Estimation of the Domain of Attraction on Controlled Nonlinear Neutral Complex Networks via Razumikhin Approach

Abstract

This paper is devoted to dealing with the issue of the estimation of the domain of attraction (DOA) for highly nonlinear neutral complex networks (HNNCNs) with time delays. Firstly, by the Razumikhin approach, we establish several novel lemmas on the estimation of DOA for highly nonlinear neutral differential systems. The cases of bounded non-differentiable delays and unbounded proportional delays are discussed, respectively. Subsequently, by utilizing the proposed lemmas, combining the Lyapunov stability theory and inequality technique, the estimation of DOA on HNNCNs with bounded delays or proportional delays is derived when the chosen control gain is sufficiently large. If initial values start from DOA, then the states of systems will exponentially or polynomially converge to the equilibrium point, which means that the local exponential or polynomial synchronization of HNNCNs is realized. Additionally, the weighted outer-coupling matrix of complex networks is not required to be symmetric, which means that the derived results can be applied to both the undirected networks and directed networks. Finally, several numerical examples are provided to illustrate the feasibility of theoretical findings.

1. Introduction

Complex networks (CNs) are composed of a great deal of nodes and complex edge associations, which can accurately simulate the practical systems such as the World Wide Web [1], air transportation networks [2], power-communication interdependent networks [3], and so on. With the rapid development of information technology, the applications of complex networks have been becoming more and more extensive in many fields including road traffic systems [4], power grids [5], and disease research [6]. Recently, great efforts have been made to investigate the CNs and fruitful work [7,8,9,10,11] about synchronization, pattern identification, statistical characteristics, and tracking control have sprung up.

Synchronization as a kind of important dynamic connective behavior has attracted widespread attention, which can be appropriately used to explain some natural phenomena. For a complex network (CN), global synchronization means that the states of each node eventually adapt to a common dynamical behavior for arbitrary initial values, and local synchronization means that the states of each node eventually adjust to a common dynamical behavior when initial values belong to a bounded domain. In order to realize global synchronization, it is necessary to impose control over addressed systems, and some control strategies are developed, such as feedback control [12], adaptive control [13], impulsive control [14], event-triggered control [15], and PID control [16]. Nevertheless, when the nonlinear node system has strong high-order nonlinearity, this network system cannot achieve global synchronization, even if the external controller is adopted and the chosen control gain is sufficiently large. Under this circumstance, the local synchronization is introduced and some results with respect to the local synchronization of CNs were acquired in [17,18,19,20,21,22]. From the aforementioned existing literature [17,18,19], it can be known that the strong nonlinearity is considered relatively rarely, and the influence of initial values on synchronization realization is also ignored. To solve this issue, the estimation of the domain of attraction (DOA) of a highly nonlinear network system is significant to achieve local synchronization.

Additionally, the estimation of DOA also plays an important role in the stability analysis and control of dynamic systems. In general, DOA can be classified into two categories. One kind originates from the invariant set. Under this case, if the solution of a system starting from an arbitrary initial state eventually converges to a set, then this set is called an attractive set, which can also be viewed as an invariant set. A body of work about attractive sets have been reported in [23,24,25,26].

The other kind of DOA greatly depends on the initial values of systems. If the trajectories of a system with its initial values from a certain region asymptotically converge to an equilibrium, periodic orbit, or chaos attractor, then this region is called DOA. Recently, numerous scholars have focused their attention on the estimation of DOA. For instance, in [27,28], the robust estimations of DOA for polynomial systems have been given with the help of parameter-dependent Lyapunov functions, where all possible uncertainties have been considered. In [29], the estimation of DOA for a controlled nonlinear system with polynomial dynamics and input constraints has been computed by solving an infinite-dimensional convex linear programming issue. Moreover, the estimations of DOA are further explored in nonlinear time-delay systems [30,31,32] by constructing a Lyapunov–Krasovskii functional.

Additionally, the estimation of DOA is investigated in some applicable systems such as the human respiratory model [33] and complex networks [34,35,36,37,38]. Particularly, in [34,35], based on the estimations of DOA, several algorithms were proposed to identify one appropriate driver node or an optimal set of driver nodes of CNs. In [36], combining a decomposition technique and Lyapunov stability theory, the estimation of DOA for the load balancing of a networked system has been discussed and the relationship between DOA and network topology has also been revealed. More recently, in [37], the estimation of the DOA of CNs without time delays has been explored, and the findings were further generalized to the controlled CNs with time delays in virtue of the Razumikhin approach in [38], where both delay-dependent DOA and delay-independent DOA have been analyzed.

