} \left( G_j\left( a_{kj}^{m}+\omega _{kj}\right) +\left( \frac{r}{r-\tau }\right) ^{\alpha }b_{kj}^{m}F_j\right) \right] +\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}\right\} V(t), \end{aligned}$$
(37)

when \({\overline{V}}(t)=V(t)\).

From (31), it follows that

$$ {}_{t_0}^{C}D_{t}^{\alpha }{\overline{V}}(t)\le 0, $$
(38)

for \(t\ge t_0\).

On the basis of Definition 3, we get

$$ {\overline{V}}(t)\le {\overline{V}}(t_0), $$
(39)

for \(t\ge t_0\). Hence for \(i=1,2,\ldots ,n,\)

$$ |x_i(t)|\le \beta _iW(t)\le \frac{\beta _{i}r^{\alpha }\Vert \varphi \Vert _C}{\beta _{\min }(t-t_0+r)^{\alpha }}, $$
(40)

where \(\beta _{\min }=\min \{\beta _i, i=1,2,\ldots ,n\}\), for \(t\ge t_0\), which implies

$$ \Vert x(t)\Vert \le \frac{\Theta r^{\alpha }\Vert \varphi \Vert _C}{(t-t_0+r)^{\alpha }}, $$

where \(\Theta =\frac{1}{\beta _{\min }}\sum \nolimits _{i=1}^{n}\beta _i\). Therefore, FMDNNs (1) can be achieved global \(O(t^{-\alpha })\) stabilization under the designed control law (30). \(\square \)

In the following, we propose the following output feedback control rule with time delays:

$$ u_i(t)=\sum \limits _{j=1}^{n}\omega _{ij}g_j(x_j(t)) +\sum \limits _{j=1}^{n}\rho _{ij}f_j(x_j(t-\tau (t))), $$
(41)

for \(i=1,2,\ldots ,n\).

Theorem 4

FMDNNs (1) with the output feedback control rule (41) can be achieved global \(O(t^{-\alpha })\) stabilization for any \(\varphi \in C_{\tau }\) if there exist a constant \(r>\tau \) and n positive constants \(\beta _i\) \((i=1,2,\ldots ,n)\) such that

$$ \sum \limits _{j=1}^{n} \beta _j \left( G_{j}\omega _{ij}+\left( \frac{r}{r-\tau }\right) ^{\alpha }F_{j} \rho _{ij}\right)\le {} \beta _i \left( 1-\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}\right) -\sum \limits _{j=1}^{n} \beta _j \left( a_{ij}^{m}G_j+\left( \frac{r}{r-\tau }\right) ^{\alpha }b_{ij}^{m}F_j\right) ,$$
(42)

for all \(i=1,2,\ldots ,n.\)

Proof

Define two Lyapunov functions as follows:

$$\begin{aligned} \left\{ \begin{array}{l} W(t)=\max \left\{ \frac{|x_i(t)|}{\beta _i},i=1,2,\ldots ,n \right\} ,\\ V(t)=(t-t_0+r)^\alpha W(t), \end{array} \right. \end{aligned}$$
(43)

and let

$$\begin{aligned} \left\{ \begin{array}{l} {\overline{W}}(t)=\sup \nolimits _{-\tau \le \theta \le t} W(\theta ), \\ {\overline{V}}(t)=\sup \nolimits _{-\tau \le \theta \le t} V(\theta ), \end{array} \right. \end{aligned}$$
(44)

for \(t\ge t_0\).

Through Theorem 1, we have

$$ {}_{t_0}^{C}D_{t}^{\alpha }V(t) \le (t-t_0+r)^{\alpha }{}_{t_0}^{C}D_{t}^{\alpha }W(t) +\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}{\overline{V}}(t). $$
(45)

It is obvious that there exists a \(k\in \{1,2,\ldots ,n\}\) such that

$$ W(t)=\frac{|x_k(t)|}{\beta _k}, $$

for given \(t\ge t_0\).

