Stability Analysis of Delayed Neural Networks via Composite-Matrix-Based Integral Inequality

Abstract

This paper revisits the problem of stability analyses for neural networks with time-varying delay. A composite-matrix-based integral inequality (CMBII) is presented, which takes the delay derivative into account. In this case, the coupling information can be fully captured in integral inequalities with the delay derivative. Based on a CMBII, a new stability criterion is derived for neural networks with time-varying delay. The effectiveness of this method is verified by a numerical example.

1. Introduction

In the past few decades, neural networks have been successfully applied in many engineering and research fields, such as image processing, pattern recognition, optimization problems, and associative memory, and have received considerable attention [1,2,3,4,5]. In this application, the designed neural networks are required to have stability properties, including asymptotic stability and exponential stability [6,7]. Time delay is a common phenomenon in many control systems [8]. In general, time delays tend to lead to unstable and poorly controlled systems. The dynamic behavior of the equilibrium point has an important effect on the application of neural networks; thus, many researchers focus a lot of time and energy on stability analyses of neural networks with time delays [9,10,11].

The Lyapunov–Krasovskii functional (LKF) method is widely recognized as a highly effective tool for analyzing system stability. At its core, this method involves identifying a positive definite function whose derivative along the system trajectory is negative definite. The crucial aspect of this approach lies in selecting a suitable LKF, as it directly influences the derivation of stability criteria. By carefully choosing an appropriate LKF, researchers can enhance the accuracy and reliability of the stability analysis process. In [12], some stability conditions of delayed neural networks are obtained, in which the constraints on the interconnection matrix of neural system are imposed. The problem of delayed neural networks is investigated, in which the assumption of the symmetry of the connection matrix is removed in [13]. In [14], an exponential stability result which establishes a relation between time delay and parameters of network is proposed. The state estimation problem is studied for neural networks with time-varying delays, in which the interconnection matrix and the activation functions are assumed to be norm-bounded [15]. In [16], an efficient method named the delay-product-type functional (DPTF) method is proposed by introducing delay-amplitude-dependent matrices. For simplicity, denote $d\left(t\right)$ as the delay amplitude and $\dot{d}\left(t\right)$ as the delay derivative. Take a DPTF $V\left(t\right)=d\left(t\right){\nu }_1^T\left(t\right)P_1{\nu }_1\left(t\right)+(h-d\left(t\right)){\nu }_2^T\left(t\right)P_2{\nu }_2\left(t\right)$ as an example, where $d\left(t\right)P_1>0$, $(h-d\left(t\right))P_2>0$ are the delay-amplitude-dependent matrices, and ${\nu }_1\left(t\right)$, ${\nu }_2\left(t\right)$ are the augmented state-related terms. Noting the derivative of $V\left(t\right)$ is

$$\begin{array}{ccc}\hfill \dot{V}\left(t\right)& =& \dot{d}\left(t\right)({\nu }_1^T\left(t\right)P_1{\nu }_1\left(t\right)-{\nu }_2^T\left(t\right)P_2{\nu }_2\left(t\right))\hfill \\ & & +2d\left(t\right){\nu }_1^T\left(t\right)P_1{\dot{\nu }}_1\left(t\right)+2(h-d\left(t\right)){\nu }_2^T\left(t\right)P_2{\dot{\nu }}_2\left(t\right),\hfill \end{array}$$

(1)

it can be seen that the coupling information among the augmented state-related terms, delay amplitude and delay derivative can be linked in the final linear matrix inequalities (LMIs). However, the conservatism suffered from incomplete state vectors ${\nu }_1\left(t\right)$ and ${\nu }_2\left(t\right)$ in the LKF is unavoidable. Thus, coupling information among the delay amplitude, delay derivative and state-related vectors has not been fully captured. On the other hand, many bounding methods such as the Jensen-based integral inequality [17], the Wirtinger-based integral inequality [18], various slack-matrix-based integral inequalities [19,20,21,22,23], Bessel–Legendre inequality (BLI) [24,25], and the Jacobi–Bessel inequality (JBI) [26] have been achieved. Therein, the BLI has the potential to obtain an analytical solution for constant delay systems. However, the inequality has drawbacks in the application to time-varying systems because the estimated boundary depends on the cross convex combination [27,28,29]. In [30], an affine Bessel–Legendre inequality (ABLI) was proposed for systems with a time-varying delay amplitude $d\left(t\right)$ by introducing some slack matrices. However, the conservatism suffered from incomplete vectors of the LKF is unavoidable in ABLI. Thus, a generalized free matrix-based integral inequality (GFMBII) was proposed in [31], which can supplement the incomplete vectors in the ABLI. Please note that the AFLI in [30] and various integral inequalities based on slack matrices in [19,20,21,22,23] play a key role in reducing conservatism. A review of the matrix structure in ABLIs can be found in [30] and in GFMBIIs in [31]. The delay-amplitude-dependent matrix is mainly concentrated on positive definite terms such as $M^T{R}^{-1}M$, and the slack matrix has not yet been fully coupled to the delay amplitude $d\left(t\right)$. Thus, introducing delay-amplitude-dependent slack matrices in integral inequalities is worthy of study. Recalling (1), the derivative $\dot{V}\left(t\right)$ contains delay-amplitude- and delay-derivative-dependent terms. However, the delay amplitude and delay derivative are separated. That is to say, the DPTF cannot fully capture the coupling information between the delay amplitude and the delay derivative. Moreover, to the authors’ best knowledge, there is no work considering the delay-derivative-dependent integral inequalities for NNs, which is the motivation behind this current work.

In this paper, we focus on a stability analysis for NNs with a time-varying delay. By introducing delay-amplitude- and delay-derivative-dependent slack matrices, a composite-matrix-based integral inequality (CMBII) is proposed. Based on the CMBII, a stability criterion is proposed for NNs with time-varying delay. The effectiveness of the stability criterion can be demonstrated by illustrating a numerical example.

Notation: In this paper, ${\mathbb{R}}^n$ represents the n-dimensional Euclidean space; $He\left[X\right]$ represents $X+X^T$; $Co\{\cdots \}$ represents a set of points; $col[X,Y]$ represents ${[X^T,Y^T]}^T$; diag$\{\dots \}$ represents a block diagonal matrix; $X^T$ represents the transposition of X; and the abbreviations in the present paper are given in

Table 1. A list of abbreviations.

CMBII	composite-matrix-based integral inequality
LKF	Lyapunov–Krasovskii functional
DPTF	delay-product-type functional
LMIs	linear matrix inequalities
BLI	Bessel–Legendre inequality
JBI	Jacobi–Bessel inequality
ABLI	affine Bessel–Legendre inequality
GFMBII	generalized free-matrix-based integral inequality
GNNs	generalized neural networks
MAUB	maximum allowable upper bound

.