It can be observed that the local synchronization [17,18] and estimation of DOA [30,31,32] have been considered based on the approach of Lyapunov–Krasovskii functional. Actually, the appropriate Lyapunov–Krasovskii functionals are difficult to find, and some sufficient conditions cannot be easily verified. Various classes of CNs are defined by some neutral-type functional differential equations involving neutral time delays. Such network models are called neutral-type CNs, where the time derivative of the states of each node involve time delay. Although the dynamic properties of neutral complex networks such as stability, periodic solution, and synchronization have been sufficiently analyzed in [39,40,41,42], the research into DOA for highly nonlinear neutral complex networks has not been examined. Since neutral term and high-order nonlinearity are included, the theory proposed in [38] by utilizing the Razumikhin approach cannot be applicable. Additionally, almost all literature only considers the estimation of DOA for systems with bounded time delays, and the case of unbounded proportional delays can be explored comparatively rarely.

Inspired by the above discussions, this paper aims to estimate the second category of DOA for highly nonlinear neutral complex networks (HNNCNs) with time delays. The main contributions of this paper are highlighted as follows.

(1) In contrast to the existing literature [37,38], we deal with the DOA issue of HNNCNs. Neutral term, high-order nonlinearity, non-differentiable bounded delays, and unbounded proportional delays are taken into account, which makes the addressed system in our paper more general.

(2) In virtue of the Razumikhin approach, two novel lemmas on the estimation of DOA for highly nonlinear neutral differential systems with bounded non-differentiable delays or unbounded proportional delays are established, respectively. Compared with the results proposed in [30,31,32], our findings are not required to choose the complicated Lyapunov–Krasovskii functional, and the acquired conditions are easier to verify. Meanwhile, since the neutral term is incorporated, Lemma 2 in [38] cannot be applied directly to this circumstance. We further develop the theory work [37,38] to neutral differential systems.

(3) By combining the presented lemmas and inequality technique, some estimations of DOA by HNNCNs with bounded non-differentiable delays or proportional delays are provided, and the local exponential synchronization or polynomial synchronization of HNNCNs is realized. Meanwhile, the convergent rate of the local synchronization error is accurately calculated and appropriate control gain is designed. Furthermore, the weighted outer-coupling matrix of complex networks may be chosen to be asymmetric, which makes the derived findings applicable to more general CNs.

The structure of this article is arranged as below. In Section 2, a class of highly nonlinear neutral complex networks are introduced, and some definitions about synchronization and DOA and one assumption are provided. Moreover, based on Razumikhin approach, several lemmas about the DOA estimation are established for neutral nonlinear differential equations. In Section 3, by utilizing the presented lemmas, some criteria on the DOA estimation of HNNCNs with bounded delays or proportional delays are derived. In Section 4, two numerical simulation examples are given to verify the effectiveness of the theoretical findings. Conclusions and future work are summarized in the last section.

Notations. Throughout this article, ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ denote the set of n-dimensional real column vectors and the set of the $n\times m$-dimensional real matrices, respectively. ${\mathbb{R}}_{+}$ indicates the set of non-negative real numbers. $\Vert ·\Vert$ means the two-norm of a vector or a matrix. ${\mathit{I}}_n$ represents an n-dimensional identity matrix. ${\lambda }_{\max }\left(P\right)$ denotes the maximum eigenvalue of symmetric matrix P. For a continuous function $\mathrm{\Psi }(t)$, its norm is represented by ${∥\mathrm{\Psi }(t)∥}_{t,{\delta }_0}=\underset{\xi \in [t-{\delta }_0,t]}{sup}\Vert \mathrm{\Psi }(\xi )\Vert$. The Kronecker product of matrices $A_1$ and $A_2$ is expressed as $A_1\otimes A_2$.