From (43) and (44), we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha }W(t)&\le {} \frac{1}{\beta _k}sgn(x_k(t)){}_{t_0}^{C}D_{t_0}^{\alpha }x_k(t)\nonumber \\&\le {} \frac{1}{\beta _k}sgn(x_k(t))\left\{ -x_k(t)+\sum \limits _{j=1}^{n} \gamma _{kj}^{a}(x_j(t))g_j(x_j(t))+\sum \limits _{j=1}^{n} \gamma _{kj}^{b}(x_j(t))f_j(x_j(t-\tau (t)))\right. \nonumber \\&\quad + \left. \sum \limits _{j=1}^{n}\omega _{kj}g_j(x_j(t))+\sum \limits _{j=1}^{n}\rho _{kj}f_j(x_j(t-\tau (t)))\right\} \nonumber \\&\le -\frac{|x_k(t)|}{\beta _k}+\frac{1}{\beta _k}\sum \limits _{j=1}^{n} \beta _j a_{kj}^{m}G_j \frac{|x_j(t)|}{\beta _j}+\frac{1}{\beta _k}\sum \limits _{j=1}^{n} \beta _j b_{kj}^{m} F_j \frac{|x_j(t-\tau (t))|}{\beta _j}\nonumber \\&\quad + \frac{1}{\beta _k} \sum \limits _{j=1}^{n} \beta _j \omega _{kj}G_j \frac{|x_j(t)|}{\beta _j}+\frac{1}{\beta _k}\sum \limits _{j=1}^{n}\beta _j \rho _{kj} F_j \frac{|x_j(t-\tau (t))|}{\beta _j}\nonumber \\&\le -[1-\frac{1}{\beta _k}\sum \limits _{j=1}^{n}\beta _j G_j(a_{kj}^{m}+\omega _{kj})]W(t)+\frac{1}{\beta _k}\sum \limits _{j=1}^{n} \beta _{j} F_j (b_{kj}^{m}+\rho _{kj}){\overline{W}}(t). \end{aligned}$$
(46)

Hence

$$ (t-t_0+r)^{\alpha }{}_{t_0}^{C}D_{t}^{\alpha }W(t) \le -\left[ 1-\sum \limits _{j=1}^{n} \frac{\beta _j}{\beta _k}\left( G_j\left( a_{kj}^{m} +\omega _{kj}\right) +\left( \frac{r}{r-\tau }\right) ^{\alpha }F_j\left(b_{kj}^{m}+\rho _{kj}\right) \right) \right] V(t),$$
(47)

where \({\overline{\theta }}\in [-\tau ,t]\) such that \({\overline{V}}(t)=(t+{\overline{\theta }}-t_0+r)^{\alpha } {\overline{W}}(t)\), when \({\overline{V}}(t)=V(t)\).

From (45) and (47), we have

$$\begin{aligned} {}_{t_0}^{C}D_{t}^{\alpha }V(t)&\le -\left[ 1-\sum \limits _{j=1}^{n} \frac{\beta _j}{\beta _k}\left( G_j\left( a_{kj}^{m}+\omega _{kj}\right) +\left( \frac{r}{r-\tau }\right) ^{\alpha }F_j\left( b_{kj}^{m}+\rho _{kj}\right) \right) \right] V(t) +\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}{\overline{V}}(t)\nonumber \\&\le \left\{ -\left[ 1-\sum \limits _{j=1}^{n} \frac{\beta _j}{\beta _k}\left( G_j\left( a_{kj}^{m}+\omega _{kj}\right) +\left( \frac{r}{r-\tau }\right) ^{\alpha }F_j\left( b_{kj}^{m}+\rho _{kj}\right) \right) \right] +\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}\right\} V(t), \end{aligned}$$
(48)

when \({\overline{V}}(t)=V(t)\).

It follows that

$$ {}_{t_0}^{C}D_{t}^{\alpha } {\overline{V}}(t)\le 0, $$
(49)

for all \(t \ge t_0\).