2. Problem Formation and Preliminaries

Consider the following generalized neural networks (GNNs) with time-varying delay

$$\dot{x}\left(t\right)=-Ax\left(t\right)+W_{0}f\left(W_{2}x\left(t\right)\right)+W_{1}f(W_{2}x(t-d\left(t\right))$$

(2)

where $x\left(t\right)=col[x_1\left(t\right),x_2\left(t\right),x_3\left(t\right),x_4\left(t\right),x_5\left(t\right),\dots ,x_n\left(t\right)]\in {\mathbb{R}}^n$ is the state vector and $f\left(W_{2}x\left(t\right)\right)=col[f_1\left(W_{21}x\left(t\right)\right),f_2\left(W_{22}x\left(t\right)\right),\dots ,f_n\left(W_{2n}x\left(t\right)\right)]$ is the activation function. $A=diag\{a_1,a_2,..a_n\}>0$, and $W_0$, $W_1$, $W_2$ are are real constant matrices with appropriate dimensions. $d\left(t\right)$ is the delay amplitude and $\dot{d}\left(t\right)$ is the delay derivative, which satisfy

$$\begin{array}{c}\hfill 0\le d\left(t\right)\le h,-\mu \le \dot{d}\left(t\right)\le \mu \end{array}$$

(3)

where h and $\mu$ are constants. The activation functions $f_i\left(W_{2i}x\left(t\right)\right),i\in \{1,2,\dots ,n\}$ satisfy $f_i\left(0\right)=0$ and

$$\begin{array}{c}\hfill l_i^{-}\le \frac{f_i\left(a\right)-f_i\left(b\right)}{a-b}\le l_i^{+},a\ne b\end{array}$$

(4)

where $l_i^{-}$ and $l_i^{+}$ are known constants that may be positive, negative or zero. For convenience, define $L^{-}=diag\{l_1^{-},l_2^{-},\dots ,l_n^{-}\}$ and $L^{+}=diag\{l_1^{+},l_2^{+},\dots ,l_n^{+}\}$. We can directly obtain the following inequalities from (4), with s, $s_1$, $s_2\in \mathbb{R}$, $U^{-}=diag\{u_1^{-},u_2^{-},\dots ,u_n^{-}\}>0$ and $U^{+}=diag\{u_1^{+},u_2^{+},\dots ,u_n^{+}\}>0$:

$$\begin{array}{c}\hfill {\lambda }_1(s,U^{-})\ge 0,{\lambda }_2(s_1,s_2,U^{+})\ge 0\end{array}$$

(5)

where

$$\begin{array}{ccc}\hfill {\lambda }_1(s,U^{-})& =& 2{[f\left(W_{2}x\left(s\right)\right)-L^{-}{W}_{2}x\left(s\right)]}^T{U}^{-}\hfill \\ & \times & [L^{+}{W}_{2}x\left(s\right)-f\left(W_{2}x\left(s\right)\right)]\hfill \\ \hfill {\lambda }_2(s_1,s_2,U^{+})& =& 2[f\left(W_{2}x\left(s_1\right)\right)-f\left(W_{2}x\left(s_2\right)\right)\hfill \\ & & -L^{-}{W}_{2}x\left(s_1\right)-x\left(s_2\right){]}^T{U}^{+}\left[L^{+}{W}_2\right(x\left(s_1\right)\hfill \\ & & -x\left(s_2\right))-f\left(W_{2}x\left(s_1\right)\right)+f\left(W_{2}x\left(s_2\right)\right)].\hfill \end{array}$$

To keep the representation simple, the following notations are used for $S\in \mathbb{N}$:

$$\begin{array}{ccc}\hfill d& =& d\left(t\right),h_d=h-d,\dot{\overline{d}}=1-\dot{d}\hfill \\ \hfill g_i(a,b)& =& {\int }_a^b{\left(\frac{s-a}{b-a}\right)}^{i}x\left(s\right)ds\hfill \\ \hfill v_{1i}\left(t\right)& =& g_i(t,t-d),v_{2i}\left(t\right)=g_i(t-d,t-h)\hfill \\ \hfill \vartheta (a,b)& =& \left\{\begin{array}{c}col\left[\begin{array}{cc}x\left(a\right)& x\left(b\right)\end{array}\right],ifS=0\hfill \\ col\left[\begin{array}{c}x\left(a\right),x\left(b\right),\frac{1}{h}{\Omega }_0,\cdots ,\frac{1}{h}{\Omega }_{S-1}\end{array}\right],ifS>0\hfill \end{array}\right.\hfill \\ \hfill {\Omega }_k& =& {\int }_{t-h}^t{L}_k\left(s\right)x\left(s\right)ds\hfill \\ \hfill L_k\left(s\right)& =& {(-1)}^k\sum _{l=0}^k\left[{(-1)}^l\left(\genfrac{}{}{0ex}{}{k}{l}\right)\left(\genfrac{}{}{0ex}{}{k+l}{l}\right)\right]{\left(\frac{s-t+h}{h}\right)}^l\hfill \\ \hfill {\pi }_S\left(k\right)& =& \left\{\begin{array}{c}\left[\begin{array}{cc}I& -I\end{array}\right],ifS=0\hfill \\ \left[\begin{array}{c}I,{(-1)}^{k+1}I,{\varsigma }_{Sk}^{0}I,\cdots ,{\varsigma }_{Sk}^{S-1}I\end{array}\right],ifS>0\hfill \end{array}\right.\hfill \\ \hfill {\varsigma }_{Sk}^i& =& \left\{\begin{array}{c}-(2i+1)(1-{(-1)}^{k+i}),ifi\le k\hfill \\ 0,ifi\ge k+1\hfill \end{array}\right.\hfill \\ \hfill \tilde{R}& =& diag\{R^{-1},\frac{1}{3}{R}^{-1},\dots ,\frac{1}{2S+1}{R}^{-1}\}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \widehat{R}& =& diag\{R,3R,\dots ,(2S+1\left)R\right\}\hfill \\ \hfill {\mathsf{\Gamma }}_S& =& col[{\pi }_S\left(0\right),{\pi }_S\left(1\right),\dots ,{\pi }_S\left(S\right)]\hfill \\ \hfill {\vartheta }_1& =& \vartheta (t-d,t),{\vartheta }_2=\vartheta (t-h,t-d)\hfill \\ \hfill f_i^{-}& =& f_i\left(s\right)-l_i^{-}s,f_i^{+}=l^{+}s-f_i\left(s\right)\hfill \\ \hfill o\left(t\right)& =& col\left[x\right(t),x(t-d),x(t-h\left)\right]\hfill \\ \hfill {\varrho }_0\left(s\right)& =& col[\dot{x}\left(s\right),x\left(s\right),f\left(W_{2}x\left(s\right)\right)]\hfill \\ \hfill {\varrho }_1\left(s\right)& =& col[{\int }_s^{t}x\left(u\right)du,{\int }_{t-d}^{s}x\left(u\right)du]\hfill \\ \hfill {\varrho }_2\left(s\right)& =& col[{\int }_s^{t-d}x\left(u\right)du,{\int }_{t-h}^{s}x\left(u\right)du]\hfill \\ \hfill {\xi }_{0S}\left(t\right)& =& col[o\left(t\right),v_{10}\left(t\right),v_{20}\left(t\right),\dots ,v_{1S}\left(t\right),v_{2S}\left(t\right)]\hfill \\ \hfill {\xi }_{1S}\left(t\right)& =& col[o\left(t\right),\frac{v_{10}\left(t\right)}{d},\frac{v_{11}\left(t\right)}{d},\dots ,\frac{v_{1S}\left(t\right)}{d}]\hfill \\ \hfill {\xi }_{2S}\left(t\right)& =& col[o\left(t\right),\frac{v_{20}\left(t\right)}{d},\frac{v_{21}\left(t\right)}{h_d},\dots ,\frac{v_{2S}\left(t\right)}{h_d}]\hfill \\ \hfill {\xi }_{3S}(t,s)& =& \left\{\begin{array}{c}\hfill \varrho \left(0\right)\left(s\right),S=0\\ \hfill {\varrho }_0\left(s\right),{\varrho }_1\left(s\right),S\ge 1\end{array}\right.\hfill \\ \hfill {\xi }_{4S}(t,s)& =& \left\{\begin{array}{c}\hfill \varrho \left(0\right)\left(s\right),S=0\\ \hfill {\varrho }_0\left(s\right),{\varrho }_5\left(s\right),S\ge 1\end{array}\right.\hfill \\ \hfill {\xi }_5\left(t\right)& =& col[f\left(W_{2}x\left(t\right)\right),f\left(W_{1}x(t-d)\right),f\left(W_{2}x(t-h)\right)\hfill \\ & & {\int }_{t-d}^{t}f\left(W_{2}x\left(s\right)\right)ds,{\int }_{t-h}^{t-d}f\left(W_{2}x\left(s\right)\right)ds]\hfill \\ \hfill {\xi }_6\left(t\right)& =& col[\dot{x}(t-d),\dot{x}(t-h)]\hfill \\ \hfill {\xi }_{7i}\left(t\right)& =& col[\frac{v_{1i}\left(t\right)}{d},\frac{v_{2i}\left(t\right)}{h_d}]\hfill \\ \hfill {\xi }_S\left(t\right)& =& col[o\left(t\right),{\xi }_5\left(t\right),{\xi }_6\left(t\right),{\xi }_{70}\left(t\right),{\xi }_{71}\left(t\right),\dots ,{\xi }_{7S}\left(t\right)]\hfill \\ \hfill e_{i,N}& =& [0_{n\times (i-1)n},I_{n\times n},0_{n\times (N-i)n}],i\in 1,2,\dots ,N.\hfill \end{array}$$

In the previous literature [23,30,31], the information of the delay derivative in the final LMIs is usually introduced by the derivative of LKFs. No integral inequalities are related to the delay derivative for NNs with time-varying delay. In order to fill the gap, a composite slack matrix integral inequality (CMBII) is proposed as follows.

Lemma 1.

For any parameter γ, any symmetric positive definite matrix R, any vector ξ, any continuously differentiable function $x:[-h,0]\to {\mathbb{R}}^n$, and slack matrices $M,N$, the following inequality holds:

$$\begin{array}{ccc}\hfill & -& {\int }_{t-h}^h{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds\hfill \\ & \le & \frac{(\mu -\gamma -\dot{d})}{\mu }{\xi }^T\left[d{M}^T\tilde{R}M+h_d{N}^T\tilde{R}N\right]\xi \hfill \\ & & +\left(\frac{\mu h-(\gamma +\dot{d})d}{\mu h}+\frac{(\gamma +\dot{d})d^2}{\mu h^2}\right)He\left[({\vartheta }_1^T{\mathsf{\Gamma }}_S^{T}M+{\vartheta }_2^T{\mathsf{\Gamma }}_S^{T}N)\xi \right]\hfill \\ & & -\frac{(\gamma +\dot{d})}{\mu h^2}\left\{h_d{\vartheta }_1^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{\vartheta }_1+d{\vartheta }_2^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{\vartheta }_2\right\}.\hfill \end{array}$$

(6)

Proof. 

For convex parameters $\alpha ,\beta \in [0,1]$ with $\alpha +\beta =1$, it yields

$$\begin{array}{ccc}& & {\int }_{t-h}^t{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds\hfill \\ & =& \alpha {\int }_{t-d}^t{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds+\alpha {\int }_{t-h}^{t-d}{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds\hfill \\ & & +\beta {\int }_{t-d}^t{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds+\beta {\int }_{t-h}^{t-d}{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds.\hfill \end{array}$$

From the inequalities in [24,31] and the relation $\alpha +\beta =1$, it yields

$$\begin{array}{ccc}\hfill & -& {\int }_{t-h}^t{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds\hfill \\ & \le & \alpha {\xi }^T\left[d{M}^T\tilde{R}M+h_d{N}^T\tilde{R}N\right]\xi \hfill \\ & & +\alpha He\left[({\vartheta }_1^T{\mathsf{\Gamma }}_S^{T}M+{\vartheta }_2^T{\mathsf{\Gamma }}_S^{T}N)\xi \right]\hfill \\ & & -\beta \left\{\frac{1}{d}{\vartheta }_1^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{\vartheta }_1+\frac{1}{h_d}{\vartheta }_2^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{\vartheta }_2\right\}.\hfill \end{array}$$

(7)

Due to $d{M}^T\tilde{R}M+h_d{N}^T\tilde{R}N>0$, one has

$$\begin{array}{ccc}\hfill & -& {\int }_{t-h}^t{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds\hfill \\ & \le & {\xi }^T\left[d{M}^T\tilde{R}M+h_d{N}^T\tilde{R}N\right]\xi \hfill \\ & & +\alpha He\left[({\vartheta }_1^T{\mathsf{\Gamma }}_S^{T}M+{\vartheta }_2^T{\mathsf{\Gamma }}_S^{T}N)\xi \right]\hfill \\ & & -\beta \left\{\frac{1}{d}{\vartheta }_1^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{\vartheta }_1+\frac{1}{h_d}{\vartheta }_2^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{\vartheta }_2\right\}.\hfill \end{array}$$

(8)

Setting

$$\begin{array}{c}\hfill \alpha =\frac{\mu h-(\gamma +\dot{d})d}{\mu h}+\frac{(\gamma +\dot{d})d^2}{\mu h^2},\beta =\frac{d{h}_d(\gamma +\dot{d})}{\mu h^2}\end{array}$$

(9)

it satisfies $0\le \alpha \le 1$, $0\le \beta \le 1$, $\alpha +\beta =1$.