2. Preliminaries and Definitions

Consider a class of highly nonlinear neutral complex network system (HNNCNS) composed of N nodes

$$\left\{\begin{array}{cc}\hfill \frac{d[y_i(t)-D{y}_i(t-\delta (t))]}{dt}=& F(y_i(t),y_i(t-\delta (t)))+l\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }g(y_j(t-\delta (t)))+U_i(t),t\ge t_0,\hfill \\ \hfill y_i(s)=& {\zeta }_i(s),s\in [t_0-{\delta }_0,t_0],1\le i\le N,\hfill \end{array}\right.$$

(1)

where $y_i(·)\in {\mathbb{R}}^n$ expressed by the state vector of the ith node; $F:{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $g:{\mathbb{R}}^n\to {\mathbb{R}}^n$ are continuous differentiable nonlinear functions; $l>0$ stands for delayed coupling strength; $\mathrm{\Gamma }=({\gamma }_{ij})\in {\mathbb{R}}^{n\times n}$ is an inner-coupling matrix; $Q={(q_{ij})}_{N\times N}$ denotes the weighted outer-coupling matrix; $D\in {\mathbb{R}}^{n\times n}$ is a neutral delayed connection weight matrix. $\delta (t)$ denotes the time-varying delay. $U_i(t)$ represents the designed controller. ${\zeta }_i(s)\in C([t_0-{\delta }_0,t_0],{\mathbb{R}}^n)$ denotes the initial value of system (1). If there is a connection from node i to node $j(j\ne i)$, then $q_{ij}>0$; otherwise, $q_{ij}=0$. Moreover, the elements of coupled matrix Q satisfy $q_{ii}=-{\sum }_{j=1,j\ne i}^N{q}_{ij}$.

Let vector $\vartheta =({\theta }_1,{\theta }_2,\dots ,{\theta }_n)\in {\mathbb{R}}^n$ be an equilibrium point of the following isolated node system

$$\frac{d[y(t)-Dy(t-\delta (t))]}{dt}=F(y(t),y(t-\delta (t))).$$

(2)

According to the definition of the equilibrium point $\vartheta$, we correspondingly obtain the following equation

$$F(\vartheta ,\vartheta )+l\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }g(\vartheta )=F(\vartheta ,\vartheta )=0.$$

(3)

Obviously, the synchronized state $y_1=y_2=\dots =y_N=\vartheta$ is also an equilibrium point of system (1) when $U_i(t)=0$. The controller $U_i(t)$ is designed as $U_i(t)=-c{z}_i(t)$, in which $z_i(t)=y_i(t)-\vartheta$ denotes the synchronization error and $c>0$ represents the control gain. Let $\mathrm{\Theta }={[{\vartheta }^T,{\vartheta }^T,\dots ,{\vartheta }^T]}^T\in {\mathbb{R}}^{nN}$ and $z(t)={[z_1^T(t),z_2^T(t),\dots ,z_N^T(t)]}^T\in {\mathbb{R}}^{nN}.$ Subsequently, subtracting Equation (3) from Equation (1) yields the following system:

$$\left\{\begin{array}{cc}\hfill \frac{d[z_i(t)-D{z}_i(t-\delta (t))]}{dt}=& F(y_i(t),y_i(t-\delta (t)))-F(\vartheta ,\vartheta )\hfill \\ \hfill & +l\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }[g(y_j(t-\delta (t)))-g(\vartheta )]+U_i(t),t\ge t_0,\hfill \\ \hfill z_i(s)=& {\zeta }_i(s)-\vartheta ={\varphi }_i(s),s\in [t_0-{\delta }_0,t_0],1\le i\le N.\hfill \end{array}\right.$$

(4)

For the above system, let $J_1=\frac{𝜕F(x,y)}{𝜕x}{|}_{(\vartheta ,\vartheta )}$, $J_2=\frac{𝜕F(x,y)}{𝜕y}{|}_{(\vartheta ,\vartheta )}$ and $J_3=\frac{𝜕g(x)}{𝜕x}{|}_{x=\vartheta }$ be the Jacobian matrices of F and g, respectively. Therefore, the synchronization error system (4) can be rewritten as:

$$\begin{array}{cc}\hfill \frac{d[z_i(t)-D{z}_i(t-\delta (t))]}{dt}=& [J_1-c{I}_n]z_i(t)+J_2{z}_i(t-\delta (t))+{\mathrm{\Phi }}_{(1)i}(t)\hfill \\ \hfill & +l\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }{J}_3{z}_j(t-\delta (t))+l\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }{\mathrm{\Phi }}_{(2)j}(t-\delta (t)),\hfill \end{array}$$

