Global Exponential Stability of Fractional Order Complex-Valued Neural Networks with Leakage Delay and Mixed Time Varying Delays

Abstract

This paper investigates the global exponential stability of fractional order complex-valued neural networks with leakage delay and mixed time varying delays. By constructing a proper Lyapunov-functional we established sufficient conditions to ensure global exponential stability of the fractional order complex-valued neural networks. The stability conditions are established in terms of linear matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

1. Introduction

In the past decades, fractional order systems have become an important and hot research field. It is well known that the stability is the primary condition of the various systems. Around 300 years back, the foundation of fractional order calculus [1,2,3,4,5,6,7,8,9,10,11], which is an extension of classical integer order calculus, was first discussed by Leibniz and L’Hospital, and its development was very slow for a long period. Compared with an integer-order model, fractional order models can offer more accurate instrument for memory description and inherited properties of several processes. In addition to the above mentioned developments, neural networks have emerged out as a valuable tool in the field of fractional calculus also for modeling various phenomena. Some researchers introduced the fractional order derivatives into neural networks. Fractional order neural networks can be used to replicate neurons in the brain and describe dynamical characteristics of actual network systems precisely. Fractional order neural networks have a wide range of applications, including parameter estimation in statistical theory, quantum motion description in physics, and network security communications [12,13,14,15,16,17,18].

In recent years, neural networks have been extensively investigated since their wide applications, such as pattern recognition, associative memories, automatic control, optimization, image processing and other areas. Stability analysis of various classes of neural network models such as Hopfield neural networks, Cohen–Grossberg neural networks, cellular neural networks, and bidirectional associative memory neural networks has been extensively investigated as stable neural networks. They all have obtained global exponential stability for some kinds of neural networks through different methods.

Through the previous decades, complex valued neural networks is a great research topic. The stability and synchronization analysis of fractional order systems are very difficult. Since calculating the fractional order derivatives of Lyapunov functional is complicated. The stability analysis methods for integer order systems such as Lyapunov functional method cannot be easily generalized to fractional order systems. Complex valued neural networks have been applied in electromagnetic, light, quantum waves and so on. As far as we know, there are a few results about complex valued neural networks and important and interesting results were proposed [19,20,21]. For Instance in [22,23], authors dealt with the existence, uniqueness and global stability of equilibrium point of fractional-order complex valued neural networks with time delays while in [24,25], the synchronization of neural network is investigated.

In reality, time delays inevitably exist in various systems. Time delay is an inherent property of a neural network due to the finite speed of the signal propagation and the information processing time among neurons. Though considering that the time delays arise frequently in practical applications, it is difficult to measure them precisely. In most situations, the delays are variable and unbounded. So it is necessary to study complex valued neural networks with mixed time delays. A few researchers declare that representative time delay, called leakage delay, may exist in the negative input term of the neural systems and greatly affect the elements of neural organizations. On the other hand, the leakage delay extensively affects the dynamical behaviors of systems [26,27,28,29,30,31,32]. So far, there exist few studies considering fractional-order complex valued neural networks with leakage delay [33,34,35].

Motivated by the above discussions, in the present endeavor, we have focused on the stability of fractional order complex valued neural networks with leakage and mixed time-varying delays. The purpose of this paper is to establish some criteria which can guarantee the globally exponential stability of complex valued neural networks with the given convergence rate by using Lyapunov function techniques. The main contributions can be listed as follows:

(i)

In this paper, by making use of a delay differential inequality, we present a new sufficient condition which guarantees global exponential stability of the unique equilibrium point of complex valued neural networks with time-varying delays.

(ii)

It is difficult to calculate fractional-order derivatives of Lyapunov functionals. In order to overcome the difficulty, we constructed a suitable functional including fractional derivative terms, and calculated its derivative to derive the stability conditions.

(iii)

By using Lyapunov functional we obtain the sufficient conditions for global exponential stability neural networks with delays, in terms of LMI, which can be easily calculated by MATLAB LMI control toolbox.

(iv)

Numerical examples are given to illustrate the effectiveness of the derived methods.

2. Preliminaries

Definition 1

([36]). The Caputo fractional derivative of order α for a function $h\left(t\right)$ is defined as follows:

$$\begin{array}{c}\hfill {\mathbb{D}}^{\alpha }h\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}{\int }_0^t{(t-s)}^{n-\alpha -1}{h}^{\left(n\right)}\left(s\right)ds,\end{array}$$

where n is the positive integer such that $n-1<\alpha <n$.

Definition 2

([37]). The zero solution of (1) is said to be globally exponentially stable in PC, if there exist constants $\alpha >0$ and $K\ge 0$ such that for any solution $y(t,t_0,\varphi )$ with the initial condition $\varphi \in PC$,

$$\begin{array}{c}\hfill ||y(t,t_0,x\left(t_0\right))||\le K||\varphi |{|}_{\rho }{e}^{-\alpha (t-t_0)},t\ge t_0.\end{array}$$

Lemma 1

([38]). If $S\in C^{n\times n}$ is a positive definite Hermitian matrix, and $\mu \left(x\right):[a,b]\to C^n$ is a scalar function with scalars $a<b$, then

$$\begin{array}{c}\hfill \left({\int }_a^b{\mu }^T\left(x\right)dx\right)S\left({\int }_a^b{\mu }^T\left(x\right)dx\right)\le (b-a){\int }_a^b{\mu }^T\left(x\right)S\mu \left(x\right)dx.\end{array}$$

Lemma 2

([39]). Given constant matrices ${\mathbb{S}}_1,{\mathbb{S}}_2,{\mathbb{S}}_3$, where ${\mathbb{S}}_1={\mathbb{S}}_1^T,{\mathbb{S}}_2={\mathbb{S}}_2^T$, and ${\mathbb{S}}_2>0$, then

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{\mathbb{S}}_1& {\mathbb{S}}_3^T\\ {\mathbb{S}}_3& -{\mathbb{S}}_2\end{array}\right]<0\end{array}$$

if and only if

$$\begin{array}{c}\hfill {\mathbb{S}}_1+{\mathbb{S}}_3^T{\mathbb{S}}_2^{-1}{\mathbb{S}}_3<0.\end{array}$$

Assumption 1.

$\mu \left(t\right)$, $\phi \left(t\right)$$\delta \left(t\right)$ are delays satisfying,

$$\begin{array}{cc}\hfill & 0\le \delta \left(t\right)\le \delta ,\dot{\delta }\left(\mathbf{t}\right)\le {\delta }_1,\hfill \\ \hfill & 0\le \phi \left(t\right)\le \phi ,\dot{\phi }\left(t\right)\le {\phi }_1,\hfill \\ \hfill & 0\le \mu \left(t\right)\le \mu ,\dot{\mu }\left(t\right)\le {\mu }_1.\hfill \end{array}$$

Assumption 2.

In the complex field, the activation functions $f(·),h(·)$ satisfy the Lipschitz conditions:

$$\begin{array}{c}\hfill |f_k\left(p\right)-f_k\left(q\right)|\le l_k|p-q|,|h_k\left(p\right)-h_k\left(q\right)|\le m_k|p-q|,\end{array}$$

where $p,q\in C$, where $l_k>0$, $m_k>0 (k=1,2,\dots ,n)$ are Lipschitz constants.

In this paper, we consider a class of fractional-order complex-valued neural networks,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathbb{D}}^{\alpha }{z}_j\left(t\right)\hfill & =-l_j{z}_j(t-\mu \left(t\right))+\sum _{\varsigma =1}^n{b}_{j\varsigma }{g}_{\varsigma }\left(z_{\varsigma }\left(t\right)\right)\hfill \\ & +\sum _{\varsigma =1}^n{c}_{j\varsigma }{g}_{\varsigma }\left(z_{\varsigma }(t-{\phi }_{\varsigma }\left(t\right))\right)\hfill \\ & +\sum _{\varsigma =1}^n{e}_{j\varsigma }{\int }_{t-{\delta }_{\varsigma }\left(t\right)}^t{g}_{\varsigma }\left(z_{\varsigma }\left(s\right)\right)ds+I_j\left(t\right),\hfill \\ z_j\left(s\right)\hfill & ={\varphi }_j\left(s\right),s\in [-\phi ,0]\hfill \end{array}\right.\end{array}$$

(1)

or

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathbb{D}}^{\alpha }z\left(t\right)\hfill & =-\mathbb{L}z(t-\mu (t\left)\right)+\mathbb{B}g\left(z\right(t\left)\right)+\mathbb{C}g\left(z\right(t-\phi \left(t\right)\left)\right)\hfill \\ & +\mathbb{E}{\int }_{t-\delta \left(t\right)}^{t}g\left(z\left(s\right)\right)ds+\mathbb{I}\left(t\right),\hfill \\ z\left(s\right)\hfill & =\varphi \left(s\right),s\in [-\phi ,0],\hfill \end{array}\right.\end{array}$$

(2)

where $0<\alpha <1$ denotes the order of fractional-order derivative, $\mu >0$ is the leakage delay, $z_{}\left(t\right)\in {\mathbb{C}}^n$ stands for the neural state, $\mathbb{L}=diag(l_1,\dots ,l_n)\in R^{n\times n}$ stands for the self-feedback connection weight matrix, $\mathbb{B}\in {\mathbb{C}}^{n\times n},\mathbb{C}\in {\mathbb{C}}^{n\times n}$ and $\mathbb{E}\in {\mathbb{C}}^{n\times n}$ all are connection weight matrices, $g(·)\in {\mathbb{C}}^n$ represent the complex valued neuron activation function.