On the basis of Definition 3, we get

$$ {\overline{V}}(t)\le {\overline{V}}(t_0), $$
(50)

for \(t\ge t_0\). Hence \(i=1,2,\ldots ,n\),

$$ |x_i(t)|\le \beta _i W(t)\le \frac{\beta _i r^{\alpha }\Vert \varphi \Vert _C}{\beta _{\min }(t-t_0+r)^{\alpha }}, $$
(51)

where \(\beta _{\min }=\min \{\beta _i, i=1,2,\ldots ,n\}\), for \(t\ge t_0\), it follows

$$ \Vert x(t)\Vert \le \frac{\Theta r^{\alpha }\Vert \varphi \Vert _C}{(t-t_0+r)^{\alpha }}, $$

where \(\Theta =\frac{1}{\beta _{\min }}\sum \nolimits _{i=1}^{n}\beta _i\). Therefore, FMDNNs (1) can be achieved global \(O(t^{-\alpha })\) stabilization under the designed control law (41). \(\square \)

Remark 3

It needs to point out that fractional-order systems can be said rarely exponential stability. While, global Mittag–Leffler stability or global \(O(t^{-\alpha })\) stability can be used to describe asymptotic stability of fractional-order systems. In consideration of the complex and rich nonlinear behaviors of fractional-order systems, especially, for the fractional-order systems with time delays, we employ global \(O(t^{-\alpha })\) stabilization for a class of FMDNNs in Theorems 1–4.

Remark 4

As a useful tool, Lyapunov function method has been introduced to fractional-order systems by borrowing ideas from classical Lyapunov function method in integer-order systems. In Theorems 1–4, a class of new fractional Lyapunov functions have been established, which consist of two Lyapunov functions [i.e., time-invariant Lyapunov function W(t) and time-varying Lyapunov function V(t)]. For this structure of Lyapunov functions, we can regard the Caputo derivative of V(t) as two parts which can be estimated by means of Leibniz rule.

Numerical examples

In this section, two numerical examples are given to show the effectiveness of the proposed theoretical results.

Example 1

Consider a two-dimensional FMDNNs as follows:

$$\begin{aligned} \left\{ \begin{array}{l} {}_{t_0}^{C}D_{t}^{\alpha }x_1(t)=-x_1(t) +a_{11}(x_1(t))g_1(x_1(t))+a_{12}(x_2(t))g_2(x_2(t))\\ \quad \quad \quad \quad \quad +b_{11}(x_1(t))f_1(x_1(t-\tau ))+b_{12}(x_2(t))f_2(x_2(t-\tau ))+u_1(t),\\ {}_{t_0}^{C}D_{t}^{\alpha }x_2(t)=-x_2(t) +a_{21}(x_1(t))g_1(x_1(t))+a_{22}(x_2(t))g_2(x_2(t))\\ \quad \quad \quad \quad \quad +b_{21}(x_1(t))f_1(x_1(t-\tau ))+b_{22}(x_2(t))f_2(x_2(t-\tau ))+u_2(t),\\ \end{array} \right. \end{aligned}$$
(52)

where \(\alpha =0.95\), \(\tau =1\), \(t_0=0\), \(g_j(\chi )=f_j(\chi )=\tanh (\chi )\) \((j=1,2)\), and

$$\begin{aligned}&a_{11}(x_1)=\left\{ \begin{array}{ll} 1.5,&{}\quad |x_1|>1,\\ 2.0, &{}\quad |x_1|< 1,\end{array}\right. \quad a_{12}(x_2)=\left\{ \begin{array}{ll} 0.1,&{}\quad |x_2|>1,\\ 0.2, &{}\quad |x_2|< 1,\end{array}\right. \quad a_{21}(x_1)=\left\{ \begin{array}{ll} 0.5,&{}\quad |x_1|>1,\\ 0.4, &{}\quad |x_1|< 1,\end{array}\right. \\&a_{22}(x_2)=\left\{ \begin{array}{ll} 1.8,&{}\quad |x_2|>1,\\ 1.5, &{}\quad |x_2|< 1,\end{array}\right. \quad b_{11}(x_1)=\left\{ \begin{array}{ll} -3.5,&{}\quad |x_1|>1,\\ -4.0, &{}\quad |x_1|< 1,\end{array}\right. \quad b_{12}(x_2)=\left\{ \begin{array}{ll} -1.8,&{}\quad |x_2|>1,\\ -1.5, &{}\quad |x_2|< 1,\end{array}\right. \\&b_{21}(x_1)=\left\{ \begin{array}{ll} 1.2,&{}\quad |x_1|>1,\\ 1.0, &{}\quad |x_1|< 1,\end{array}\right. \quad b_{22}(x_2)=\left\{ \begin{array}{ll} -1.8,&{}\quad |x_2|>1.\\ -1.5, &{}\quad |x_2|< 1.\end{array}\right. \end{aligned}$$