By substituting (9) into (7), it yields

$$\begin{array}{ccc}\hfill & -& {\int }_{t-h}^h{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds\hfill \\ & \le & \frac{\mu h-(\gamma +\dot{d})d}{\mu h}+\frac{(\gamma +\dot{d})d^2}{\mu h^2}\hfill \\ & \times & {\xi }^T\left[d{M}^T\tilde{R}M+h_d{N}^T\tilde{R}N\right]\xi \hfill \\ & & +\left(\frac{\mu h-(\gamma +\dot{d})d}{\mu h}+\frac{(\gamma +\dot{d})d^2}{\mu h^2}\right)He\left[({\vartheta }_1^T{\mathsf{\Gamma }}_S^{T}M+{\vartheta }_2^T{\mathsf{\Gamma }}_S^{T}N)\xi \right]\hfill \\ & & -\frac{(\gamma +\dot{d})}{\mu h^2}\left\{h_d{\vartheta }_1^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{\vartheta }_1+d{\vartheta }_2^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{\vartheta }_2\right\}\hfill \end{array}$$

(10)

Due to $d{M}^T\tilde{R}M+h_d{N}^T\tilde{R}N>0$, $0\le \frac{d{h}_d}{h^2}\le 1$ and $0\le \frac{\gamma +\dot{d}}{\mu }\le 1$, one has

$$\begin{array}{c}\hfill \frac{\mu h-(\gamma +\dot{d})d}{\mu h}+\frac{(\gamma +\dot{d})d^2}{\mu h^2}\le \frac{\mu -\gamma -\dot{d}}{\mu }.\end{array}$$

(11)

Combining (11) and (9), it yields (6). This completes the proof. □

Remark 1.

A new integral inequality is proposed in Lemma 1 by combining delay-amplitude- and delay-derivative-dependent slack matrices, which can be named CMBII. It should be pointed out that the delay derivative is first introduced in CMBII by the intermediate convex parameters $\alpha ,\beta$ satisfying $\alpha +\beta =1$, which has not been considered in the previous literature. The other merits can be concluded as follows. (i) Delay-amplitude- and derivative-dependent slack matrices are introduced to link the coupling information among system states, delay amplitude and delay derivative. Thus, compared with Lemma 1, more coupling information can be utilized without additional decision variables. (ii) Parameter γ is introduced to avoid some zero terms. For example, if $\gamma =0$ and $\dot{d}=0$, the last term $h_d{\vartheta }_1^T{\mathsf{\Gamma }}_S^T\tilde{R}{\mathsf{\Gamma }}_S{\vartheta }_1+d{\vartheta }_2^T{\mathsf{\Gamma }}_S^T\tilde{R}{\mathsf{\Gamma }}_S{\vartheta }_2$ will disappear. Similarly, if $\gamma =0$ and $\dot{d}=\mu$, the first term $d{M}^T\tilde{R}M+h_d{N}^T\tilde{R}N$ will disappear. Moreover, it can also increase flexibility.

Remark 2.

In the previous works, the information of the delay derivative is introduced in the final LMIs by the derivative of LKF. It is well known that DPTF is an efficient way to introduce delay information. However, recalling the derivative of the DPTF (1), the delay amplitude and the delay derivative are separated in each term. Moreover, the augmented vectors ${\nu }_1\left(t\right)$ and ${\nu }_2\left(t\right)$ and their derivatives cannot contain all state-related terms. Due to the two limitations, the coupling information among the delay amplitude, the delay derivative and state-related vectors has not been fully captured. Fortunately, the two limitations can be simultaneously solved in CMBII, which can be concluded as follows. (I) The delay amplitude and delay derivative are coupled in CMBII. (II) Any augmented vector ξ can cover all the state-related terms in LKF and its derivative. In these two cases, CMBII can fully capture system information, which leads to less conservatism.

The main aim of the present paper is to derive a less conservative stability condition of GNNs (2) with time-varying delay. We will apply CMBII to reach this aim as follows.

3. Main Results

In order to make full use of the information of the delay derivative, the following stability criterion for system (2) is established based on the CMBII.

Theorem 1.

For given scalars γ, h and μ, system (2) is asymptotically stable if there exists positive definite symmetric matrices P, $Q_1$, $Q_2$ and R and matrices M and N such that the following inequalities hold

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}\tilde{\mathsf{\Omega }}(0,\dot{d})& \sqrt{\frac{(\mu -\gamma -\dot{d})}{\mu }}h{e}_8^T{L}_2^T& \sqrt{\frac{(\mu -\gamma -\dot{d})}{\mu }}h{N}^T\\ *& -Z& 0\\ *& *& -\widehat{R}\end{array}\right]<0\end{array}$$

(12)

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}\tilde{\mathsf{\Omega }}(h,\dot{d})& \sqrt{\frac{(\mu -\gamma -\dot{d})}{\mu }}h{e}_7^T{L}_1^T& \sqrt{\frac{(\mu -\gamma -\dot{d})}{\mu }}h{M}^T\\ *& -Z& 0\\ *& *& -\widehat{R}\end{array}\right]<0\end{array}$$

(13)

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}\widehat{\mathsf{\Omega }}(0,\dot{d})& \sqrt{\frac{(\mu -\gamma -\dot{d})}{\mu }}h{e}_8^T{L}_2^T& \sqrt{\frac{(\mu -\gamma -\dot{d})}{\mu }}h{N}^T\\ *& -Z& 0\\ *& *& -\widehat{R}\end{array}\right]<0\end{array}$$