A constant point $z^{*}=(z_1^{*},z_2^{*},\dots ,z_n^{*})$ is said to be an equilibrium point of (1) if $z^{*}$ is a solution of (1), that is,

$$\begin{array}{cc}\hfill 0& =-l_j{z}_j^{*}+\sum _{\varsigma =1}^n{b}_{j\varsigma }{g}_{\varsigma }\left(z_{\varsigma }^{*}\right)+\sum _{\varsigma =1}^n{c}_{j\varsigma }{g}_{\varsigma }\left(z_{\varsigma }^{*}\right)+\sum _{\varsigma =1}^n{e}_{j\varsigma }{\int }_{t-{\delta }_{\varsigma }\left(t\right)}^t{g}_{\varsigma }\left(z_{\varsigma }^{*}\right)ds+I_j.\hfill \end{array}$$

(3)

Let us perturb the system (1) through shifting of coordinates as ${\tilde{\pi }}_j\left(t\right)=z_j\left(t\right)-z_j^{*}$, then we obtain

$$\begin{array}{cc}\hfill {\mathbb{D}}^{\alpha }{\tilde{\pi }}_j\left(t\right)& =-l_j{\tilde{\pi }}_j(t-\mu \left(t\right))+\sum _{\varsigma =1}^n{b}_{j\varsigma }\left[g_{\varsigma }(z_{\varsigma }\left(t\right)+z_{\varsigma }^{*})-g_{\varsigma }\left(z_{\varsigma }^{*}\right)\right]\hfill \\ & +\sum _{\varsigma =1}^n{c}_{j\varsigma }\left[g_{\varsigma }(z_{\varsigma }(t-{\phi }_{\varsigma }\left(t\right))+z_{\varsigma }^{*})-g_{\varsigma }\left(z_{\varsigma }^{*}\right)\right]\hfill \\ & +\sum _{\varsigma =1}^n{e}_{j\varsigma }{\int }_{t-{\delta }_{\varsigma }\left(t\right)}^t{g}_{\varsigma }[(z_{\varsigma }\left(s\right)+z_{\varsigma }^{*})-g_{\varsigma }\left(z_{\varsigma }^{*}\right)]ds\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\mathbb{D}}^{\alpha }{\tilde{\pi }}_j\left(t\right)& =-l_j{\tilde{\pi }}_j(t-\mu \left(t\right))+\sum _{\varsigma =1}^n{b}_{j\varsigma }{h}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }\left(t\right)\right)\hfill \\ & +\sum _{\varsigma =1}^n{c}_{j\varsigma }{f}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }(t-{\phi }_{\varsigma }\left(t\right))\right)\hfill \\ & +\sum _{\varsigma =1}^n{e}_{j\varsigma }{\int }_{t-{\delta }_{\varsigma }\left(t\right)}^t{h}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }\left(s\right)\right)ds,\hfill \end{array}$$

(5)

where

$$\begin{array}{cc}\hfill \left[g_{\varsigma }(z_{\varsigma }\left(t\right)+z_{\varsigma }^{*})-g_{\varsigma }\left(z_{\varsigma }^{*}\right)\right]& =h_{\varsigma }\left({\tilde{\pi }}_{\varsigma }\left(t\right)\right),\hfill \\ \hfill \left[g_{\varsigma }(z_{\varsigma }(t-{\phi }_{\varsigma }\left(t\right))+z_{\varsigma }^{*})-g_{\varsigma }\left(z_{\varsigma }^{*}\right)\right]& =f_{\varsigma }({\tilde{\pi }}_{\varsigma }(t-{\phi }_{\varsigma }\left(t\right)).\hfill \end{array}$$

It is obvious that the functions $f(·),h(·)$ also satisfy the Assumption 2.

3. Mian Results

Theorem 1.

If ${\alpha }_{\mathfrak{j}}(\mathfrak{j}=\mathfrak{1},\mathfrak{2},\dots \dots ,\mathfrak{n})$ is a positive constant such that

$$\begin{array}{c}\hfill {\mathsf{\Lambda }}_1=[-2l_j+3\sum _{\varsigma =1}^n{\mathbb{L}}_{\varsigma }^2]+b_{ij}^2\frac{{\alpha }_j}{{\alpha }_{\varsigma }},{\mathsf{\Lambda }}_2=\frac{{\alpha }_j}{{\alpha }_{\varsigma }}[c_{j\varsigma }^2+e_{j\varsigma }^2].\end{array}$$

(6)

Then the equilibrium point $z^{*}$ of system (1) is globally exponentially stable.

Proof. 

Let $y_j\left(t\right)=\frac{1}{2}{\alpha }_j{\tilde{\pi }}_j^2\left(t\right)$, and calculating ${\mathbb{D}}^{\alpha }{y}_j\left(t\right)$

$$\begin{array}{cc}\hfill {\mathbb{D}}^{\alpha }{y}_j\left(t\right)& ={\alpha }_j{\tilde{\pi }}_j\left(t\right)\left[{\mathbb{D}}^{\alpha }{\tilde{\pi }}_j\left(t\right)\right]\hfill \\ \hfill & ={\alpha }_j{\tilde{\pi }}_j\left(t\right)[-l_j{\tilde{\pi }}_j(t-\mu \left(t\right))+\sum _{\varsigma =1}^n{b}_{j\varsigma }{h}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }\left(t\right)\right)\hfill \\ & +\sum _{\varsigma =1}^n{c}_{j\varsigma }{f}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }(t-{\phi }_{\varsigma }\left(t\right))\right)+\sum _{\varsigma =1}^n{e}_{j\varsigma }{\int }_{t-{\delta }_{\varsigma }\left(t\right)}^t{h}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }\left(s\right)\right)ds]\hfill \\ \hfill & =-{\alpha }_j{l}_j{\tilde{\pi }}_j\left(t\right){\tilde{\pi }}_j(t-\mu \left(t\right))+\sum _{\varsigma =1}^n{\alpha }_j{\tilde{\pi }}_j\left(t\right)b_{j\varsigma }{h}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }\left(t\right)\right)\hfill \\ & +\sum _{\varsigma =1}^n{\alpha }_j{\tilde{\pi }}_j\left(t\right)c_{j\varsigma }{f}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }(t-{\phi }_{\varsigma }\left(t\right))\right)\hfill \\ & +\sum _{\varsigma =1}^n{\alpha }_j{\tilde{\pi }}_j\left(t\right)e_{j\varsigma }{\int }_{t-{\delta }_{\varsigma }\left(t\right)}^t{h}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }\left(s\right)\right)ds\hfill \\ \hfill & \le -{\alpha }_j{\tilde{\pi }}_j^2\left(t\right)l_j+\sum _{\varsigma =1}^n{\alpha }_j|{\tilde{\pi }}_j\left(t\right)|b_{j\varsigma }|{\mathbb{L}}_{\varsigma }|{\tilde{\pi }}_{\varsigma }\left(t\right)|\hfill \\ & +\sum _{\varsigma =1}^n{\alpha }_j|{\tilde{\pi }}_j\left(t\right)||c_{j\varsigma }||{\mathbb{L}}_{\varsigma }||{\overline{\tilde{\pi }}}_{\varsigma }\left(t\right)|\hfill \\ & +\sum _{\varsigma =1}^n{\alpha }_j|{\tilde{\pi }}_j\left(t\right)||e_{j\varsigma }||{\mathbb{L}}_{\varsigma }||{\overline{\tilde{\pi }}}_{\varsigma }\left(t\right)|\hfill \\ \hfill & \le -{\alpha }_j{\tilde{\pi }}_j^2\left(t\right)l_j+\frac{1}{2}\sum _{\varsigma =1}^n{\alpha }_j[{\mathbb{L}}_{\varsigma }^2{\tilde{\pi }}_j^2\left(t\right)+b_{j\varsigma }^2{\tilde{\pi }}_{\varsigma }^2\left(t\right)]\hfill \\ & +\frac{1}{2}\sum _{\varsigma =1}^n{\alpha }_j[{\mathbb{L}}_{\varsigma }^2{\tilde{\pi }}_j^2\left(t\right)+c_{j\varsigma }^2{\overline{\tilde{\pi }}}_{\varsigma }^2\left(t\right)]\hfill \\ & +\frac{1}{2}\sum _{\varsigma =1}^n{\alpha }_j[|{\tilde{\pi }}_j^2\left(t\right){\mathbb{L}}_{\varsigma }^2+e_{j\varsigma }^2{\overline{\tilde{\pi }}}_{\varsigma }^2\left(t\right)]\hfill \\ \hfill & =[-{\alpha }_j{l}_j+\frac{3}{2}\sum _{\varsigma =1}^n{\alpha }_j{\mathbb{L}}_{\varsigma }^2]{\tilde{\pi }}_j^2\left(t\right)+\frac{1}{2}\left[{\alpha }_j{b}_{j\varsigma }^2\right]{\tilde{\pi }}_{\varsigma }^2\left(t\right)\hfill \\ & +\frac{1}{2}[\sum _{\varsigma =1}^n{\alpha }_j{c}_{j\varsigma }^2+\sum _{\varsigma =1}^n{\alpha }_j{e}_{j\varsigma }^2]{\overline{\tilde{\pi }}}_{\varsigma }^2\hfill \\ \hfill & =\sum _{\varsigma =1}^n\{[-2l_j+3\sum _{\varsigma =1}^n{\mathbb{L}}_{\varsigma }^2]+b_{j\varsigma }^2\frac{{\alpha }_j}{{\alpha }_{\varsigma }}\}\frac{1}{2}{\alpha }_{\varsigma }{\tilde{\pi }}_{\varsigma }^2\left(t\right)\hfill \\ \hfill & +\sum _{\varsigma =1}^n\{\frac{{\alpha }_j}{{\alpha }_{\varsigma }}{c}_{j\varsigma }^2+\frac{{\alpha }_j}{{\alpha }_{\varsigma }}{e}_{j\varsigma }^2\}\frac{1}{2}{\alpha }_{\varsigma }{\overline{\tilde{\pi }}}_{\varsigma }^2\left(t\right).\hfill \end{array}$$

As we know that ${\mathsf{\Lambda }}_1=[-2l_j+3{\sum }_{\varsigma =1}^n{L}_{\varsigma }^2]+b_{j\varsigma }^2\frac{{\alpha }_j}{{\alpha }_{\varsigma }}$, ${\mathsf{\Lambda }}_2=\frac{{\alpha }_j}{{\alpha }_{\varsigma }}[c_{j\varsigma }^2+e_{j\varsigma }^2]$, then it can be rewritten as

$$\begin{array}{c}\hfill D^{\alpha }{y}_j\left(t\right)\le {\mathsf{\Lambda }}_{1}y\left(t\right)+{\mathsf{\Lambda }}_2\overline{y}\left(t\right).\end{array}$$

If the matrix $\mathsf{\Lambda }=-({\mathsf{\Lambda }}_1+{\mathsf{\Lambda }}_2)$ is an M-matrix, then there exist constants $\lambda >0, s_j>0$$(j=1,2,\dots ,n)$ then

$$\begin{array}{cc}\hfill \frac{1}{2}{\alpha }_{min}{\tilde{\pi }}_j^2\left(t\right)\le y_j\left(t\right)& =\frac{1}{2}{\alpha }_j{\tilde{\pi }}_j^2\left(t\right)\hfill \\ \hfill & \le s_j\sum _{\varsigma =1}^n{\overline{y}}_{\varsigma }\left(t_0\right){\eta }^{-\lambda (t-t_0)}\hfill \\ \hfill & =s_j\sum _{\varsigma =1}^n\frac{1}{2}{\alpha }_{\varsigma }{\overline{\tilde{\pi }}}_{\varsigma }^2\left(t_0\right){\eta }^{-\lambda (t-t_0)}\hfill \\ \hfill & \le \frac{1}{2}{s}_j{\alpha }_{max}\sum _{\varsigma =1}^n{\overline{\tilde{\pi }}}_{\varsigma }^2\left(t_0\right){\eta }^{-\lambda (t-t_0)}\hfill \\ \hfill {\tilde{\pi }}_j^2\left(t\right)& \le \left(\frac{{\alpha }_{max}}{{\alpha }_{min}}{s}_j\right)\sum _{\varsigma =1}^n{\overline{\tilde{\pi }}}_{\varsigma }^2\left(t_0\right){\eta }^{-\lambda (t-t_0)}\hfill \end{array}$$

that is,

$$\begin{array}{c}\hfill ||z_j\left(t\right)-z_j^{*}||\le {\left(\frac{{\alpha }_{max}}{{\alpha }_{min}}{s}_j\right)}^{\frac{1}{2}}||{\overline{z}}_j\left(t_0\right)-z_j^{*}||{\eta }^{\frac{-\lambda (t-t_0)}{2}}.\end{array}$$