It is obvious that we can get \(G_j=F_j=1\), \(j=1,2.\)

Figure 1 shows the results of time response of (52) without external controller, which implies that the state trajectory of (52) can not convergence to the origin.

Fig. 1
figure 1

Transient behavior of \(x_{1}(t)\) and \(x_{2}(t)\) for (52) without external controller

Assume that there exist two positive constants \(\beta _1\) and \(\beta _2\) to satisfy

$$\begin{aligned} \left\{ \begin{array}{l} \beta _1 p_{11}+\beta _2 p_{12} \le \beta _1\left( 1-\left( a_{11}^{m}G_1+b_{11}^{m}F_1\right) \right) +\beta _2 \left( a_{12}^{m}G_2+b_{12}^{m}F_2\right) ,\\ \beta _2 p_{22}+\beta _1 p_{21} \le \beta _2 \left( 1-\left( a_{22}^{m}G_2+b_{22}^{m}F_2\right) \right) +\beta _1 \left( a_{21}^{m}G_1+b_{21}^{m}F_1\right) , \end{array}\right. \end{aligned}$$
(53)

it follows that there exists a positive constant r such that

$$\begin{aligned} \left\{ \begin{array}{l} \beta _1 p_{11}+\beta _2 p_{12} \le \beta _1\left( 1-\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}-\left( a_{11}^{m}G_1 +\left( \frac{r}{r-\tau }\right) ^{\alpha }b_{11}^{m}F_1\right) \right) -\beta _2 \left( a_{12}^{m}G_2+\left( \frac{r}{r-\tau }\right) ^{\alpha } b_{12}^{m}F_2\right) ,\\ \beta _2 p_{22}+\beta _1 p_{21} \le \beta _2 \left( 1-\frac{1+\alpha }{r^{\alpha }\Gamma (2-\alpha )}- \left( a_{22}^{m}G_2+\left( \frac{r}{r-\tau }\right) ^{\alpha }b_{22}^{m}F_2\right) \right) -\beta _1 \left( a_{21}^{m}G_1+\left( \frac{r}{r-\tau }\right) ^{\alpha } b_{21}^{m}F_1\right) , \end{array}\right. \end{aligned}$$

which implies the conditions of Theorem 1 hold.

From Example 1, we have

$$\begin{aligned} \left\{ \begin{array}{l} \beta _1 (p_{11}+5)+\beta _2 (p_{12}+2) \le 0.\\ \beta _2 (p_{22}+2.6)+\beta _1 (p_{21}+1.7) \le 0, \end{array}\right. \end{aligned}$$

then we can choose \(p_{11}=-5\), \(p_{12}=-2\), \(p_{21}=-2\), \(p_{22}=-3\), i.e., the state feedback controller without time delays can be designed as follows:

$$\begin{aligned} \left\{ \begin{array}{l} u_1(t)=-5x_1(t)-2x_2(t).\\ u_2(t)=-2x_1(t)-3x_2(t). \end{array}\right. \end{aligned}$$
(54)

According to Theorem 1, system (52) can be achieved global \(O(t^{-\alpha })\) stabilization. From Fig. 2, we can get that the state trajectory of the resulting closed-loop system of (52) with the designed control law (54) is globally \(O(t^{-\alpha })\) stable.