(5)

where

$$\begin{array}{cc}\hfill & {\mathrm{\Phi }}_{(1)i}(t)=F(y_i(t),y_i(t-\delta (t)))-F(\vartheta ,\vartheta )-J_1{z}_i(t)-J_2{z}_i(t-\delta (t)),\hfill \\ \hfill & {\mathrm{\Phi }}_{(2)i}(t)=g(y_i(t-\delta (t)))-g(\vartheta )-J_3{z}_i(t-\delta (t)).\hfill \end{array}$$

Suppose that functions ${\psi }_i:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ with ${\psi }_i(0)=0,1\le i\le 3$ are nondecreasing and continuous. Meanwhile, ${\mathrm{\Phi }}_{(1)i}(t),{\mathrm{\Phi }}_{(2)i}(t)$ satisfy the following conditions:

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{(1)i}^T(t){\mathrm{\Phi }}_{(1)i}(t)& \le {\psi }_1(\Vert z_i(t)\Vert )z_i^T(t)z_i(t)+{\psi }_2(\Vert z_i(t){\Vert }_{t,\delta (t)})z_i^T(t-\delta (t))z_i(t-\delta (t)),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{(2)i}^T(t){\mathrm{\Phi }}_{(2)i}(t)& \le {\psi }_3(\Vert z_i(t){\Vert }_{t,\delta (t)})z_i^T(t-\delta (t))z_i(t-\delta (t)).\hfill \end{array}$$

(7)

Remark 1.

For network system (1), since functions $F(x(t),x(t-\delta (t)))$ and $g(x(t-\delta (t))$ are continuous and differentiable, they satisfy local the Lipschitz condition, which guarantees the existence and uniqueness of the local solution of the network system (1). In order to overcome the difficulties introduced by high-order nonlinearity, some necessary assumptions on nonlinear functions $F(x(t),x(t-\delta (t)))$ and $g(x(t-\delta (t))$ are made in inequalities (6) and (7).

Definition 1.

Complex network (1) is said to achieve global exponential synchronization, if positive constants $K_0$ and ${\eta }_1$ exist such that

$$\Vert z(t;\varphi )\Vert =\Vert y(t;\zeta )-\mathrm{\Theta }\Vert \le K_0{e}^{-{\eta }_1(t-t_0)},$$

where $\zeta ={[{\zeta }_1^T,{\zeta }_2^T,\dots ,{\zeta }_N^T]}^T$, ${\zeta }_i\in C([t_0-{\delta }_0,t_0],{\mathbb{R}}^n)$, $i=1,2,\dots ,N$.

Definition 2.

Complex network (1) is said to achieve local exponential synchronization, if positive constants $K_0,{\eta }_1$ and r exist such that

$$\Vert z(t;\varphi )\Vert =\Vert y(t;\zeta )-\mathrm{\Theta }\Vert \le K_0{e}^{-{\eta }_1(t-t_0)},$$

where $\zeta ={[{\zeta }_1^T,{\zeta }_2^T,\dots ,{\zeta }_N^T]}^T$, ${\Vert \zeta -\mathrm{\Theta }\Vert }_{t_0,{\delta }_0}<r$, ${\zeta }_i\in C([t_0-{\delta }_0,t_0],{\mathbb{R}}^n)$, $i=1,2,\dots ,N$.

Definition 3.

Complex network (1) is said to achieve global polynomial synchronization, if positive constants $K_0$ and ${\eta }_1$ exist such that

$$\Vert z(t;\varphi )\Vert =\Vert y(t;\zeta )-\mathrm{\Theta }\Vert \le K_0{(1+t)}^{-{\eta }_1},$$

where $\zeta ={[{\zeta }_1^T,{\zeta }_2^T,\dots ,{\zeta }_N^T]}^T$, ${\zeta }_i\in C([t_0-{\delta }_0,t_0],{\mathbb{R}}^n)$, $i=1,2,\dots ,N$.

Definition 4.