(7)

This implies that the unique equilibrium point of Equation (1) is globally exponentially stable. □

Remark 1.

$$\begin{array}{cc}\hfill {\mathbb{D}}^{\alpha }{\tilde{\pi }}_j\left(t\right)& =-l_j{\tilde{\pi }}_j(t-\mu \left(t\right))+\sum _{\varsigma =1}^n{b}_{j\varsigma }{h}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }\left(t\right)\right)\hfill \\ & +\sum _{\varsigma =1}^n{c}_{j\varsigma }{f}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }(t-{\phi }_{\varsigma }\left(t\right))\right)\hfill \\ & +\sum _{\varsigma =1}^n{e}_{j\varsigma }{\int }_{t-{\delta }_{\varsigma }\left(t\right)}^t{h}_{\varsigma }\left({\tilde{\pi }}_{\varsigma }\left(s\right)\right)ds+u_{\varsigma }\left(t\right)\hfill \end{array}$$

(8)

or equivalently

$$\begin{array}{cc}\hfill {\mathbb{D}}^{\alpha }\tilde{\pi }\left(t\right)& =-\mathbb{L}\tilde{\pi }(t-\mu \left(t\right))+\mathbb{B}h\left(\tilde{\pi }\left(t\right)\right)+\mathbb{C}f\left(\tilde{\pi }(t-\phi \left(t\right))\right)\hfill \\ & +\mathbb{E}{\int }_{t-\delta \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds+\mathbb{U}\left(t\right).\hfill \end{array}$$

(9)

To investigate the global exponential stability of the system, a simple sliding motion is proposed as

$$\begin{array}{c}\hfill s\left(t\right)=D^{\alpha -1}\tilde{\pi }\left(t\right).\end{array}$$

It is worth noting that the first-order differential order of the sliding surface is equal to the equation of the system. That is calculating the first-order derivative on both sides of above equation, we can get

$$\begin{array}{c}\hfill \dot{s}\left(t\right)=D^{\alpha }\tilde{\pi }\left(t\right).\end{array}$$

A state feedback sliding mode control design for the system was constructed as follows

$$\begin{array}{c}\hfill \mathbb{U}\left(t\right)=-\mathbb{K}s\left(t\right),\end{array}$$

(10)

where $\mathbb{K}$ is the state feedback gain. Then,

$$\begin{array}{cc}\hfill {\mathbb{D}}^{\alpha }\tilde{\pi }\left(t\right)& =-\mathbb{L}\tilde{\pi }(t-\mu )+\mathbb{B}h\left(\tilde{\pi }\left(t\right)\right)+\mathbb{C}f\left(\tilde{\pi }(t-\phi \left(t\right))\right)\hfill \\ & +\mathbb{E}{\int }_{t-\delta \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds-\mathbb{K}s\left(t\right).\hfill \end{array}$$

(11)

Theorem 2.

The system (11) is globally exponentially stable if there exist positive definite Hermitian matrices ${\mathbb{R}}_1$, ${\mathbb{R}}_2$, ${\mathbb{R}}_3$, ${\mathbb{R}}_4$, ${\mathbb{R}}_5$, ${\mathbb{R}}_6$, ${\mathbb{R}}_7$, ${\mathbb{R}}_8$, ${\mathbb{R}}_9$, ${\mathbb{R}}_{10}$, positive diagonal matrices ${\mathbb{T}}_1$, ${\mathbb{T}}_2$, ${\mathbb{T}}_3$, ${\mathbb{T}}_4$ and a positive scalar α, satisfying the following condition:

$$\begin{array}{c}\hfill \mathsf{\Omega }={\left({\mathsf{\Omega }}_{i,j}\right)}_{13\times 13}<0,\end{array}$$

(12)

where

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_{1,1}& ={\mathbb{R}}_4+\delta {\mathbb{R}}_6+\phi {\mathbb{R}}_8+{\mathbb{R}}_9+{\mathbb{R}}_{10}-{\mathbb{H}}_1{\mathbb{T}}_1{\mathbb{H}}_2-{\mathbb{L}}_1{\mathbb{T}}_3{\mathbb{L}}_2,\hfill \\ \hfill {\mathsf{\Omega }}_{1,2}& ={\mathbb{T}}_1({\mathbb{H}}_1+{\mathbb{H}}_2),{\mathsf{\Omega }}_{1,3}={\mathbb{T}}_3({\mathbb{L}}_1+{\mathbb{L}}_2),\hfill \\ \hfill {\mathsf{\Omega }}_{2,2}& ={\mathbb{R}}_2+\phi {\mathbb{R}}_5-{\mathbb{T}}_1,{\mathsf{\Omega }}_{3,3}={\mathbb{R}}_3+\delta {\mathbb{R}}_7-{\mathbb{T}}_3,{\mathsf{\Omega }}_{3,13}={\mathbb{B}}^T{\mathbb{R}}_1,\hfill \\ \hfill {\mathsf{\Omega }}_{4,4}& =-{\eta }^{-2\alpha \phi }{\mathbb{R}}_9(1-{\phi }_1)-{\mathbb{H}}_1{\mathbb{T}}_2{\mathbb{H}}_2,{\mathsf{\Omega }}_{4,7}={\mathbb{T}}_2({\mathbb{H}}_1+{\mathbb{H}}_2),\hfill \\ \hfill {\mathsf{\Omega }}_{5,5}& =-{\eta }^{-2\alpha \mu }{\mathbb{R}}_4(1-{\mu }_1),{\mathsf{\Omega }}_{5,13}=-{\mathbb{L}}^T{\mathbb{R}}_1,\hfill \\ \hfill {\mathsf{\Omega }}_{6,6}& =-{\eta }^{-2\alpha \delta }{\mathbb{R}}_{10}(1-{\delta }_1)-{\mathbb{L}}_1{\mathbb{T}}_4{\mathbb{L}}_2,{\mathsf{\Omega }}_{6,8}={\mathbb{T}}_4({\mathbb{L}}_1+{\mathbb{L}}_2),\hfill \\ \hfill {\mathsf{\Omega }}_{7,7}& =-{\mathbb{R}}_2(1-{\phi }_1){\eta }^{-2\alpha \phi }-{\mathbb{T}}_2,{\mathsf{\Omega }}_{7,13}={\mathbb{C}}^T{\mathbb{R}}_1,\hfill \\ \hfill {\mathsf{\Omega }}_{8,8}& =-{\eta }^{-2\alpha \delta }{\mathbb{R}}_3(1-{\delta }_1)-{\mathbb{T}}_4,{\mathsf{\Omega }}_{9,9}=-\frac{{\mathbb{R}}_6}{\delta },\hfill \\ \hfill {\mathsf{\Omega }}_{10,10}& =-\frac{{\mathbb{R}}_7}{\delta },{\mathsf{\Omega }}_{10,13}={\mathbb{E}}^T{\mathbb{R}}_1,{\mathsf{\Omega }}_{11,11}=-\frac{{\mathbb{R}}_8}{\phi },\hfill \\ \hfill {\mathsf{\Omega }}_{12,12}& =-\frac{{\mathbb{R}}_5}{\phi },{\mathsf{\Omega }}_{13,13}=2\alpha {\mathbb{R}}_1-2{\mathbb{R}}_1\mathbb{K}.\hfill \end{array}$$

Proof. 

Consider the following Lyapunov function

$$\begin{array}{c}\hfill \mathbb{V}={\mathbb{V}}_1+{\mathbb{V}}_2+{\mathbb{V}}_3+{\mathbb{V}}_4+{\mathbb{V}}_5+{\mathbb{V}}_6+{\mathbb{V}}_7+{\mathbb{V}}_8+{\mathbb{V}}_9+{\mathbb{V}}_{10},\end{array}$$

(13)

where

$$\begin{array}{cc}\hfill {\mathbb{V}}_1(t,\tilde{\pi }\left(t\right))& ={\eta }^{2\alpha t}{s}^T\left(t\right){\mathbb{R}}_{1}s\left(t\right),\hfill \\ \hfill {\mathbb{V}}_2\left(t,\tilde{\pi }\left(t\right)\right)& ={\int }_{t-\phi \left(t\right)}^t{\eta }^{2\alpha s}{f}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{2}f\left(\tilde{\pi }\left(s\right)\right)ds,\hfill \\ \hfill {\mathbb{V}}_3\left(t,\tilde{\pi }\left(t\right)\right)& ={\int }_{t-\delta \left(t\right)}^t{\eta }^{2\alpha s}{h}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{3}h\left(\tilde{\pi }\left(s\right)\right)ds,\hfill \\ \hfill {\mathbb{V}}_4(t,\tilde{\pi }\left(t\right))& ={\int }_{t-\mu \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_4\tilde{\pi }\left(s\right)ds,\hfill \\ \hfill {\mathbb{V}}_5(t,\tilde{\pi }\left(t\right))& ={\int }_{-\phi \left(t\right)}^0{\int }_{t+\theta }^t{\eta }^{2\alpha s}{f}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{5}f\left(\tilde{\pi }\left(s\right)\right)dsd\theta ,\hfill \\ \hfill {\mathbb{V}}_6(t,\tilde{\pi }\left(t\right))& ={\int }_{-\delta \left(t\right)}^0{\int }_{t+\theta }^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_6\tilde{\pi }\left(s\right)dsd\theta ,\hfill \\ \hfill {\mathbb{V}}_7(t,\tilde{\pi }\left(t\right))& ={\int }_{-\delta \left(t\right)}^0{\int }_{t+\theta }^t{\eta }^{2\alpha s}{h}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{7}h\left(\tilde{\pi }\left(s\right)\right)dsd\theta ,\hfill \\ \hfill {\mathbb{V}}_8(t,\tilde{\pi }\left(t\right))& ={\int }_{-\phi \left(t\right)}^0{\int }_{t+\theta }^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_8\tilde{\pi }\left(s\right)dsd\theta ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathbb{V}}_9(t,\tilde{\pi }\left(t\right))& ={\int }_{t-\phi \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_9\tilde{\pi }\left(s\right)ds,\hfill \\ \hfill {\mathbb{V}}_{10}(t,\tilde{\pi }\left(t\right))& ={\int }_{t-\delta \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_{10}\tilde{\pi }\left(s\right)ds.\hfill \end{array}$$