(14)

where

$$\begin{array}{ccc}\hfill \widehat{\mathsf{\Omega }}(0,\dot{d})& =& -h^2{a}_2\left(\dot{d}\right)+\tilde{\mathsf{\Omega }}(0,\dot{d})\hfill \\ \hfill a_2\left(\dot{d}\right)& =& {\overline{\gamma }}_1^T{Q}_1{\overline{\gamma }}_1-\dot{\overline{d}}{\overline{\gamma }}_2^T{Q}_1{\overline{\gamma }}_2+He\left[{\gamma }_3^T{Q}_1{\overline{\gamma }}_4\right]\hfill \\ & & +\dot{\overline{d}}{\overline{\gamma }}_5^T{Q}_2{\overline{\gamma }}_5-{\overline{\gamma }}_6^T{Q}_2{\overline{\gamma }}_6+He\left[{\gamma }_7^T{Q}_2{\overline{\gamma }}_8\right]\hfill \\ & & +\frac{\gamma +\dot{d}}{\mu h}He[E_{1S}^T{\mathsf{\Gamma }}_S^{T}M+E_{2S}^T{\mathsf{\Gamma }}_S^{T}N\hfill \\ & & +e_7^T{L}_1{e}_7+e_8^T{L}_2{e}_8]\hfill \\ \hfill \tilde{\mathsf{\Omega }}(d,\dot{d})& =& {\mathsf{\Omega }}_1(d,\dot{d})+{\mathsf{\Omega }}_2(d,\dot{d})+{\mathsf{\Omega }}_3(d,\dot{d})\hfill \\ & & +{\mathsf{\Omega }}_4\left(\dot{d}\right)+{\mathsf{\Omega }}_5(d,\dot{d})+{\mathsf{\Omega }}_6\hfill \\ \hfill {\mathsf{\Omega }}_1(d,\dot{d})& =& He\left[{\mathsf{\Pi }}_1^T\left(d\right)P_{0S}{\mathsf{\Pi }}_2\left(\dot{d}\right)\right]+\dot{d}{\mathsf{\Pi }}_3^T{P}_{1S}{\mathsf{\Pi }}_3\hfill \\ & & +He\left[{\mathsf{\Pi }}_3^T{P}_{1S}{\mathsf{\Pi }}_4(d,\dot{d})\right]-\dot{d}{\mathsf{\Pi }}_5^T{P}_{2S}{\mathsf{\Pi }}_5\hfill \\ \hfill {\mathsf{\Omega }}_2(d,\dot{d})& =& {\gamma }_1^T{Q}_1{\gamma }_1-\dot{\overline{d}}{\gamma }_2^T{Q}_1{\gamma }_2+He\left[{\gamma }_3^T{Q}_1{\gamma }_4\right]\hfill \\ & & +\dot{\overline{d}}{\gamma }_5^T{Q}_2{\gamma }_5-{\gamma }_6^T{Q}_2{\gamma }_6+He\left[{\gamma }_7^T{Q}_2{\gamma }_8\right]\hfill \\ \hfill {\mathsf{\Omega }}_3(d,\dot{d})& =& h^2{e}_a^{T}R{e}_a+\left(\frac{\mu h-(\gamma +\dot{d})d}{\mu }+\frac{(\gamma +\dot{d})d^2}{\mu h}\right)\hfill \\ & \times & He[E_{1S}^T{\mathsf{\Gamma }}_S^{T}M+E_{2S}^T{\mathsf{\Gamma }}_S^{T}N)]\hfill \\ & & -\frac{(\gamma +\dot{d})}{\mu h}\left\{h_d{E}_{1S}^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{E}_{1S}+d{E}_{S2}^T{\mathsf{\Gamma }}_S^T\widehat{R}{\mathsf{\Gamma }}_S{E}_{2S}\right\}\hfill \\ \hfill {\mathsf{\Omega }}_4\left(\dot{d}\right)& =& He[{\rho }_{31}^T{W}_2{e}_a+\dot{\overline{d}}{\rho }_{32}^T{W}_2{e}_9+{\rho }_{33}^T{W}_2{e}_{10}]\hfill \\ \hfill {\mathsf{\Omega }}_5(d,\dot{d})& =& h^2{e}_4^{T}Z{e}_4+\left(\frac{\mu h-(\gamma +\dot{d})d}{\mu }+\frac{(\gamma +\dot{d})d^2}{\mu h}\right)\hfill \\ & \times & He[e_7^T{L}_1{e}_7+e_8^T{L}_2{e}_8]\hfill \\ & & -\frac{(\gamma +\dot{d})}{\mu h}(h_d{e}_7^{T}Z{e}_7+d{e}_8^{T}Z{e}_8)\hfill \\ & & +\left(\frac{\mu h-(\gamma +\dot{d})d}{\mu }+\frac{(\gamma +\dot{d})d^2}{\mu h}\right)\hfill \\ & \times & He[e_7^T{L}_1{e}_7+e_8^T{L}_2{e}_8]\hfill \\ & & -\frac{(\gamma +\dot{d})}{\mu h}\left\{h_d{e}_7^{T}Z{e}_7+d{e}_8^{T}Z{e}_8\right\}.\hfill \\ \hfill {\mathsf{\Omega }}_6& =& \sum _{i=1}^{3}He[{(e_{3+i}-L^{-}{W}_2{e}_i)}^T{U}_i^{-}(L^{+}{W}_2{e}_i\hfill \\ & & -e_{3+i}\left)\right]+\sum _{i=1}^{2}He\left[\right[e_{3+i}-e_{4+i}\hfill \\ & & -L^{-}{W}_2(e_i-e_{i+1}){]}^T{U}_i^{+}\hfill \\ & \times & [L^{+}{W}_2(e_i-e_{i+1})-e_{3+i}+e_{4+i}]]\hfill \\ & & +He[{[e_4-e_6-L^{-}{W}_2(e_1-e_3)]}^T{U}_3^{+}\hfill \\ & \times & [L^{+}{W}_2(e_1-e_3)-e_4+e_6]]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathsf{\Pi }}_1\left(d\right)& =& col[e_o,e_{u_0},e_{v_0},\cdots ,e_{u_S},e_{v_S}]\hfill \\ \hfill {\mathsf{\Pi }}_2\left(\dot{d}\right)& =& col[{\dot{e}}_0,{\dot{e}}_{u_0},{\dot{e}}_{v_0},\cdots ,{\dot{e}}_{u_S},{\dot{e}}_{v_S}]\hfill \\ \hfill {\mathsf{\Pi }}_3& =& col[e_0,e_{13},\cdots ,e_{2S+11}]\hfill \\ \hfill {\mathsf{\Pi }}_4(d,\dot{d})& =& col[d{\dot{e}}_0,{\dot{e}}_{u_0}-\dot{d}{e}_{11},{\dot{e}}_{u_1}-\dot{d}{e}_{13},\hfill \\ & & \cdots ,{\dot{e}}_{u_S}-\dot{d}{e}_{2S+11}]\hfill \\ \hfill {\mathsf{\Pi }}_5& =& col[e_a,e_{12},e_{14},\cdots ,e_{2S+14}]\hfill \\ \hfill {\mathsf{\Pi }}_6& =& col[h_d{\dot{e}}_0,{\dot{e}}_{v_0}+\dot{d}{e}_{12},{\dot{e}}_{v_1}+\dot{d}{e}_{14},\hfill \\ & & \cdots ,{\dot{e}}_{v_S}+\dot{d}{e}_{2S+12}]\hfill \\ \hfill {\gamma }_1& =& col[e_a,e_1,e_4,0,d{e}_{11}]\hfill \\ \hfill {\gamma }_2& =& col[e_9,e_2,e_5,d{e}_{11},0]\hfill \\ \hfill {\gamma }_3& =& col[0,0,0,e_1,-\dot{\overline{d}}{e}_2,0]\hfill \\ \hfill {\gamma }_4& =& col[e_1-e_2,d{e}_{11},e_7,d^2{e}_{13},d^2(e_{11}-e_{13})]\hfill \\ \hfill {\gamma }_5& =& col[e_9,e_2,e_5,0,h_d{e}_{12}]\hfill \\ \hfill {\gamma }_6& =& col[e_{10},e_3,e_6,h_d{e}_{12},0]\hfill \\ \hfill {\gamma }_7& =& col[0,0,0,\dot{\overline{d}}{e}_2,-e_3]\hfill \\ \hfill {\gamma }_8& =& col[e_2-e_3,h_d{e}_{12},e_8,h_d^2{e}_{14},h_d^2(e_{12}-e_{14})]\hfill \\ \hfill {\overline{\gamma }}_1& =& col[0,0,0,0,e_{11}]\hfill \\ \hfill {\overline{\gamma }}_2& =& col[0,0,0,e_{11},0]\hfill \\ \hfill {\overline{\gamma }}_4& =& col[0,0,0,e_{13},(e_{11}-e_{13})]\hfill \\ \hfill {\overline{\gamma }}_5& =& col[0,0,0,0,e_{12}]\hfill \\ \hfill {\overline{\gamma }}_6& =& col[e_{10},0,0,e_{12},0]\hfill \\ \hfill {\overline{\gamma }}_8& =& col[0,0,0,e_{14},(e_{12}-e_{14})]\hfill \\ \hfill {\rho }_{31}& =& M_1(e_4-L^{-}{W}_2{e}_1)+M_2(L^{+}{W}_2{e}_1-e_4)\hfill \\ \hfill {\rho }_{32}& =& M_3(e_5-L^{-}{W}_2{e}_2)+M_4(L^{+}{W}_2{e}_2-e_5)\hfill \\ \hfill {\rho }_{33}& =& M_5(e_6-L^{-}{W}_2{e}_3)+M_6(L^{+}{W}_2{e}_3-e_6)\hfill \\ \hfill {\epsilon }_{S1}& =& col[e_1,e_2,e_{11},2e_{13},(S+1)e_{11+2S}]\hfill \\ \hfill {\epsilon }_{S2}& =& col[e_2,e_3,e_{12},2e_{14},(S+1)e_{12+2S}]\hfill \\ \hfill e_{u_i}& =& d{e}_{11+2i},e_{v_i}=h_d{e}_{12+2i}\hfill \\ \hfill {\dot{e}}_{u_i}& =& \left\{\begin{array}{c}{e}_1-\dot{\overline{d}}{e}_2,i=1\hfill \\ e_1-i\dot{\overline{d}}{e}_{11+2(i-1)}-i\dot{d}{e}_{11+2i},i\ge 1\hfill \end{array}\right.\hfill \\ \hfill e_a& =& -A{e}_1+W_0{e}_4+W_1{e}_5\hfill \\ \hfill e_0& =& [e_1,e_2,e_2],{\dot{e}}_0=col[e_a,\dot{\overline{d}}{e}_9,e_{10}]\hfill \\ \hfill {\dot{e}}_{v_i}& =& \left\{\begin{array}{c}\dot{\overline{d}}{e}_2-e_3,i=1\hfill \\ \dot{\overline{d}}{e}_2-i{e}_{12+2(i-1)}-i\dot{d}{e}_{12+2i},i\ge 1\hfill \end{array}\right.\hfill \\ \hfill e_i& =& e_{i,10+2(S+1)}\hfill \end{array}$$