Complex network (1) is said to achieve local polynomial synchronization, and if there exists positive constants $K_0,{\eta }_1$ and r such that

$$\Vert z(t;\varphi )\Vert =\Vert y(t;\zeta )-\mathrm{\Theta }\Vert \le K_0{(1+t)}^{-{\eta }_1},$$

where $\zeta ={[{\zeta }_1^T,{\zeta }_2^T,\dots ,{\zeta }_N^T]}^T$, ${\Vert \zeta -\mathrm{\Theta }\Vert }_{t_0,{\delta }_0}<r$, ${\zeta }_i\in C([t_0-{\delta }_0,t_0],{\mathbb{R}}^n)$, $i=1,2,\dots ,N$.

Definition 5

([38]). A set $R_D$ is said to be a DOA if all the trajectories of network (1) converge to the equilibrium point Θ for the initial values $\zeta (s)={[{\zeta }_1^T(s),{\zeta }_2^T(s),\dots ,{\zeta }_N^T(s)]}^T\in R_D$, then we call it a DOA, which is equivalent to

$$R_D=\left\{\zeta \right|\underset{t\to \infty }{lim}\Vert y(t;\zeta )-\mathrm{\Theta }\Vert =0\}.$$

In general, some difficulties are encountered during the process of estimation of DOA. For simplicity, we may choose the ball-like set or cube-like set as a DOA. In this article, we attempt to find an open ball-like subset of $R_D$ as described below

$$R_0(r)=\{\zeta \in R_D{|\Vert \zeta -\mathrm{\Theta }\Vert }_{t_0,{\delta }_0}<r\},$$

(8)

which implies that the states of each node converge to the equilibrium point $\mathrm{\Theta }$ and synchronization is achieved when the upper bound of the initial error $\zeta -\mathrm{\Theta }$ is less than r. Correspondingly, the upper bound r is called the radius of the ball-like set $R_0(r)$. This paper intends to seek the maximum radius $r_0.$

In what follows, several necessary assumptions and lemmas are presented to derive the main results.

Assumption 1.

Suppose that there is a constant $k\in (0,1)$ such that $\Vert D\Vert \le k,$ which means that

$$|Dx(t-\delta (t))|\le k|x(t-\delta (t))|.$$

According to Lemma 2 in the literature [43], by choosing $p=2$, we immediately acquire the following lemma.

Lemma 1.

Let assumption 1 hold and $x(t)$ denote a solution of Equation (1). If

$$e^{\eta (t-t_0)}{|x(t)-Dx(t-\delta (t))|}^2\le {(1+k)}^2\underset{t_0-{\delta }_0\le \theta \le t_0}{sup}{\left|x(\theta )\right|}^2,$$

where $0<\eta <-\frac{2}{{\delta }_0}lnk$ for $\forall t\ge t_0$, then

$$e^{\eta (t-t_0)}{\left|x(t)\right|}^2\le \frac{{(1+k)}^2}{{[1-k{(e^{\eta {\delta }_0})}^{\frac{1}{2}}]}^2}\underset{t_0-{\delta }_0\le \theta \le t_0}{sup}{\left|x(\theta )\right|}^2.$$

Proof. 

According to the inequality ${|a+b|}^2\le {(1+ϵ)(|a|}^2+\frac{{\left|b\right|}^2}{ϵ}),0<ϵ<1$, we can obtain that

$$\begin{array}{c}\hfill {\left|x(t)\right|}^2\le \frac{1}{1-ϵ}{|x(t)-Dx(t-\delta (t))|}^2+\frac{k^2}{ϵ}\underset{t_0-{\delta }_0\le \theta \le t_0}{sup}{\left|x(\theta )\right|}^2.\end{array}$$

(9)

Choose a positive constant $ϵ$ satisfying $k^2{e}^{\eta {\delta }_0}<ϵ<1.$ Noting that $e^{\eta (t-t_0)}{|x(t)-Dx(t-\delta (t))|}^2\le {(1+k)}^2\underset{t_0-{\delta }_0\le \theta \le t_0}{sup}{\left|x(\theta )\right|}^2,$ we can calculate that