Fig. 2
figure 2

Transient behavior of \(x_{1}(t)\) and \(x_{2}(t)\) for (52) with state feedback control rule \(u_1(t)=-5x_1(t)-2x_2(t), u_2(t)=-2x_1(t)-3x_2(t)\)

Similarly, select the state feedback controller with time delays designed as follows:

$$\begin{aligned} \left\{ \begin{array}{l} u_1(t)=-7x_1(t)-3x_2(t)+0.5x_1(t-1)-0.1x_2(t-1).\\ u_2(t)=-3x_1(t)-5x_2(t)+0.1x_1(t-1)-0.5x_2(t-1). \end{array}\right. \end{aligned}$$
(55)

Then it follows from Theorem 2 that system (52) can be achieved global \(O(t^{-\alpha })\) stabilization. From Fig. 3, we can get that the state trajectory of the resulting closed-loop system of (52) with the designed control law (55) is globally \(O(t^{-\alpha })\) stable.

Fig. 3
figure 3

Transient behavior of \(x_{1}(t)\) and \(x_{2}(t)\) for (52) with state feedback control rule \(u_1(t)=-7x_1(t)-3x_2(t)+0.5x_1(t-1)-0.1x_2(t-1)\), \(u_2(t)=-3x_1(t)-5x_2(t)+0.1x_1(t-1)-0.5x_2(t-1)\)

Example 2

Consider an one-dimensional FMDNNs as follows:

$$ {}_{t_0}^{C}D_{t}^{\alpha }x(t)=-x(t)+a(x(t))g(x(t)) +b(x(t))f(x(t-\tau ))+u(t), $$
(56)

where \(\alpha =0.5\), \(\tau =1\), \(t_0=0\), \(g(\chi )=\sin (\chi )\), \(f(\chi )=\tanh (\chi )\), and

$$\begin{aligned} a(x)=\left\{ \begin{array}{ll} 1.0,&{}\quad |x|>1,\\ 1.2, &{}\quad |x|< 1,\end{array}\right. \quad b(x)=\left\{ \begin{array}{ll} -4.8,&{}\quad |x|>1.\\ -3.5, &{}\quad |x|< 1.\end{array}\right. \end{aligned}$$

It is obvious that we can obtain \(G=F=1.\)

Figure 4 shows the results of time response of (56) without external controller, which implies that the state trajectory of (56) can not convergence to the origin.

Fig. 4
figure 4

Transient behavior of x(t) for (56) without external controller

In order to apply Theorem 3, the following inequality needs to be satisfied

$$ \omega +5 \le 0, $$
(57)

then we can choose \(\omega\) = −5, i.e., the state feedback controller without time delays can be designed as follows:

$$ u(t)=-5\sin (x). $$
(58)

It follows from Theorem 3 that system (56) can be achieved global \(O(t^{-\alpha })\) stabilization. From Fig. 5, we can get that the state trajectory of the resulting closed-loop system of (56) with the designed control law (58) is globally \(O(t^{-\alpha })\) stable.

Fig. 5
figure 5

Transient behavior of x(t) for (56) with output feedback control rule \(u(t)=-5\sin (x)\)

Similarly, select the output feedback controller with time delays designed as follows:

$$ u(t)=-7\sin (x)-\tanh (x-1). $$
(59)

Then it follows from Theorem 4 that system (56) can be achieved global \(O(t^{-\alpha })\) stabilization. From Fig. 6, we can get that the state trajectory of the resulting closed-loop system of (56) with the designed control law (59) is globally \(O(t^{-\alpha })\) stable.

Fig. 6
figure 6

Transient behavior of x(t) for (56) with output feedback control rule \(u(t)=-7\sin (x)-\tanh (x-1)\)

Concluding remarks

In this article, we exploit the global \(O(t^{-\alpha })\) stabilization for a class of fractional-order memristive neural networks with time delays. The main theoretical results of this article are that the linear state feedback control law and the output feedback control law are constructed to stabilize the fractional systems. In addition, some sufficient conditions ensuring to stabilize fractional-order systems are also given in terms of algebraic inequalities according to a new fractional Lyapunov function and a fractional-order differential inequality skill. The article provides a novel way to construct a Lyapunov function and a new method to deal with fractional-order inequalities, which may be applied to discuss other properties or analyze other more complex systems such as the fractional-order form of the model explored in the literatures Chandrasekar and Rakkiyappan (2016), Lou et al. (2013), Shang (2014, 2015, 2016), Wang et al. (2003), Yang and Tong (2016) and so on. Future research will focus on these issues.