and for the other symbols, see Lemma 1.

Proof. 

To better show the CMBII of the present paper, we construct the following LKF candidate, which is the same as that in [31]

$$\begin{array}{c}\hfill V\left(t\right)=\sum _{i=1}^5{V}_i\left(t\right)\end{array}$$

(15)

$$\begin{array}{ccc}\hfill V_1\left(t\right)& =& {\xi }_{0S}^T\left(t\right)P_{0S}{\xi }_{0S}\left(t\right)+d{\xi }_{1S}^T\left(t\right)P_{1S}{\xi }_{1S}\left(t\right)\hfill \\ & & +h_d{\xi }_{2S}^T\left(t\right)P_{2S}{\xi }_{2S}\left(t\right)\hfill \\ \hfill V_2\left(t\right)& =& {\int }_{t-d}^t{\xi }_{3S}^T(t,s)Q_{1S}{\xi }_{3S}(t,s)ds\hfill \\ & & +{\int }_{t-h}^{t-d}{\xi }_{4S}^T(t,s)Q_{2S}{\xi }_{4S}(t,s)\hfill \\ \hfill V_3\left(t\right)& =& h{\int }_{t-h}^t{\int }_s^t{\dot{x}}^T\left(u\right)R\dot{x}\left(u\right)duds\hfill \\ \hfill V_4\left(t\right)& =& 2\sum _{i=1}^n{\int }_0^{W_{2i}x\left(t\right)}[m_{1i}{f}_i^{-}\left(s\right)+m_{2i}{f}_i^{+}\left(s\right)]ds\hfill \\ & & +2\sum _{i=1}^n{\int }_0^{W_{2i}x(t-d)}[m_{3i}{f}_i^{-}\left(s\right)+m_{4i}{f}_i^{+}\left(s\right)]ds\hfill \\ & & +2\sum _{i=1}^n{\int }_0^{W_{2i}x(t-h)}[m_{5i}{f}_i^{-}\left(s\right)+m_{6i}{f}_i^{+}\left(s\right)]ds\hfill \\ \hfill V_5\left(t\right)& =& h{\int }_{t-h}^t{\int }_s^t{f}^T\left(W_{2}x\left(u\right)\right)Zf\left(W_{2}x\left(u\right)\right)duds.\hfill \end{array}$$