Now, we can calculate the time derivative of $\mathbb{V}$ along the trajectories of system (11), then we have

$$\begin{array}{cc}\hfill {\dot{\mathbb{V}}}_1(t,\tilde{\pi }\left(t\right))& =2\alpha {\eta }^{2\alpha t}{s}^T\left(t\right){\mathbb{R}}_{1}s\left(t\right)+2{\eta }^{2\alpha t}{s}^T\left(t\right){\mathbb{R}}_1{\mathbb{D}}^{\alpha }\tilde{\pi }\left(t\right),\hfill \\ \hfill {\dot{\mathbb{V}}}_2(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{f}^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{2}f\left(\tilde{\pi }\left(t\right)\right)\hfill \\ \hfill & -{\eta }^{2\alpha (t-\phi )}{f}^T\left(\tilde{\pi }(t-\phi \left(t\right))\right){\mathbb{R}}_{2}f\left(\tilde{\pi }(t-\phi \left(t\right))\right)(1-{\phi }_1),\hfill \\ \hfill {\dot{\mathbb{V}}}_3(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{h}^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{3}h\left(\tilde{\pi }\left(t\right)\right)\hfill \\ \hfill & -{\eta }^{2\alpha (t-\delta )}{h}^T\left(\tilde{\pi }(t-\delta \left(t\right))\right){\mathbb{R}}_{3}h\left(\tilde{\pi }(t-\delta \left(t\right))\right)(1-{\delta }_1),\hfill \\ \hfill {\dot{\mathbb{V}}}_4(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_4\tilde{\pi }\left(t\right)\hfill \\ \hfill & -{\eta }^{2\alpha (t-\mu )}{\tilde{\pi }}^T(t-\mu \left(t\right)){\mathbb{R}}_4\tilde{\pi }(t-\mu \left(t\right))(1-{\mu }_1),\hfill \\ \hfill {\dot{\mathbb{V}}}_5(t,\tilde{\pi }\left(t\right))& =\phi {\eta }^{2\alpha t}{f}^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{5}f\left(\tilde{\pi }\left(t\right)\right)\hfill \\ \hfill & -{\int }_{t-\phi \left(t\right)}^t{\eta }^{2\alpha s}{f}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{5}f\left(\tilde{\pi }\left(s\right)\right)ds,\hfill \\ \hfill {\dot{\mathbb{V}}}_6(t,\tilde{\pi }\left(t\right))& =\delta {\eta }^{2\alpha t}{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_6\tilde{\pi }\left(t\right)\hfill \\ \hfill & -{\int }_{t-\delta \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_6\tilde{\pi }\left(s\right)ds,\hfill \\ \hfill {\dot{\mathbb{V}}}_7(t,\tilde{\pi }\left(t\right))& =\delta {\eta }^{2\alpha t}{h}^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{7}h\left(\tilde{\pi }\left(t\right)\right)\hfill \\ \hfill & -{\int }_{t-\delta \left(t\right)}^t{\eta }^{2\alpha s}{h}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{7}h\left(\tilde{\pi }\left(s\right)\right)ds,\hfill \\ \hfill {\dot{\mathbb{V}}}_8(t,\tilde{\pi }\left(t\right))& =\phi {\eta }^{2\alpha t}{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_8\tilde{\pi }\left(t\right)\hfill \\ & -{\int }_{t-\phi \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_8\tilde{\pi }\left(s\right)ds,\hfill \\ \hfill {\dot{\mathbb{V}}}_9(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_9\tilde{\pi }\left(t\right)\hfill \\ & -{\eta }^{2\alpha (t-\phi (t\left)\right)}{\tilde{\pi }}^T(t-\phi \left(t\right)){\mathbb{R}}_9\tilde{\pi }(t-\phi \left(t\right))(1-{\phi }_1),\hfill \\ \hfill {\dot{\mathbb{V}}}_{10}(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_{10}\tilde{\pi }\left(t\right)\hfill \\ & -{\eta }^{2\alpha (t-\delta (t\left)\right)}{\tilde{\pi }}^T(t-\delta \left(t\right)){\mathbb{R}}_{10}\tilde{\pi }(t-\delta \left(t\right))(1-{\delta }_1).\hfill \end{array}$$

We have the following inequalities according Lemma 1, we obtain

$$\begin{array}{ccc}\hfill & -{\int }_{t-\phi \left(t\right)}^t{\eta }^{2\alpha s}{f}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{5}f\left(\tilde{\pi }\left(s\right)\right)ds\hfill & \hfill \\ & \le -\frac{1}{\phi }{\eta }^{2\alpha t}{\left[{\int }_{t-\phi \left(t\right)}^{t}f\left(\tilde{\pi }\left(s\right)\right)ds\right]}^T{\mathbb{R}}_5\left[{\int }_{t-\phi \left(t\right)}^{t}f\left(\tilde{\pi }\left(s\right)\right)ds\right],\hfill \\ \hfill & -{\int }_{t-\delta \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_6\tilde{\pi }\left(s\right)ds\hfill \\ & \le \frac{1}{\delta }{\eta }^{2\alpha t}{\left[{\int }_{t-\delta \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right]}^T{\mathbb{R}}_6\left[{\int }_{t-\delta \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\int }_{t-\delta \left(t\right)}^t{\eta }^{2\alpha s}{h}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{7}h\left(\tilde{\pi }\left(s\right)\right)ds\hfill \\ & \le \frac{1}{\delta }{\eta }^{2\alpha t}{\left[{\int }_{t-\delta \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds\right]}^T{\mathbb{R}}_7\left[{\int }_{t-\delta \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds\right],\hfill \\ \hfill & -{\int }_{t-\phi \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_8\tilde{\pi }\left(s\right)ds\hfill \\ & \le \frac{1}{\phi }{\eta }^{2\alpha t}{\left[{\int }_{t-\phi \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right]}^T{\mathbb{R}}_8\left[{\int }_{t-\phi \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right].\hfill \end{array}$$

Since ${\mathbb{T}}_1,{\mathbb{T}}_2,{\mathbb{T}}_3$ and ${\mathbb{T}}_4$, diagonal matrices, we can get from Assumption 2 that

$$\begin{array}{cc}\hfill & {\tilde{\pi }}^T\left(t\right){\mathbb{H}}_1{\mathbb{T}}_1{\mathbb{H}}_2\tilde{\pi }\left(t\right)-{\tilde{\pi }}^T\left(t\right){\mathbb{T}}_1({\mathbb{H}}_1+{\mathbb{H}}_2)f(\tilde{\pi }\left(t\right))+f^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{T}}_{1}f\left(\tilde{\pi }\left(t\right)\right)\le 0,\hfill \\ \hfill & {\tilde{\pi }}^T(t-\phi \left(t\right)){\mathbb{H}}_1{\mathbb{T}}_2{\mathbb{H}}_2\tilde{\pi }(t-\phi \left(t\right))-{\tilde{\pi }}^T(t-\phi \left(t\right)){\mathbb{T}}_2({\mathbb{H}}_1+{\mathbb{H}}_2)\hfill \\ & f(\tilde{\pi }(t-\phi \left(t\right)))+f^T\left(\tilde{\pi }(t-\phi \left(t\right))\right){\mathbb{T}}_{2}f\left(\tilde{\pi }(t-\phi \left(t\right))\right)\le 0,\hfill \\ \hfill & {\tilde{\pi }}^T\left(t\right){\mathbb{L}}_1{\mathbb{T}}_3{\mathbb{L}}_2\tilde{\pi }\left(t\right)-{\tilde{\pi }}^T\left(t\right){\mathbb{T}}_3({\mathbb{L}}_1+{\mathbb{L}}_2)h(\tilde{\pi }\left(t\right))+h^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{T}}_{3}h\left(\tilde{\pi }\left(t\right)\right)\le 0,\hfill \\ \hfill & {\tilde{\pi }}^T(t-\delta \left(t\right)){\mathbb{L}}_1{\mathbb{T}}_4{\mathbb{L}}_2\tilde{\pi }(t-\delta \left(t\right))-{\tilde{\pi }}^T(t-\delta \left(t\right)){\mathbb{T}}_4({\mathbb{L}}_1+{\mathbb{L}}_2)\hfill \\ & h(\tilde{\pi }(t-\delta \left(t\right)))+h^T\left(\tilde{\pi }(t-\delta \left(t\right))\right){\mathbb{T}}_{4}h\left(\tilde{\pi }(t-\delta \left(t\right))\right)\le 0.\hfill \end{array}$$

We obtain the following estimate the derivation of (13)

$$\begin{array}{cc}\hfill \dot{\mathbb{V}}(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}[s^T\left(t\right)2\alpha {\mathbb{R}}_{1}s\left(t\right)+2s^T\left(t\right){\mathbb{R}}_1{D}^{\alpha }\tilde{\pi }\left(t\right)\hfill \\ & +f^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{2}f\left(\tilde{\pi }\left(t\right)\right)\hfill \\ & -{\eta }^{-2\alpha \phi }{f}^T\left(\tilde{\pi }(t-\phi \left(t\right))\right){\mathbb{R}}_{2}f\left(\tilde{\pi }(t-\phi \left(t\right))\right)(1-{\phi }_1)\hfill \\ & +h^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{3}h\left(\tilde{\pi }\left(t\right)\right)\hfill \\ & -{\eta }^{-2\alpha \delta }{h}^T\left(\tilde{\pi }(t-\delta \left(t\right))\right){\mathbb{R}}_{3}h\left(\tilde{\pi }(t-\delta \left(t\right))\right)(1-{\delta }_1)\hfill \\ & +{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_4\tilde{\pi }\left(t\right)\hfill \\ & -{\eta }^{-2\alpha \mu }{\tilde{\pi }}^T(t-\mu \left(t\right)){\mathbb{R}}_4\tilde{\pi }(t-\mu \left(t\right))(1-{\mu }_1)\hfill \\ & +\phi f^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{5}f\left(\tilde{\pi }\left(t\right)\right)\hfill \\ & -{\left({\int }_{t-\phi \left(t\right)}^{t}f\left(\tilde{\pi }\left(s\right)\right)ds\right)}^T\frac{{\mathbb{R}}_5}{\phi }({\int }_{t-\phi \left(t\right)}^{t}f\left(\tilde{\pi }\left(s\right)\right)ds)ds\hfill \\ & +\delta {\tilde{\pi }}^T\left(t\right){\mathbb{R}}_6\overline{\pi }\left(t\right)\hfill \\ & -{\left({\int }_{t-\delta \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right)}^T\frac{{\mathbb{R}}_6}{\delta }\left({\int }_{t-\delta \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right)\hfill \\ & +\delta h^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{7}h\left(\tilde{\pi }\left(t\right)\right)\hfill \\ & -{\left({\int }_{t-\delta \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds\right)}^T\frac{{\mathbb{R}}_7}{\delta }\left({\int }_{t-\delta \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds\right)\hfill \\ & +\phi {\tilde{\pi }}^T\left(t\right){\mathbb{R}}_8\tilde{\pi }\left(t\right)\hfill \\ & -{\left({\int }_{t-\phi }^t\tilde{\pi }\left(s\right)ds\right)}^T\frac{{\mathbb{R}}_8}{\phi }\left({\int }_{t-\phi }^t\tilde{\pi }\left(s\right)ds\right)\hfill \\ & +{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_9\widetilde{\pi \left(t\right)}\hfill \\ & -{\eta }^{-2\alpha \phi }{\tilde{\pi }}^T(t-\phi \left(t\right)){\mathbb{R}}_9\zeta (t-\phi \left(t\right))(1-{\phi }_1)\hfill \\ & +{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_{10}\tilde{\pi }\left(t\right)\hfill \\ & -{\eta }^{-2\alpha \delta }{\tilde{\pi }}^T(t-\delta \left(t\right)){\mathbb{R}}_{10}\tilde{\pi }(t-\delta \left(t\right))(1-{\delta }_1)],\hfill \end{array}$$

$$\begin{array}{cc}\hfill \dot{\mathbb{V}}(t,\tilde{\pi }\left(t\right))& ={\eta }^{2\alpha t}{\beta }^T\left(t\right)\mathsf{\Omega }\beta \left(t\right)<0,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \beta \left(t\right)=& [{\tilde{\pi }}^T\left(t\right)f^T\left(\tilde{\pi }\left(t\right)\right)h^T\left(\tilde{\pi }\left(t\right)\right){\tilde{\pi }}^T(t-\phi \left(t\right)){\tilde{\pi }}^T(t-\mu \left(t\right)){\tilde{\pi }}^T(t-\delta \left(t\right))\hfill \\ & f^T(\tilde{\pi }(t-\phi \left(t\right))h^T\left(\tilde{\pi }(t-\delta \left(t\right))\right)({\int }_{t-\delta \left(t\right)}^t\tilde{\pi }\left(s\right)ds{)}^T\hfill \\ & {\left({\int }_{t-\delta \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds\right)}^T{\left({\int }_{t-\phi \left(t\right)}^t\tilde{\pi }\left(s\right)ds{)}^T({\int }_{t-\phi \left(t\right)}^{t}f\left(\tilde{\pi }\left(s\right)\right)ds\right)}^T{s}^T\left(t\right){]}^T.\hfill \end{array}$$