$$\begin{array}{cc}\hfill \underset{t_0\le s\le t}{sup}(e^{\eta (t-t_0)}{\left|x(s)\right|}^2)\le & \frac{1}{1-ϵ}\underset{t_0\le s\le t}{sup}(e^{\eta (s-t_0)}{|x(s)-Dx(s-\delta (s))|}^2)\hfill \\ \hfill & +\frac{k^2}{ϵ}\underset{t_0\le s\le t}{sup}(e^{\eta (s-t_0)}\underset{s-{\delta }_0\le \theta \le s}{sup}{\left|x(\theta )\right|}^2)\hfill \\ \hfill & \le \frac{{(1+k)}^2}{1-ϵ}\underset{t_0-{\delta }_0\le \theta \le t_0}{sup}{\left|x(\theta )\right|}^2+\frac{k^2{e}^{\eta {\delta }_0}}{ϵ}\underset{t_0-{\delta }_0\le s\le t}{sup}(e^{\eta (s-t_0)}{\left|x(t)\right|}^2),\hfill \end{array}$$

which implies that

$$\underset{t_0-{\delta }_0\le s\le t}{sup}(e^{\eta (t-t_0)}{\left|x(s)\right|}^2)\le \frac{{(1+k)}^2}{1-ϵ}\underset{t_0-{\delta }_0\le \theta \le t_0}{sup}{\left|x(\theta )\right|}^2+\frac{k^2{e}^{\eta {\delta }_0}}{ϵ}\underset{t_0-{\delta }_0\le s\le t}{sup}(e^{\eta (s-t_0)}{\left|x(t)\right|}^2).$$

By taking $ϵ={(k^2{e}^{\eta {\delta }_0})}^{\frac{1}{2}}$, we immediately acquire that

$$\underset{t_0-{\delta }_0\le s\le t}{sup}(e^{\eta (t-t_0)}{\left|x(s)\right|}^2)\le \frac{{(1+k)}^2}{{[1-k{(e^{\eta {\delta }_0})}^{\frac{1}{2}}]}^2}\underset{t_0-{\delta }_0\le \theta \le t_0}{sup}{\left|x(\theta )\right|}^2.$$

□

Since a neutral term is introduced, we cannot directly establish the Halanay inequality to deal with the estimation of DOA of network (1) similar to the literature [38]. In order to solve this issue, the Razumikhin approach is developed and one novel lemma is derived.

Lemma 2.

Let Assumption 1 hold. Suppose that $H_1:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a nonincreasing continuous function while $H_2:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a nondecreasing continuous function. Let $V:{\mathbb{R}}^n\to [0,+\infty )$ be a differentiable function that satisfies

$${\alpha }_1{\left|x\right|}^2\le V(x)\le {\alpha }_2{\left|x\right|}^2,{\alpha }_2\ge {\alpha }_1>0.$$

$x(t)$ is the solution of the equation

$$\frac{d[x-Dx(t-\delta (t))]}{dt}=f(x(t),x(t-\delta (t)),t).$$

(10)

Let $\varpi (t)=x(t)-Dx(t-\delta (t)),\omega (t)=V(\varpi (t))$ and $u(t)=V(x(t)).$ Functions $\omega (t)$ and $u(t)$ satisfy the following inequality

$$\dot{\omega }(t)\le [-H_1(u(t))+H_2{(\Vert u(t)\Vert }_{t,\delta (t)})]\omega (t),$$

(11)

for all $t\ge t_0$ when $u(t+s)<q\omega (t)$, $s\in [-{\delta }_0,0],$ where $\overline{q}>{(1-k)}^{-2}$, $\overline{q}=\frac{{\alpha }_{1}q}{{\alpha }_2}$. If $H_1(0)>H_2(0)$ and $M={\Vert u(t)\Vert }_{t_0,{\delta }_0}<\frac{r_0}{M_0}$, then there exists a sufficiently small positive constant ${\eta }_1$ such that

$$u(t)\le r_0{e}^{-{\eta }_1(t-t_0)},t\ge t_0,$$

(12)

where a series of parameters satisfy $r_0=sup\left\{r\right|H_1(r)>H_2(r)\}$, $M_0=\frac{{\alpha }_2^2{(1+k)}^2}{{\alpha }_1^2{(1-k)}^2}$, $r_1=M_{0}M$, $\lambda =H_1(r_1)-H_2(r_1)>0$ and ${\eta }_1\in \left(0,min\left\{\lambda ,\frac{1}{{\delta }_0}ln\left[\frac{\overline{q}}{{(1+\sqrt{\overline{q}}k)}^2}\right]\right\}\right)$.

Proof. 