Setting

$$\begin{array}{ccc}\hfill S_i& =& {\int }_a^b{\int }_{u_1}^b\cdots {\int }_{u_i-1}^{b}d{u}_i\cdots d{u}_{2}d{u}_1\hfill \\ \hfill {\mathsf{\Omega }}_{0^i}& =& {\int }_a^b{\int }_{u_1}^b\cdots {\int }_{u_i-1}^{b}x\left(u_i\right)d{u}_i\cdots d{u}_{2}d{u}_1\hfill \end{array}$$

it yields

$$\begin{array}{c}\hfill \frac{1}{S_k}{\mathsf{\Omega }}_{0^k}=\frac{k}{b-a}{g}_{(a,b)}^{k-1}.\end{array}$$

Setting $a=t-d,b=t$, we have

$$\begin{array}{ccc}\hfill {\vartheta }_1& =& col[x\left(t\right),x(t-d),\frac{u_0\left(t\right)}{d},\frac{2u_1\left(t\right)}{d},\cdots ,\frac{(S+1)u_S\left(t\right)}{d}]\hfill \\ & =& {\epsilon }_{S1}{\xi }_S\left(t\right).\hfill \end{array}$$

Setting $a=t-h,b=t-d$, we have

$$\begin{array}{c}\hfill {\vartheta }_2={\epsilon }_{S2}{\xi }_S\left(t\right).\end{array}$$

The derivative of $V\left(t\right)$ can be computed by

$$\begin{array}{ccc}\hfill {\dot{V}}_1\left(t\right)& =& 2{\xi }_{0S}^T\left(t\right)P_{0S}{\dot{\xi }}_{0S}\left(t\right)+\dot{d}{\xi }_{1S}^T\left(t\right)P_{1S}{\xi }_{1S}\left(t\right)\hfill \\ & & +2d{\xi }_{1S}^T\left(t\right)P_{1S}{\dot{\xi }}_{1S}\left(t\right)-\dot{d}{\xi }_{2S}^T\left(t\right)P_{2S}{\xi }_{2S}\left(t\right)\hfill \\ & & +2h_d{\xi }_{2S}^T\left(t\right)P_{2S}{\dot{\xi }}_{2S}\left(t\right)\hfill \\ & =& {\xi }_S^T\left(t\right){\mathsf{\Omega }}_1(d,\dot{d}){\xi }_S\left(t\right)\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\dot{V}}_2\left(t\right)& =& {\xi }_{3S}^T(t,t)Q_1{\xi }_{3S}(t,t)\hfill \\ & & -\dot{\overline{d}}{\xi }_{3S}^T(t,t-d)Q_1{\xi }_{3S}(t,t-d)\hfill \\ & & +2{\int }_{t-d}^t{\xi }_{3S}^T(t,s)Q_1\frac{d{\xi }_{3S}(t,s)}{dt}ds\hfill \\ & & +\dot{\overline{d}}{\xi }_{4S}^T(t,t-d)Q_2{\xi }_{4S}(t,t-d)\hfill \\ & & -{\xi }_{4S}^T(t,t-h)Q_2{\xi }_{4S}(t,t-h)\hfill \\ & & +2{\int }_{t-h}^{t-d}{\xi }_{4S}^T(t,s)Q_2\frac{d{\xi }_{4S}(t,s)}{dt}ds\hfill \\ & =& {\xi }_S^T\left(t\right){\mathsf{\Omega }}_2(d,\dot{d}){\xi }_S\left(t\right)\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill {\dot{V}}_3\left(t\right)& =& h^2{\dot{x}}^T\left(t\right)R\dot{x}\left(t\right)-h{\int }_{t-h}^t{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {\dot{V}}_4\left(t\right)& =& {\xi }_S^T\left(t\right){\mathsf{\Omega }}_4\left(\dot{d}\right){\xi }_S\left(t\right)\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill {\dot{V}}_5\left(t\right)& =& h^2{f}^T\left(W_{2}x\left(t\right)\right)Zf\left(W_{2}x\left(t\right)\right)\hfill \\ & & -h{\int }_{t-h}^t{f}^T\left(W_{2}x\left(s\right)\right)Zf\left(W_{2}x\left(s\right)\right)ds\hfill \end{array}$$

(20)

where ${\mathsf{\Omega }}_1(d,\dot{d})$, ${\mathsf{\Omega }}_2(d,\dot{d})$ and ${\mathsf{\Omega }}_4\left(\dot{d}\right)$ are the same as those in Proposition 1. By differentiating $v_{1i}\left(t\right)$ and $v_{2i}\left(t\right)$ in ${\xi }_{1S}\left(t\right)$ and ${\xi }_{2S}\left(t\right)$, it yields

$$\begin{array}{c}\hfill {\dot{g}}_i(a,b)=\left\{\begin{array}{c}\dot{b}x\left(b\right)-\dot{a}x\left(a\right),i=0\hfill \\ \dot{b}x\left(b\right)-\frac{i\dot{a}}{b-a}{g}^{i-1}-\frac{i(\dot{b}-\dot{a})}{b-a}{g}_i,i\ge 1\hfill \end{array}\right.\end{array}$$

With the help of Lemma 1, one has

$$\begin{array}{c}\hfill {\dot{V}}_3\left(t\right)\le {\xi }_S^T\left(t\right)[{\tilde{\mathsf{\Omega }}}_3(d,\dot{d})+{\mathsf{\Omega }}_3(d,\dot{d})]{\xi }_S\left(t\right)\end{array}$$

(21)

where ${\tilde{\mathsf{\Omega }}}_3(d,\dot{d})=\frac{(\mu -\gamma -\dot{d})}{\mu }h[d{M}^T\tilde{R}M+h_d{N}^T\tilde{R}N]$. Applying Lemma 1 with $S=0$, it yields

$$\begin{array}{c}\hfill {\dot{V}}_5\left(t\right)\le {\xi }_S^T\left(t\right)[{\mathsf{\Omega }}_5(d,\dot{d})+{\widehat{\mathsf{\Omega }}}_5(d,\dot{d})]{\xi }_S\left(t\right)\end{array}$$

(22)

where

$${\widehat{\mathsf{\Omega }}}_5(d,\dot{d})=\frac{(\mu -\gamma -\dot{d})}{\mu }h\left[d{e}_7^T{L}_1^T{Z}^{-1}{L}_1{e}_7+h_d{e}_8^T{L}_2^T{Z}^{-1}{L}_2{e}_8\right].$$