From above condition, we have

$$\begin{array}{c}\hfill \dot{\mathbb{V}}(t,\tilde{\pi }\left(t\right))\le 0.\end{array}$$

(14)

Based on the Lyapunov method, then system (11) is asymptotically stable. Furthermore, we prove the exponential stability of the neural networks. Thus, we know that $V\left(t\right)$ is monotone non increasing in t for $t\in [t_0,\infty )$, i.e., $\mathbb{V}(t,\tilde{\pi }\left(t\right))\le \mathbb{V}(t_0,\tilde{\pi }\left(t_0\right))$.

On the other hand,

$$\begin{array}{cc}\hfill \mathbb{V}(t_0,\tilde{\pi }\left(t_0\right))& ={\eta }^{2\alpha t_0}{s}^T\left(t_0\right){\mathbb{R}}_{1}s\left(t_0\right)\hfill \\ \hfill & +{\int }_{t_0-\phi \left(t_0\right)}^{t_0}{\eta }^{2\alpha s}{f}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{2}f\left(\tilde{\pi }\left(s\right)\right)ds\hfill \\ \hfill & +{\int }_{t_0-\delta \left(t_0\right)}^{t_0}{\eta }^{2\alpha s}{h}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{3}h\left(\tilde{\pi }\left(s\right)\right)ds\hfill \\ & +{\int }_{t_0-\mu \left(t_0\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_4\tilde{\pi }\left(s\right)ds\hfill \\ & +{\int }_{\phi \left(t_0\right)}^0{\int }_{t_0+\theta }^{t_0}{\eta }^{2\alpha s}{f}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{5}f\left(\tilde{\pi }\left(s\right)\right)dsd\theta \hfill \\ & +{\int }_{\delta \left(t_0\right)}^0{\int }_{t_0+\theta }^{t_0}{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_6\tilde{\pi }\left(s\right)dsd\theta \hfill \\ & +{\int }_{\delta \left(t_0\right)}^0{\int }_{t_0+\theta }^{t_0}{\eta }^{2\alpha s}{h}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{7}h\left(\tilde{\pi }\left(s\right)\right)dsd\theta \hfill \\ & +{\int }_{\phi \left(t_0\right)}^0{\int }_{t_0+\theta }^{t_0}{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_8\tilde{\pi }\left(s\right)dsd\theta \hfill \\ & +{\int }_{t_0-\phi \left(t_0\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_9\tilde{\pi }\left(s\right)ds\hfill \\ & +{\int }_{t_0-\delta \left(t_0\right)}^{t_0}{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_{10}\tilde{\pi }\left(s\right)ds,\hfill \\ \hfill & \le {\eta }^{2\alpha t_0}\{{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_1\right)+[{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_2\right){\beta }_1+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_3\right){\beta }_2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_4\right){\beta }_3\hfill \\ & +{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_9\right){\beta }_1+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_{10}\right){\beta }_2]l_{\mathbb{R}}^2+[({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_5\right)+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_8\right)){\beta }_4\hfill \\ & +({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_6\right)+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_7\right)){\beta }_5{\left]\right\}||\varphi ||}^2,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\beta }_1& =\left[\frac{1-{\eta }^{-2\alpha \phi }}{2\alpha }\right],{\beta }_2=\left[\frac{1-{\eta }^{-2\alpha \delta }}{2\alpha }\right],{\beta }_3=\left[\frac{1-{\eta }^{-2\alpha \mu }}{2\alpha }\right],\hfill \\ \hfill {\beta }_4& =\frac{1}{2\alpha }\left[\phi -\left[\frac{1-{\eta }^{-2\alpha \phi }}{2\alpha }\right]\right],{\beta }_5=\frac{1}{2\alpha }\left[\delta -\left[\frac{1-{\eta }^{-2\alpha \delta }}{2\alpha }\right]\right].\hfill \end{array}$$

For any variable t, form the chosen Lyapunov functional, we can get the following inequality:

$$\begin{array}{c}\hfill V(t,\tilde{\pi }\left(t\right))\ge {\lambda }_m\left({\mathbb{R}}_1\right){\eta }^{2\alpha t}||s\left(t\right){||}^2\end{array}$$

$$\begin{array}{c}\hfill V(t,\tilde{\pi }\left(t\right))\le V(t_0,\tilde{\pi }\left(t_0\right)).\end{array}$$

That’s to say,

$$\begin{array}{cc}\hfill ||s\left(t\right)||& \le \sqrt{\frac{M_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha t},\hfill \\ \hfill ||D^{\alpha -1}\tilde{\pi }\left(t\right)||& \le \sqrt{\frac{M_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha t}.\hfill \end{array}$$

(15)

For the ith component of vector $\eta \left(t\right)$, we have

$$\begin{array}{c}\hfill -\sqrt{\frac{M_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha t}\le s^{\gamma -1}{\tilde{\pi }}_i\left(s\right)\le \sqrt{\frac{M_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha t}.\end{array}$$

Taking Laplace transform and Laplace inverse transformation, we have the following inequality

$$\begin{array}{c}\hfill |\tilde{\pi }\left(t\right)|\le \sqrt{\frac{M_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||\frac{1}{t^{1-\gamma }}{E}_{1,\gamma }(-\alpha t).\end{array}$$

Thereupon,

$$\begin{array}{c}\hfill ||\tilde{\pi }\left(t\right)||\le \sqrt{\frac{M_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||\frac{1}{t^{1-\gamma }}{E}_{1,\gamma }(-\alpha t).\end{array}$$

For $\gamma =1$, we have

$$\begin{array}{c}\hfill ||\tilde{\pi }\left(t\right)||\le \sqrt{\frac{M_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{(-\alpha t)},\end{array}$$

where,

$$\begin{array}{cc}\hfill M_1& ={\eta }^{2\alpha t_0}\{({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_1\right)+({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_2\right){\beta }_1+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_3\right){\beta }_2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_4\right){\beta }_3\hfill \\ & +{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_9\right){\beta }_1+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_{10}\right){\beta }_2)l_{\mathbb{R}}^2\hfill \\ & +\left[({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_5\right)+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_8\right)){\beta }_4+({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_6\right)+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_7\right)){\beta }_5\right]\},\hfill \end{array}$$

$$\begin{array}{c}\hfill ||\tilde{\pi }\left(t\right)||\le \sqrt{\frac{M_2}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha (t-t_0)},\end{array}$$

(16)

and

$$\begin{array}{cc}\hfill M_2& =({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_1\right)+({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_2\right){\beta }_1+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_3\right){\beta }_2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_4\right){\beta }_3\hfill \\ & +{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_9\right){\beta }_1+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_{10}\right){\beta }_2)l_{\mathbb{R}}^2\hfill \\ & +\left[({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_5\right)+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_8\right)){\beta }_4+({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_6\right)+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_7\right)){\beta }_5\right].\hfill \end{array}$$

From the inequalities (15) and (16), the sliding mode surface $s\left(t\right)$ and the system state trajectory $\tilde{\pi }\left(t\right)$ converge exponentially to zero. Thus, system (11) is globally exponentially stable. □

Remark 2.

In Theorem 2, Hermitian matrices ${\mathbb{R}}_1$, ${\mathbb{R}}_2$, ${\mathbb{R}}_3$, ${\mathbb{R}}_4$, ${\mathbb{R}}_5$, ${\mathbb{R}}_6$, ${\mathbb{R}}_7$, ${\mathbb{R}}_8$, ${\mathbb{R}}_9$, ${\mathbb{R}}_{10}$, positive diagonal matrices ${\mathbb{T}}_1$, ${\mathbb{T}}_2$, ${\mathbb{T}}_3$, ${\mathbb{T}}_4$ and positive scalars $0\le \delta <1$, satisfying the following.

Theorem 3.

The system (11) is globally exponentially stable if there exist positive definite Hermitian matrices ${\mathbb{R}}_1$, ${\mathbb{R}}_2$, ${\mathbb{R}}_3$, ${\mathbb{R}}_4$, ${\mathbb{R}}_5$, ${\mathbb{R}}_6$, ${\mathbb{R}}_7$, ${\mathbb{R}}_8$, ${\mathbb{R}}_9$, ${\mathbb{R}}_{10}$, positive diagonal matrices ${\mathbb{T}}_1$, ${\mathbb{T}}_2$, ${\mathbb{T}}_3$, ${\mathbb{T}}_4$ and positive scalars $0\le \delta <1$, satisfying the following LMI:

$$\begin{array}{c}\hfill \zeta ={\left({\zeta }_{i,j}\right)}_{13\times 13}<0,\end{array}$$