Since $r_1=M_{0}M<r_0$, we can choose two small enough constants ${\epsilon }_0>0$ and ${\eta }_1\in \left(0,min\left\{\lambda ,\frac{1}{{\delta }_0}ln\left[\frac{q}{{(1+\sqrt{q}k)}^2}\right]\right\}\right)$ satisfying

$$H_1(r_2)-H_2(r_2)>{\eta }_1,$$

(13)

and

$$\frac{{\alpha }_2^2{(1+k)}^2}{{\alpha }_1^2{[1-k{(e^{{\eta }_1{\delta }_0})}^{\frac{1}{2}}]}^2}\le M_0+{\epsilon }_0,$$

(14)

where $r_2=(M_0+{\epsilon }_0)M<r_0$. Now, we will claim that

$$e^{{\eta }_1(t-t_0)}\omega (t)\le \frac{{\alpha }_2{(1+k)}^2}{{\alpha }_1}M,\forall t\ge t_0-{\delta }_0.$$

(15)

Assume that inequality (15) is not true, then there exists one constant $\rho >t_0$ such that

$$e^{{\eta }_1(t-t_0)}\omega (t)\le e^{{\eta }_1(\rho -t_0)}\omega (\rho )=\frac{{\alpha }_2{(1+k)}^2}{{\alpha }_1}M,\forall t\in [t_0,\rho ].$$

(16)

Accordingly, there exists a sufficient constant $h>0$ satisfying

$$e^{{\eta }_1(\rho +h-t_0)}\omega (\rho +h)-e^{{\eta }_1(\rho -t_0)}\omega (\rho )>0.$$

(17)

By employing Lemma 1, we have that

$$e^{{\eta }_1(t-t_0)}u(t)\le \frac{{\alpha }_2^2{(1+k)}^2}{{\alpha }_1^2{[1-k{(e^{{\eta }_1{\delta }_0})}^{\frac{1}{2}}]}^2}M=\frac{{\alpha }_2{e}^{{\eta }_1(\rho -t_0)}}{{\alpha }_1{[1-k{(e^{{\eta }_1{\delta }_0})}^{\frac{1}{2}}]}^2}\omega (\rho ),\forall t\in [t_0,\rho ].$$

(18)

In particular,

$$u(\rho +\theta )\le \frac{{\alpha }_2{e}^{{\eta }_1{\delta }_0}}{{\alpha }_1{[1-k{(e^{{\eta }_1{\delta }_0})}^{\frac{1}{2}}]}^2}\omega (\rho )<q\omega (\rho ),\forall \theta \in [-{\delta }_0,0].$$

(19)

which signifies that the inequality (11) holds at $t=\rho$. In virtue of Lemma 1 and Equation (16), it follows that $u(t)\le \frac{{\alpha }_2^2{(1+k)}^2}{{\alpha }_1^2{[1-k{(e^{{\eta }_1{\delta }_0})}^{\frac{1}{2}}]}^2}M{e}^{-{\eta }_1(t-t_0)}\le (M_0+{\epsilon }_0)M=r_2,t\in [t_0,\rho ]$. When $t=\rho +h$, by the Newton–Leibniz formula, we can obtain that

$$\begin{array}{cc}\hfill & e^{{\eta }_1(\rho +h-t_0)}\omega (\rho +h)-e^{{\eta }_1(\rho -t_0)}\omega (\rho )\hfill \\ \hfill =& {\int }_{\rho }^{\rho +h}{\left[e^{{\eta }_1(s-t_0)}\omega (s)\right]}^{\prime }ds\hfill \\ \hfill =& {\int }_{\rho }^{\rho +h}{e}^{{\eta }_1(s-t_0)}[{\eta }_1\omega (s)+\dot{\omega }(s)]ds\hfill \\ \hfill \le & {\int }_{\rho }^{\rho +h}{e}^{{\eta }_1(s-t_0)}{\eta }_1\omega (s)ds\hfill \\ \hfill & +{\int }_{\rho }^{\rho +h}{e}^{{\eta }_1(s-t_0)}\left(-H_1(u(s))+H_2{(\Vert u\Vert }_{t,\delta (t)})\right)\omega (s)ds\hfill \\ \hfill \le & {\int }_{\rho }^{\rho +h}{e}^{{\eta }_1(s-t_0)}\left[{\eta }_1\omega (s)+\left(-H_1(r_2)+H_2(r_2)\right)\omega (s)\right]ds\hfill \\ \hfill \le & 0,\hfill \end{array}$$

which yields a contradiction with Equation (17). Therefore, inequality (15) holds. Accordingly, we have that

$$\begin{array}{c}\hfill u(t)\le \frac{{\alpha }_2^2{(1+k)}^2}{{\alpha }_1^2{[1-k{(e^{{\eta }_1{\delta }_0})}^{\frac{1}{2}}]}^2}M{e}^{-{\eta }_1(t-t_0)}\le (M_0+{\epsilon }_0)M{e}^{-{\eta }_1(t-t_0)}\le r_0{e}^{-{\eta }_1(t-t_0)}.\end{array}$$