In addition, it follows from (5) that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\lambda }_1(t,U_1^{-})\ge 0\hfill \\ {\lambda }_1(t-d,U_2^{-})\ge 0\hfill \\ {\lambda }_1(t-h,U_3^{-})\ge 0\hfill \\ {\lambda }_2(t,t-d,U_1^{+})\ge 0\hfill \\ {\lambda }_2(t-d,t-h,U_2^{+})\ge 0\hfill \\ {\lambda }_2(t,t-h,U_3^{+})\ge 0\hfill \end{array}\right.\end{array}$$

further yielding

$$\begin{array}{c}\hfill {\xi }_S^T\left(t\right){\mathsf{\Omega }}_6{\xi }_S\left(t\right)\ge 0.\end{array}$$

(23)

According to the above-mentioned discussions, we have

$$\begin{array}{c}\hfill \dot{V}\left(t\right)\le {\xi }_S^T\left(t\right)\overline{\mathsf{\Omega }}(d,\dot{d}){\xi }_S\left(t\right)\end{array}$$

(24)

where $\overline{\mathsf{\Omega }}(d,\dot{d})=\tilde{\mathsf{\Omega }}(d,\dot{d})+{\widehat{\mathsf{\Omega }}}_3\left(d\right)+{\widehat{\mathsf{\Omega }}}_5\left(d\right).$ Define

$$\overline{\mathsf{\Omega }}(d,\dot{d})=d^2{a}_2\left(\dot{d}\right)+d{a}_1+a_0$$

where $a_2\left(\dot{d}\right)$ can be found in Theorem 1 and $a_1$ and $a_0$ are appropriate matrices, respectively. By using Lemma 2 of ref. [8], it yields $\overline{\mathsf{\Omega }}(d,{\mu }_2)<0$, which infers the asymptotical stability of NNs (2). □

Remark 3.

In previous works, such as [9,10,11,21,22,31], thet delay derivative $\dot{d}$ is introduced by the derivative of the LKF. However, the delay derivative $\dot{d}$ has not been fully captured. Instead of the LKFs, the delay derivative $\dot{d}$ is introduced by using CMBII. In this case, the delay derivative $\dot{d}$, the delay amplitude d, the slack matrices M and N and the augmented vector ${\xi }_S\left(t\right)$ can be simultaneously introduced beside the positive definite matrices. That is, the system information can be efficiently coupled. Moreover, the parameter γ is introduced to avoid zero terms so more coupling information can be utilized. For example, for $\sqrt{\frac{h(\mu -\gamma -\dot{d})}{\mu }}$ in (12), it can be seen that the coupling information between $\dot{d}$ and ${\xi }_S\left(t\right)$ linked by N will disappear if we set $\gamma =0$ and $\dot{d}=\mu$. Meanwhile, any parameter $\gamma \le 0$ (due to $\mu -\gamma -\dot{d}\ge 0$) can increase the flexibility of Theorem 1. As a result, Theorem 1 is less conservative.

4. Numerical Example

In this section, we will illustrate the validity of the proposed stability criterion through a numerical example.

Consider a local field neural network of the form (2) with $W_2=I$, where the parameters are from [31]

$$\begin{array}{ccc}\hfill A& =& diag\{1.2769,0.6231,0.9230,0.4480\}\hfill \\ \hfill W_0& =& \left[\begin{array}{cccc}-0.0373& 0.4852& -0.3351& 0.2336\\ -1.6033& 0.5988& -0.3224& 1.2352\\ 0.3394& -0.086& -0.3824& -0.5785\\ -0.1311& 0.3253& -0.9534& -0.5015\end{array}\right]\hfill \\ \hfill W_1& =& \left[\begin{array}{cccc}0.8674& -1.2405& -0.5325& 0.022\\ 0.0474& -0.9164& 0.0360& 0.9816\\ 1.8495& 2.6117& -0.3788& 0.8428\\ -2.0413& 0.5179& 1.1734& -0.2775\end{array}\right]\hfill \\ \hfill L^{-}& =& diag\{0,0,0,0\}\hfill \\ \hfill L^{+}& =& diag\{0.1137,0.1279,0.7994,0.2368\}.\hfill \end{array}$$

It should be pointed out that the cited references in

Table 2. MAUBs of h for different $\mu$.

$\mathit{\mu }$	0.1	0.5	0.9	NVs
Th.3, [9]	3.9337	3.5307	3.2627	$42n^2+27n$
Th. 3, [21]	4.4167	3.5986	3.3755	$79n^2+15n$
Th. 1, [10]	4.5086	3.8091	3.2895	$153n^2+22n$
Pr.1 [22]	4.5382	3.9313	3.4763	$60n^2+22n$
Pr. 3, [11] ($N=3$)	4.5468	4.0253	3.6246	$198n^2+26n$
Th. 1, [31] ($S=0$)	4.4924	3.7680	3.2251	$44.5n^2+22.5n$
Th 1, [31] ($S=1$)	4.5426	3.9438	3.4688	$83.5n^2+26.5n$
Th 1, [31] ($S=2$)	4.5470	3.9749	3.5052	$112.5n^2+28.5n$
Th.1 ($S=0,\gamma =0$)	4.7417	3.9507	3.4291	$44.5n^2+22.5n$
Th 1 ($S=1,\gamma =0$)	4.8154	4.1028	3.5174	$83.5n^2+26.5n$
Th 1 ($S=2,\gamma =0$)	4.9049	4.2145	3.7101	$112.5n^2+28.5n$
Th 1 ($S=2,\gamma =-0.02$)	4.9204	4.3411	3.8517	$112.5n^2+28.5n$
Th 1 ($S=2,\gamma =-0.05$)	4.9517	4.3741	3.9144	$112.5n^2+28.5n$

means that the maximum allowable upper bounds (MAUBs) are obtained by using the methods in the mentioned references. The MAUBs of the delay amplitude calculated for different results and different $\mu$ values are listed in Table 2. It is shown in Table 2 that a larger bound can be obtained by using Theorem 2, which confirms that the conservatism of the results obtained in this paper is lower than the existing results [9,10,11,21,22,31]. Comparing Theorem 1 of this paper with the results of [31], it can be seen that CMBII is effective in reducing conservatism. Moreover, it can further seen that $\gamma$ can increase the flexibility of the stability condition. It should be pointed out the CMBII can be also applied to the synchronization of time-varying neural networks such as [32,33], which have not considered the delay derivative information.

5. Conclusions

This paper studied the problem of a stability analysis of time-varying delay neural networks. By considering delay derivatives in integral inequalities, composite-matrix-based integral inequalities (CMBIIs) have been proposed. Using a CMBII, a new stability condition for neural networks with time-varying delay has been proposed. Among them, all augmented vectors and their derivatives in LKF are included in the CMBII and fully coupled to the delay amplitude and delay derivative through the slack matrix, resulting in less conservative results. The advantages of the proposed criterion are verified by numerical examples.