(17)

where

$$\begin{array}{cc}\hfill {\zeta }_{1,1}& ={\mathbb{R}}_4+\delta {\mathbb{R}}_6+\phi {\mathbb{R}}_8+{\mathbb{R}}_9+{\mathbb{R}}_{10}-{\mathbb{H}}_1{\mathbb{T}}_1{\mathit{H}}_2--{\mathbb{L}}_1{\mathbb{T}}_3{\mathit{L}}_2,\hfill \\ \hfill {\zeta }_{1,2}& ={\mathbb{T}}_1({\mathbb{H}}_1+{\mathbb{H}}_2),{\mathsf{\Omega }}_{1,3}={\mathbb{T}}_3({\mathbb{L}}_1+{\mathbb{L}}_2),\hfill \\ \hfill {\zeta }_{2,2}& ={\mathbb{R}}_2+\phi {\mathbb{R}}_5-{\mathbb{T}}_1,{\zeta }_{3,3}={\mathbb{R}}_3+\delta {\mathbb{R}}_7-{\mathbb{T}}_3,{\zeta }_{3,13}={\mathbb{B}}^T{\mathbb{R}}_1,\hfill \\ \hfill {\zeta }_{4,4}& =-{\mathbb{M}}_9(1-{\phi }_1)-{\mathbb{H}}_1{\mathbb{T}}_2{\mathbb{H}}_2,{\zeta }_{4,7}={\mathbb{T}}_2({\mathbb{H}}_1+{\mathbb{H}}_2),\hfill \\ \hfill {\zeta }_{5,5}& =-{\mathbb{M}}_4(1-{\mu }_1),{\zeta }_{5,13}=-{\mathbb{L}}^T{\mathbb{R}}_1,\hfill \\ \hfill {\zeta }_{6,6}& =-{\mathbb{M}}_{10}(1-{\delta }_1)-{\mathbb{L}}_1{\mathbb{T}}_4{\mathbb{L}}_2,{\zeta }_{6,8}={\mathbb{T}}_4({\mathbb{L}}_1+{\mathbb{L}}_2),\hfill \\ \hfill {\zeta }_{7,7}& =-{\mathbb{M}}_2(1-{\phi }_1)-{\mathbb{T}}_2,{\zeta }_{7,13}={\mathbb{C}}^T{\mathbb{R}}_1,\hfill \\ \hfill {\zeta }_{8,8}& =-{\mathbb{M}}_3(1-{\delta }_1)-{\mathbb{T}}_4,{\zeta }_{9,9}=-\frac{{\mathbb{R}}_6}{\delta },\hfill \\ \hfill {\zeta }_{10,10}& =-\frac{{\mathbb{R}}_7}{\delta },{\zeta }_{10,13}={\mathbb{E}}^T{\mathbb{R}}_1,{\zeta }_{11,11}=-\frac{{\mathbb{R}}_8}{\phi },\hfill \\ \hfill {\zeta }_{12,12}& =-\frac{{\mathbb{R}}_5}{\phi },{\zeta }_{13,13}={\mathbb{M}}_1-2{\mathbb{R}}_1\mathbb{K}.\hfill \end{array}$$

Proof. 

First we consider the following terms, where $0<\delta ={\eta }^{-2\alpha \rho }<1$

$$\begin{array}{cc}\hfill 0<2\alpha {\mathbb{R}}_1& ={\mathbb{M}}_1,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill 0<{\mathbb{M}}_2<\delta {\mathbb{R}}_2& ={\eta }^{-2\alpha \rho }{\mathbb{R}}_2\le {\eta }^{-2\alpha \phi }{\mathbb{R}}_2,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill 0<{\mathbb{M}}_3<\delta {\mathbb{R}}_3& ={\eta }^{-2\alpha \rho }{\mathbb{R}}_3\le {\eta }^{-2\alpha \delta }{\mathbb{R}}_3,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill 0<{\mathbb{M}}_4<\delta {\mathbb{R}}_4& ={\eta }^{-2\alpha \rho }{\mathbb{R}}_4\le {\eta }^{-2\alpha \mu }{\mathbb{R}}_4,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill 0<{\mathbb{M}}_9<\delta {\mathbb{R}}_9& ={\eta }^{-2\alpha \rho }{\mathbb{R}}_9\le {\eta }^{-2\alpha \phi }{\mathbb{R}}_9,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill 0<{\mathbb{M}}_{10}<\delta {\mathbb{R}}_{10}& ={\eta }^{-2\alpha \rho }{\mathbb{R}}_{10}\le {\eta }^{-2\alpha \delta }{\mathbb{R}}_{10}.\hfill \end{array}$$

(23)

Using (14) and (18)–(23), we get

$$\begin{array}{cc}\hfill \dot{\mathbb{V}}(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}[s^T\left(t\right)({\mathbb{M}}_1-2{\mathbb{R}}_1\mathbb{K})s\left(t\right)\hfill \\ & +s^T\left(t\right)(-2{\mathbb{R}}_1\mathbb{L})\tilde{\pi }(t-\mu \left(t\right))\hfill \\ & +s^T\left(t\right)2{\mathbb{R}}_1\mathbb{B}h\left(\tilde{\pi }\left(t\right)\right)+s^T\left(t\right)2{\mathbb{R}}_{1}Cf(\tilde{\pi }(t-\phi \left(t\right))\hfill \\ & +s^T\left(t\right)2{\mathbb{R}}_1\mathbb{E}{\int }_{t-\phi \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds+f^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{2}f\left(\tilde{\pi }\left(t\right)\right)\hfill \\ & -f^T\left(\tilde{\pi }(t-\phi \left(t\right))\right){\mathbb{M}}_{2}f\left(\tilde{\pi }(t-\phi \left(t\right))\right)(1-{\phi }_1)\hfill \\ & +h^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{3}h\left(\tilde{\pi }\left(t\right)\right)\hfill \\ & -h^T\left(\tilde{\pi }(t-\delta \left(t\right))\right){\mathbb{M}}_{3}h\left(\tilde{\pi }(t-\delta \left(t\right))\right)(1-{\delta }_1)\hfill \\ & +{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_4\tilde{\pi }\left(t\right)-{\tilde{\pi }}^T(t-\mu \left(t\right)){\mathbb{M}}_4\tilde{\pi }(t-\mu \left(t\right))(1-{\mu }_1)\hfill \\ & +\phi f^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{5}f\left(\tilde{\pi }\left(t\right)\right)+\delta {\tilde{\pi }}^T\left(t\right){\mathbb{R}}_6\tilde{\pi }\left(t\right)\hfill \\ & -\frac{{\mathbb{R}}_5}{\phi }{\left[{\int }_{t-\phi \left(t\right)}^{t}f\left(\tilde{\pi }\left(s\right)\right)ds\right]}^T\left[{\int }_{t-\phi \left(t\right)}^t{f}^T\left(\tilde{\pi }\left(s\right)\right)ds\right]\hfill \\ & -\frac{{\mathbb{R}}_6}{\delta }{\left[{\int }_{t-\delta \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right]}^T\left[{\int }_{t-\delta \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right]\hfill \\ & +\delta h^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{7}h\left(\tilde{\pi }\left(t\right)\right)\hfill \\ & -\frac{{\mathbb{R}}_7}{\delta }{\left[{\int }_{t-\delta \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds\right]}^T\left[{\int }_{t-\delta \left(t\right)}^{t}h\left(\tilde{\pi }\left(s\right)\right)ds\right]\hfill \\ & +\phi \tilde{\pi }\left(t\right){\mathbb{R}}_8\tilde{\pi }\left(t\right)\hfill \\ & -\frac{{\mathbb{R}}_8}{\phi }{\left[{\int }_{t-\phi \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right]}^T\left[{\int }_{t-\phi \left(t\right)}^t\tilde{\pi }\left(s\right)ds\right]\hfill \\ & +{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_9\tilde{\pi }\left(t\right)\hfill \\ & -{\mathbb{M}}_9{\tilde{\pi }}^T(t-\phi \left(t\right))(1-{\phi }_1)\tilde{\pi }(t-\phi \left(t\right))\hfill \\ & +{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_{10}\tilde{\pi }\left(t\right)\hfill \\ & -{\mathbb{M}}_{10}{\tilde{\pi }}^T(t-\delta \left(t\right))(1-{\delta }_1)\tilde{\pi }(t-\delta \left(t\right))]\hfill \end{array}$$

$$\begin{array}{c}\hfill \dot{\mathbb{V}}(t,\tilde{\pi }\left(t\right))\le {\eta }^{2\alpha t}{\beta }^T\left(t\right)\zeta \beta \left(t\right).\end{array}$$

(24)

The remaining part of the proof is similar to that of Theorem 2.

This implies that the equilibrium point of the system (11) is globally exponentially stable. The proof is completed. □

Remark 3.

When $\mathbb{E}=0$, the fractional-order complex-valued neural networks (11) correspondingly reduces to

$$\begin{array}{cc}\hfill {\mathbb{D}}^{\alpha }\tilde{\pi }\left(t\right)& =-\mathbb{L}\tilde{\pi }(t-\mu )+\mathbb{B}h\left(\tilde{\pi }\left(t\right)\right)+\mathbb{C}f\left(\tilde{\pi }(t-\phi \left(t\right))\right)-\mathbb{K}s\left(t\right).\hfill \end{array}$$

(25)

Theorem 4.

The system (25) is globally exponentially stable if there exist Hermitian matrices ${\mathbb{R}}_1$, ${\mathbb{R}}_2$,${\mathbb{R}}_3$,${\mathbb{R}}_4$,${\mathbb{R}}_9$,${\mathbb{R}}_{10}$, positive diagonal matrices ${\mathbb{T}}_1$, ${\mathbb{T}}_2$, ${\mathbb{T}}_3$, ${\mathbb{T}}_4$ satisfying the following LMI:

$$\begin{array}{c}\hfill \varrho ={\left({\varrho }_{i,j}\right)}_{9\times 9}<0,\end{array}$$

(26)

where

$$\begin{array}{cc}\hfill {\varrho }_{1,1}& ={\mathbb{R}}_4+{\mathbb{R}}_9+{\mathbb{R}}_{10}-{\mathbb{H}}_1{\mathbb{T}}_1{\mathit{H}}_2-{\mathbb{L}}_1{\mathbb{T}}_3{\mathit{L}}_2,{\varrho }_{1,2}={\mathbb{T}}_3({\mathbb{L}}_1+{\mathbb{L}}_2),\hfill \\ \hfill {\varrho }_{1,3}& ={\mathbb{T}}_1({\mathbb{H}}_1+{\mathbb{H}}_2),{\varrho }_{1,6}={\mathbb{H}}_3\mathbb{U}{\mathbb{H}}_3\hfill \\ \hfill {\varrho }_{2,2}& ={\mathbb{R}}_2-{\mathbb{T}}_3,{\varrho }_{2,9}=B^T{R}_1,{\varrho }_{3,3}={\mathbb{R}}_2-{\mathbb{T}}_1,\hfill \\ \hfill {\varrho }_{4,4}& =-{\eta }^{-2\alpha \phi }{\mathbb{R}}_2(1-{\phi }_1)-{\mathbb{T}}_2,{\varrho }_{4,7}={({\mathbb{H}}_1+{\mathbb{H}}_2)}^T{\mathbb{T}}_2,\hfill \\ \hfill {\varrho }_{4,9}& =C^T{R}_1,{\varrho }_{5,5}=-{\eta }^{-2\alpha \delta }{\mathbb{R}}_3(1-{\delta }_1)-T_4,\hfill \\ \hfill {\varrho }_{5,8}& ={({\mathbb{L}}_1+{\mathbb{L}}_2)}^T{\mathbb{T}}_4{\varrho }_{6,6}=-{\eta }^{-2\alpha \mu }{\mathbb{R}}_4(1-{\mu }_1),\hfill \\ \hfill {\varrho }_{6,9}& =-L^T{R}_1,{\varrho }_{7,7}=-{\eta }^{-2\alpha \phi }{\mathbb{R}}_9(1-{\phi }_1)-{\mathbb{H}}_1{\mathbb{T}}_2{\mathbb{H}}_2,\hfill \\ \hfill {\varrho }_{8,8}& =-{\eta }^{-2\alpha \delta }{\mathbb{R}}_{10}(1-{\delta }_1)-{\mathbb{L}}_1{\mathbb{T}}_4{\mathbb{L}}_2,{\varrho }_{9,9}=2\alpha {\mathbb{R}}_1-2{\mathbb{R}}_1\mathbb{K}.\hfill \end{array}$$

Proof. 