References

  • Bao G, Zeng ZG (2013) Multistability of periodic delayed recurrent neural network with memristors. Neural Comput Appl 23(7):1963–1967

    Article  Google Scholar 

  • Chandrasekar A, Rakkiyappan R (2016) Impulsive controller design for exponential synchronization of delayed stochastic memristor-based recurrent neural networks. Neurocomputing 173(3):1348–1355

    Article  Google Scholar 

  • Chen BS, Chen JJ (2015) Razumikhin-type stability theorems for functional fractional-order differential systems and applications. Appl Math Comput 254:63–69

    Google Scholar 

  • Chen BS, Chen JJ (2016) Global \(O(t^{-\alpha })\) stability and global asymptotical periodicity for a non-autonomous fractional-order neural networks with time-varying delays. Neural Netw 73:47–57

    Article  Google Scholar 

  • Chen JJ, Zeng ZG, Jiang P (2014) Global Mittag–Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw 51(3):1–8

    Article  Google Scholar 

  • Chen LP, Wu RC, Cao JD, Liu JB (2015) Stability and synchronization of memristor-based fractional-order delayed neural networks. Neural Netw 71:37–44

    Article  Google Scholar 

  • Chua LO (1971) Memristor—the missing circuit element. IEEE Trans Circuit Theory 18(5):507–519

    Article  Google Scholar 

  • Guo ZY, Wang J, Yan Z (2013) Global exponential dissipativity and stabilization of memristor-based recurrent neural networks with time-varying delays. Neural Netw 48(6):158–172

    Article  Google Scholar 

  • Guo ZY, Wang J, Yan Z (2015) Global exponential synchronization of two memristor-based recurrent neural networks with time delays via static or dynamic coupling. IEEE Trans Syst Man Cybern Syst 45(2):235–249

    Article  Google Scholar 

  • Huang YC (2010) On a stabilization problem of nonlinear programming neural networks. Neural Process Lett 31(2):93–103

    Article  Google Scholar 

  • Huang JJ, Li CD, Han Q (2009) Stabilization of delayed chaotic neural networks by periodically intermittent control. Circuits Syst Signal Process 28(4):567–579

    Article  Google Scholar 

  • Li HL, Jiang YL, Wang ZL, Zhang L, Teng ZD (2015) Global Mittag–Leffler stability of coupled system of fractional-order differential equations on network. Appl Math Comput 270:269–277

    Google Scholar 

  • Liang S, Wu RC, Chen LP (2015) Comparison principles and stability of nonlinear fractional-order cellular neural networks with multiple time delays. Neurocomputing 168:618–625

    Article  Google Scholar 

  • Liu QX, Liu JK, Chen YM (2015) Non-diminishing relative error of the predictor–corrector algorithm for certain fractional differential equations. Math Comput Simul 117:10–19

    Article  Google Scholar 

  • Lou XY, Ye Q, Cui BT (2013) Impulsive stabilization of fuzzy neural networks with time-varying delays. Arab J Math 2(1):65–79

    Article  Google Scholar 

  • Mathiyalagan K, Anbuvithya R, Sakthivel R, Park JuH, Prakash P (2015) Reliable stabilization for memristor-based reccurent neural networks with time-varying delays. Neurocomputing 153:140–147

    Article  Google Scholar 

  • Rakkiyappan R, Velmurugan G, Cao JD (2014) Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays. Nonlinear Dyn 78(4):2823–2836

    Article  Google Scholar 

  • Rakkiyappan R, Sivasamy R, Cao JD (2015a) Stochastic sampled-data stabilizaiton of neural-network-based control systems. Nonlinear Dyn 81:1823–1839

    Article  Google Scholar 

  • Rakkiyappan R, Velmurugan G, Cao JD (2015b) Stability analysis of memristor-based fractional-order neural networks with different memductance functions. Cogn Neurodyn 9(2):145–177

    Article  Google Scholar 

  • Shang YL (2014) Average consensus in multi-agent systems with uncertain topologies and multiple time-varying delays. Linear Algebra Appl 459(3):411–429