Consider the following Lyapunov function

$$\begin{array}{c}\hfill \mathbb{V}={\mathbb{V}}_1+{\mathbb{V}}_2+{\mathbb{V}}_3+{\mathbb{V}}_4+{\mathbb{V}}_5+{\mathbb{V}}_6,\end{array}$$

(27)

where

$$\begin{array}{cc}\hfill {\mathbb{V}}_1(t,\tilde{\pi }\left(t\right))& ={\eta }^{2\alpha t}{s}^T\left(t\right){\mathbb{R}}_{1}s\left(t\right),\hfill \\ \hfill {\mathbb{V}}_2(t,\tilde{\pi }\left(t\right))& ={\int }_{t-\phi \left(t\right)}^t{\eta }^{2\alpha s}{f}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{2}f\left(\tilde{\pi }\left(s\right)\right)ds,\hfill \\ \hfill {\mathbb{V}}_3(t,\tilde{\pi }\left(t\right))& ={\int }_{t-\delta \left(t\right)}^t{\eta }^{2\alpha s}{h}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{3}h\left(\tilde{\pi }\left(s\right)\right)ds,\hfill \\ \hfill {\mathbb{V}}_4(t,\tilde{\pi }\left(t\right))& ={\int }_{t-\mu \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_4\tilde{\pi }\left(s\right)ds,\hfill \\ \hfill {\mathbb{V}}_5(t,\tilde{\pi }\left(t\right))& ={\int }_{t-\phi \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_9\tilde{\pi }\left(s\right)ds,\hfill \\ \hfill {\mathbb{V}}_6(t,\tilde{\pi }\left(t\right))& ={\int }_{t-\delta \left(t\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_{10}\tilde{\pi }\left(s\right)ds.\hfill \end{array}$$

Now, we can calculate the time derivative of $\mathbb{V}$ along the trajectories of system (25), then we have

$$\begin{array}{cc}\hfill {\dot{\mathbb{V}}}_1(t,\tilde{\pi }\left(t\right))& =2\alpha {\eta }^{2\alpha t}{s}^T\left(t\right){\mathbb{R}}_{1}s\left(t\right)+2{\eta }^{2\alpha t}{s}^T\left(t\right){\mathbb{R}}_1{\mathbb{D}}^{\alpha }\tilde{\pi }\left(t\right),\hfill \\ \hfill {\dot{\mathbb{V}}}_2(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{f}^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{2}f\left(\tilde{\pi }\left(t\right)\right)\hfill \\ & -{\eta }^{2\alpha (t-\phi )}{f}^T(\tilde{\pi }(t-\phi \left(t\right)){\mathbb{R}}_{2}f\left(\tilde{\pi }(t-\phi \left(t\right))\right)(1-{\phi }_1),\hfill \\ \hfill {\dot{\mathbb{V}}}_3(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{h}^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{R}}_{3}h\left(\tilde{\pi }\left(t\right)\right)\hfill \\ & -{\eta }^{2\alpha (t-\delta )}{h}^T(\tilde{\pi }(t-\delta \left(t\right)){\mathbb{R}}_{3}h\left(\tilde{\pi }(t-\delta \left(t\right))\right)(1-{\delta }_1),\hfill \\ \hfill {\dot{\mathbb{V}}}_4(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_4\tilde{\pi }\left(t\right)\hfill \\ & -{\eta }^{2\alpha (t-\mu )}{\tilde{\pi }}^T(t-\mu \left(t\right)){\mathbb{R}}_4\tilde{\pi }(t-\mu \left(t\right))(1-{\mu }_1),\hfill \\ \hfill {\dot{\mathbb{V}}}_5(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_9\tilde{\pi }\left(t\right)\hfill \\ & -{\eta }^{2\alpha (t-\phi )}{\tilde{\pi }}^T(t-\phi \left(t\right)){\mathbb{R}}_9\tilde{\pi }(t-\phi \left(t\right))(1-{\phi }_1),\hfill \\ \hfill {\dot{\mathbb{V}}}_6(t,\tilde{\pi }\left(t\right))& \le {\eta }^{2\alpha t}{\tilde{\pi }}^T\left(t\right){\mathbb{R}}_{10}\tilde{\pi }\left(t\right)\hfill \\ & -{\eta }^{2\alpha (t-\delta )}{\tilde{\pi }}^T(t-\delta \left(t\right)){\mathbb{R}}_{10}\tilde{\pi }(t-\delta \left(t\right))(1-{\delta }_1).\hfill \end{array}$$

Since ${\mathbb{T}}_1,{\mathbb{T}}_2,{\mathbb{T}}_3$ and ${\mathbb{T}}_4$, diagonal matrices, we can get from Assumption 2 that

$$\begin{array}{cc}\hfill & {\tilde{\pi }}^T\left(t\right){\mathbb{H}}_1{\mathbb{T}}_1{\mathbb{H}}_2\tilde{\pi }\left(t\right)-{\tilde{\pi }}^T\left(t\right){\mathbb{T}}_1({\mathbb{H}}_1+{\mathbb{H}}_2)f(\tilde{\pi }\left(t\right))+f^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{T}}_{1}f\left(\tilde{\pi }\left(t\right)\right)\le 0,\hfill \\ \hfill & {\tilde{\pi }}^T(t-\phi \left(t\right)){\mathbb{H}}_1{\mathbb{T}}_2{\mathbb{H}}_2\tilde{\pi }(t-\phi \left(t\right))-{\tilde{\pi }}^T(t-\phi \left(t\right)){\mathbb{T}}_2({\mathbb{H}}_1+{\mathbb{H}}_2)\hfill \\ & f(\tilde{\pi }(t-\phi \left(t\right)))+f^T\left(\tilde{\pi }(t-\phi \left(t\right))\right){\mathbb{T}}_{2}f\left(\tilde{\pi }(t-\phi \left(t\right))\right)\le 0,\hfill \\ \hfill & {\tilde{\pi }}^T\left(t\right){\mathbb{L}}_1{\mathbb{T}}_3{\mathbb{L}}_2\tilde{\pi }\left(t\right)-{\tilde{\pi }}^T\left(t\right){\mathbb{T}}_3({\mathbb{L}}_1+{\mathbb{L}}_2)h(\tilde{\pi }\left(t\right))+h^T\left(\tilde{\pi }\left(t\right)\right){\mathbb{T}}_{3}h\left(\tilde{\pi }\left(t\right)\right)\le 0,\hfill \\ \hfill & {\tilde{\pi }}^T(t-\delta \left(t\right)){\mathbb{L}}_1{\mathbb{T}}_4{\mathbb{L}}_2\tilde{\pi }(t-\delta \left(t\right))-{\tilde{\pi }}^T(t-\delta \left(t\right)){\mathbb{T}}_4({\mathbb{L}}_1+{\mathbb{L}}_2)\hfill \\ & h(\tilde{\pi }(t-\delta \left(t\right)))+h^T\left(\tilde{\pi }(t-\delta \left(t\right))\right){\mathbb{T}}_{4}h\left(\tilde{\pi }(t-\delta \left(t\right))\right)\le 0.\hfill \end{array}$$

Adding above, we obtain the following estimate the derivation of (27)

$$\begin{array}{c}\hfill \dot{\mathbb{V}}(t,\tilde{\pi }\left(t\right))\le {\eta }^{2\alpha t}{\delta }^T\left(t\right)\varrho \delta \left(t\right)\end{array}$$

(28)

$$\begin{array}{cc}\hfill \delta \left(t\right)=& [{\tilde{\pi }}^T\left(t\right),h^T\left(\tilde{\pi }\left(t\right)\right),f^T\left(\tilde{\pi }\left(t\right)\right),f^T\left(\tilde{\pi }(t-\phi \left(t\right))\right),\hfill \\ & h^T\left(\tilde{\pi }(t-\delta \left(t\right))\right),{\tilde{\pi }}^T(t-\mu \left(t\right)),{\tilde{\pi }}^T(t-\phi \left(t\right),{\tilde{\pi }}^T(t-\delta \left(t\right)),s^T{\left(t\right)]}^T.\hfill \end{array}$$

From above condition, we have

$$\begin{array}{c}\hfill \dot{\mathbb{V}}(t,\tilde{\pi }\left(t\right))\le 0.\end{array}$$

(29)

Based on the Lyapunov method, system (25) is asymptotically stable. Furthermore, we prove the exponential stability of the neural networks. Thus, we know that $V\left(t\right)$ is monotone non-increasing in t for $t\in [t_0,\infty )$, i.e., $V\left(t\right)\le V\left(t_0\right)$.

$$\begin{array}{c}\hfill \mathbb{V}(t,\tilde{\pi }\left(t\right))\le \mathbb{V}(t_0,\tilde{\pi }\left(t_0\right)).\end{array}$$

From condition (29), we have $\dot{V}\le 0$, and then for any solution $\mathbb{V}\left(t,\tilde{\pi }\left(t\right)\right)$. On the other hand,

$$\begin{array}{cc}\hfill \mathbb{V}(t_0,\tilde{\pi }\left(t_0\right))& ={\eta }^{2\alpha t_0}{s}^T\left(t_0\right){\mathbb{R}}_{1}s\left(t_0\right)\hfill \\ & +{\int }_{t_0-\phi \left(t_0\right)}^{t_0}{\eta }^{2\alpha s}{f}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{2}f\left(\tilde{\pi }\left(s\right)\right)ds\hfill \\ & +{\int }_{t_0-\delta \left(t_0\right)}^{t_0}{\eta }^{2\alpha s}{h}^T\left(\tilde{\pi }\left(s\right)\right){\mathbb{R}}_{3}h\left(\tilde{\pi }\left(s\right)\right)ds\hfill \\ & +{\int }_{t_0-\mu \left(t_0\right)}^t{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_4\tilde{\pi }\left(s\right)ds\hfill \\ & +{\int }_{t_0-\phi \left(t_0\right)}^{t_0}{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_9\tilde{\pi }\left(s\right)ds\hfill \\ & +{\int }_{t_0-\delta \left(t_0\right)}^{t_0}{\eta }^{2\alpha s}{\tilde{\pi }}^T\left(s\right){\mathbb{R}}_{10}\tilde{\pi }\left(s\right)ds,\hfill \\ \hfill & \le {\eta }^{2\alpha t_0}\{{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_1\right)+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_2\right){\beta }_1{l}_{\mathbb{R}}^2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_3\right){\beta }_2{l}_{\mathbb{R}}^2\hfill \\ & +{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_4\right){\beta }_3{l}_{\mathbb{R}}^2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_9\right){\beta }_1{l}_{\mathbb{R}}^2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_{10}\right){\beta }_2{l}_{\mathbb{R}}^2{\}||\varphi ||}^2,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\beta }_1=\left[\frac{1-{\eta }^{-2\alpha \phi }}{2\alpha }\right],{\beta }_2=\left[\frac{1-{\eta }^{-2\alpha \delta }}{2\alpha }\right],{\beta }_3=\left[\frac{1-{\eta }^{-2\alpha \mu }}{2\alpha }\right].\end{array}$$

$$\begin{array}{c}\hfill ||s\left(t\right)||\le \sqrt{\frac{{\sigma }_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha t},\end{array}$$