    Article  Google Scholar 

  • Shang YL (2015) Group consensus of multi-agent systems in directed networks with noises and time delays. Int J Syst Sci 46(14):2481–2492

    Article  Google Scholar 

  • Shang YL (2016) Consensus seeking over Markovian switching networks with time-varying delays and uncertain topologies. Appl Math Comput 273:1234–1245

    Google Scholar 

  • Stamova I (2014) Global Mittag–Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays. Nonlinear Dyn 77(4):1251–1260

    Article  Google Scholar 

  • Strukov DB, Snider GS, Stewart DR, Williams RS (2008) The missing memristor found. Nature 453(7191):80–83

    Article  Google Scholar 

  • Tour JM, He T (2008) The fourth element. Nature 453(7191):42–43

    Article  Google Scholar 

  • Velmurugan G, Rakkiyappan R (2016) Hybrid projective synchronization of fractional-order memristor-based neural networks with time delays. Nonlinear Dyn 83(1):419–432

    Article  Google Scholar 

  • Velmurugan G, Rakkiyappan R, Cao JD (2016) Finite-time synchronization of fractional-order memristor-based neural networks with time delays. Neural Netw 73:36–46

    Article  Google Scholar 

  • Wang X, Li CD, Huang TW, Duan SK (2003) Global exponential stability of a class of memristive neural networks with time-varying delays. Neural Comput Appl 24(7):1707–1715

    Google Scholar 

  • Wang H, Yu YG, Wen GG, Zhang S (2014) Stability analysis of fractional-order neural networks with time delay. Neural Netw 55:98–109

    Article  Google Scholar 

  • Wen SP, Zeng ZG (2012) Dynamics analysis of a class of memristor-based recurrent networks with time-varying delays in the presence of strong external stimuli. Neural Process Lett 35(1):47–59

    Article  Google Scholar 

  • Wu AL, Zeng ZG (2012) Danamic behaviors of memristor-based recurrent neural networks with time-varying delays. Neural Netw 36(8):1–10

    Article  Google Scholar 

  • Wu AL, Zeng ZG (2016) Boundness, Mittag-L-effler stability asymptotical \(\omega \)-periodicity of fractional-order fuzzy neural networks. Neural Netw 74:73–84

    Article  Google Scholar 

  • Wu AL, Wen SP, Zeng ZG (2012) Synchronization control of a class of memristor-based recurrent neural networks. Inf Sci 183(2):106–116

    Article  Google Scholar 

  • Wu AL, Zeng ZG, Song XG (2016) Global Mittag–Leffler stabilization of fractional-order bidirectional associative memory neural networks. Neurocomputing 177:489–496

    Article  Google Scholar 

  • Yang W, Tong SC (2016) Robust stabilization of switched fuzzy systems with actuator dead zone. Neurocomputing 173:1028–1033

    Article  Google Scholar 

  • Zhang LL, Huang LH, Cai ZW (2016) Finite-time stabilization control for discontinuous time-delayed networks: new switching design. Neutal Netw 75:84–96

    Article  Google Scholar 

  • Zhao H, Li LX, Peng HP, Kurths J, Xiao JH, Yang YX (2015) Anti-synchronization for stochastic memristor-based neural networks with non-modeled dynamics via adaptive control approach. Eur Phys J B 88:109

    Article  Google Scholar 

Download references

Authors' contributions

LL wrote the draft of this article. AW provided the idea of this article. XS offered the composition and polished English expression. All authors read and approved the final manuscript.

Acknowledgements

The authors thank the Associate Editor and the reviewers for their suggestions which have been very useful to improve the quality of article. The work is supported by the Natural Science Foundation of China under Grant 61304057.

Competing interests

The authors declare that they have no competing interests.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ailong Wu.

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, L., Wu, A. & Song, X. Global \(O(t^{-\alpha })\) stabilization of fractional-order memristive neural networks with time delays. SpringerPlus 5, 1034 (2016). https://doi.org/10.1186/s40064-016-2374-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s40064-016-2374-3

Keywords

  • Fractional-order systems
  • Memristive neural networks
  • Global \(O(t^{-\alpha })\) stabilization

Mathematics Subject Classification

  • 37B25
  • 34A08