(30)

replace s(t) by $D^{\alpha -1}$, we get

$$\begin{array}{c}\hfill ||D^{\alpha -1}\tilde{\pi }\left(t\right)||\le \sqrt{\frac{{\sigma }_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha t},\end{array}$$

for the ith component of vector $\eta \left(t\right)$, we have

$$\begin{array}{c}\hfill -\sqrt{\frac{{\sigma }_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha t}\le s^{\gamma -1}{\tilde{\pi }}_i\left(s\right)\le \sqrt{\frac{{\sigma }_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha t}.\end{array}$$

Taking Laplace transform and Laplace inverse transformation, we have the following inequality:

$$\begin{array}{c}\hfill |\tilde{\pi }\left(t\right)|\le \sqrt{\frac{{\sigma }_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||\frac{1}{t^{1-\gamma }}{E}_{1,\gamma }(-\alpha t).\end{array}$$

Thereupon,

$$\begin{array}{c}\hfill ||\tilde{\pi }\left(t\right)||\le \sqrt{\frac{{\sigma }_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||\frac{1}{t^{1-\gamma }}{E}_{1,\gamma }(-\alpha t).\end{array}$$

For $\gamma =1$, we have

$$\begin{array}{c}\hfill ||\tilde{\pi }\left(t\right)||\le \sqrt{\frac{{\sigma }_1}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{(-\alpha t)},\end{array}$$

where,

$$\begin{array}{cc}\hfill {\sigma }_1& ={\eta }^{2\alpha t_0}({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_1\right)+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_2\right){\beta }_1{l}_{\mathbb{R}}^2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_3\right){\beta }_2{l}_{\mathbb{R}}^2\hfill \\ & +{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_4\right){\beta }_3{l}_{\mathbb{R}}^2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_9\right){\beta }_1{l}_{\mathbb{R}}^2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_{10}\right){\beta }_2{l}_{\mathbb{R}}^2)\hfill \end{array}$$

$$\begin{array}{c}\hfill ||\tilde{\pi }\left(t\right)||\le \sqrt{\frac{{\sigma }_2}{{\lambda }_m\left({\mathbb{R}}_1\right)}}||\varphi ||{\eta }^{-\alpha (t-t_0)}\end{array}$$

(31)

and

$$\begin{array}{cc}\hfill {\sigma }_2& =({\lambda }_{\mathbb{M}}\left({\mathbb{R}}_1\right)+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_2\right){\beta }_1{l}_{\mathbb{R}}^2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_3\right){\beta }_2{l}_{\mathbb{R}}^2\hfill \\ & +{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_4\right){\beta }_3{l}_{\mathbb{R}}^2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_9\right){\beta }_1{l}_{\mathbb{R}}^2+{\lambda }_{\mathbb{M}}\left({\mathbb{R}}_{10}\right){\beta }_2{l}_{\mathbb{R}}^2).\hfill \end{array}$$

From the inequalities (30) and (31), the sliding mode surface $s\left(t\right)$ and the system state trajectory $\tilde{\pi }\left(t\right)$ converge exponentially to zero. Thus, system (25) is globally exponentially stable. □

4. Numerical Examples

Example 1.

Consider the fractional order complex-valued neural networks (11), with parameters

$$\begin{array}{cc}\hfill \mathbb{L}& =\left[\begin{array}{cc}0.94& 0\\ 0& 0.94\end{array}\right],\mathbb{B}=\left[\begin{array}{cc}0.9+0.9i& -0.9+0.8i\\ 0.8+0.4i& 0.2+0.7i\end{array}\right],\hfill \\ \hfill \mathbb{C}& =\left[\begin{array}{cc}-0.7+0.4i& -0.5-0.6i\\ -0.6-0.5i& 0.5+0.6i\end{array}\right],\mathbb{E}=\left[\begin{array}{cc}0.2+0.6i& -0.5-0.7i\\ 0.8+0.8i& -0.6-0.7i\end{array}\right].\hfill \end{array}$$

Choose scalars, ${\delta }_1=0.88,{\phi }_1=0.28,\alpha =0.27,\delta =0.36,{\mu }_1=0.65,\phi =0.87.$ Utilizing Matlab, the LMI (12) of Theorem 2 has the following feasible solutions:

$$\begin{array}{cc}\hfill {\mathbb{R}}_1& =\left[\begin{array}{cc}3.6748+0.0000i& -11.4759-4.4816i\\ -11.4759+4.4816i& 51.7224+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_2& =\left[\begin{array}{cc}24.2135+0.0000i& -10.4759-4.4816i\\ -10.4759+4.4816i& 42.7224+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_3& =\left[\begin{array}{cc}8.0048+0.0000i& -5.4759-1.6143i\\ -5.4759+1.6143i& 17.7224+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_4& =\left[\begin{array}{cc}23.6748+0.0000i& -8.4759-2.9916i\\ -8.4759+2.9916i& 37:7224+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_5& =\left[\begin{array}{cc}16.3740+0.0000i& -5.2878-1.8567i\\ -5.2878+1.8567i& 37.8758+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_6& =\left[\begin{array}{cc}10.5478+0.0000i& -2.7546-0.7654i\\ -2.7546+0.7654i& 15.6774+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_7& =\left[\begin{array}{c}13.0757+0.0000i-6.9876-4.7658i\\ -6.9876+4.7658i20.7764+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_8& =\left[\begin{array}{cc}3.6748+0.0000i& -1.4759-0.6356i\\ -1.4759+0.6356i& 7.4554+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_9& =\left[\begin{array}{c}19.6748+0.0000i-2.7756-0.6816i\\ -2.7756+0.6816i25.9864+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_{10}& =\left[\begin{array}{cc}17.3498+0.0000i& -3.4759-0.4816i\\ -3.4759+0.4816i& 22.7624+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{T}}_1& =\left[\begin{array}{cc}43.6748& 0\\ 0& 54.6543\end{array}\right],\hfill \\ \hfill {\mathbb{T}}_2& =\left[\begin{array}{cc}34.1745& 0\\ 0& 38.7224\end{array}\right],\hfill \\ \hfill {\mathbb{T}}_3& =\left[\begin{array}{cc}78.5777& 0\\ 0& 87.6767\end{array}\right],\hfill \\ \hfill {\mathbb{T}}_4& =\left[\begin{array}{cc}36.6748& 0\\ 0& 56.7667\end{array}\right].\hfill \end{array}$$

The control gain matrix is obtained as follows:

$$\begin{array}{cc}\hfill \mathbb{K}& =\left[\begin{array}{cc}65.3498+0.0000i& -7.6585-0.566i\\ -7.6585+0.566i& 68.7624+0.0000i\end{array}\right].\hfill \end{array}$$

The above results shows that the fractional order complex-valued neural networks (11) is globally exponentially stable.

Example 2.

Consider the following fractional order complex-valued neural networks (25), with parameters are

$$\begin{array}{cc}\hfill \mathbb{L}& =\left[\begin{array}{cc}0.67& 0\\ 0& 0.67\end{array}\right],\mathbb{B}=\left[\begin{array}{cc}0.6+0.9i& -0.1+0.8i\\ 0.8+0.2i& 0.6+0.7i\end{array}\right]\hfill \\ \hfill \mathbb{C}& =\left[\begin{array}{cc}0.1+0.6i& -0.3-0.2i\\ 0.6+0.8i& -0.7-0.7i\end{array}\right].\hfill \end{array}$$

Choose, scalars are ${\delta }_1=0.84,{\phi }_1=0.18,\alpha =0.27,\delta =0.35,{\mu }_1=0.15,\phi =0.67.$ Utilizing Matlab, the LMI (26) of Theorem 4 has the following feasible solutions

$$\begin{array}{cc}\hfill {\mathbb{R}}_1& =\left[\begin{array}{cc}1.6678+0.0000i& -13.8769-1.4816i\\ -13.8769+1.4816i& 11.7224+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_2& =\left[\begin{array}{cc}14.6595+0.0000i& -11.4439-28.4876i\\ -11.4439+28.4876i& 28.7224+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_3& =\left[\begin{array}{cc}8.1148+0.0000i& -1.6789-1.6183i\\ -1.6789+1.6183i& 17.7224+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_4& =\left[\begin{array}{cc}23.6748+0.0000i& -6.1259-8.9565i\\ -6.1259-8.9565i& 37.1224+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_9& =\left[\begin{array}{c}16.6748+0.0000i-2.2356-0.5516i\\ -2.2356+0.5516i43.9787+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{R}}_{10}& =\left[\begin{array}{cc}18.3788+0.0000i& -2.4759-0.8965i\\ -2.4759+0.8965i& 22.7624+0.0000i\end{array}\right],\hfill \\ \hfill {\mathbb{T}}_1& =\left[\begin{array}{cc}27.1148& 0\\ 0& 44.1233\end{array}\right],\hfill \\ \hfill {\mathbb{T}}_2& =\left[\begin{array}{cc}67.4445& 0\\ 0& 76.8724\end{array}\right],\hfill \\ \hfill {\mathbb{T}}_3& =\left[\begin{array}{cc}54.5877& 0\\ 0& 76.6546\end{array}\right],\hfill \\ \hfill {\mathbb{T}}_4& =\left[\begin{array}{cc}86.5332& 0\\ 0& 97.5678\end{array}\right].\hfill \end{array}$$

The control gain matrix is obtained as follows:

$$\begin{array}{cc}\hfill \mathbb{K}& =\left[\begin{array}{cc}6.3498+0.0000i& -1.5685-0.566i\\ -1.5685+0.566i& 6.7624+0.0000i\end{array}\right].\hfill \end{array}$$

The above results shows that the fractional order complex-valued neural networks (25) is globally exponentially stable.

5. Conclusions

In this paper, we have investigated the global exponential stability of fractional order complex valued neural networks with leakage delay and mixed time varying delays. By constructing a proper Lyapunov functional and using fractional-order differential inequality and some other inequality techniques, we derived sufficient conditions to ensure the global exponential stability for the discussed fractional-order systems. Based on the Lyapunov functional containing some novel single and double integral terms, the stability conditions are derived in terms of LMIs which can be very efficiently solved by using Matlab LMI control toolbox. Finally illustrative examples have been provided to show the effectiveness of the obtained results.