(20)

□

If D is a zero matrix, which implies that $k=0$, then the restrictive condition ${\alpha }_1{\left|x\right|}^2\le V(x)\le {\alpha }_2{\left|x\right|}^2$ can be removed. Similarly to the proof of Lemma 2, we immediately derive the following assertion.

Corollary 1.

Suppose that $H_1:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a nonincreasing continuous function while $H_2:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a nondecreasing continuous function. Let $V:{\mathbb{R}}^n\to [0,+\infty )$ be a differentiable function and $x(t)$ be the solution of the equation

$$\frac{dx}{dt}=f(x(t),x(t-\delta (t)),t).$$

(21)

Function $u(t)=V(x(t))$ satisfies the following inequality

$$\dot{u}(t)\le [-H_1(u(t))+H_2{(\Vert u\Vert }_{t,\delta (t)})]u(t),$$

(22)

for all $t\ge t_0$ when $u(t+s)<qu(t)$, $q>1$, $s\in [-{\delta }_0,0].$ If $H_1(0)>H_2(0)$ and $M={\Vert u\Vert }_{t_0,{\delta }_0}<r_0$, then there exists a sufficiently small positive constant ${\eta }_1$ such that

$$u(t)\le r_0{e}^{-{\eta }_1(t-t_0)},t\ge t_0,$$

(23)

where a series of parameters $r_0,\lambda ,{\eta }_1$ satisfy $r_0=sup\left\{r\right|H_1(r)>H_2(r)\}$, $\lambda =H_1(r_0)-H_2(r_0)$ and ${\eta }_1\in \left(0,min\left\{\lambda ,\frac{1}{{\delta }_0}lnq\right\}\right)$.

Remark 2.

In [38], a Halanay inequality is proposed based on the Razumikhin approach while the above corollary directly establishes the Razumikhin theory on the estimation of DOA for a class of time-delay system. Noting that $q>1$, the above result accords with the one in [38] in essence.

It can be observed that the considered delay $\delta (t)$ is bounded in Lemma 2. In what follows, we will discuss the case of the unbounded delay. This is especially true if $\delta (t)=(1-\sigma )t,0<\sigma <1$ is a proportional delay, combining with the results in [44], then we can acquire the following assertion by choosing $q=2$.

Lemma 3.

Let Assumption 1 hold. If $x(t)$ satisfies

$${(1+t)}^{\eta }{|x(t)-Dx(\sigma t)|}^2\le \frac{{\alpha }_2}{{\alpha }_1}{(1+k)}^2{(1+t_0)}^{\eta }\underset{\sigma t_0\le s\le t_0}{sup}{\left|x(s)\right|}^2,$$

where $t_0\ge 0$ and $0<\eta <\frac{2lnk}{ln\sigma }$, then

$${(1+t)}^{\eta }{\left|x(t)\right|}^2\le \frac{{\alpha }_2{(1+k)}^2{(1+t_0)}^{\eta }}{{\alpha }_1{[1-k{(\frac{1}{\sigma })}^{\frac{\eta }{2}}]}^2}\underset{\sigma t_0\le s\le t_0}{sup}{\left|x(s)\right|}^2.$$

Proof. 

According to the inequality ${|a+b|}^2\le {(1+ϵ)(|a|}^2+\frac{{\left|b\right|}^2}{ϵ}),0<ϵ<1$, we can obtain that

$${(1+t)}^{\eta }{\left|x(t)\right|}^2\le {(1+t)}^{\eta }\left(\frac{{|x(t)-Dx(\sigma t)|}^2}{1-ϵ}+\frac{k^2|x(\sigma (t){|}^2)}{ϵ}\right).$$

Noting that

$${(1+t)}^{\eta }{|x(t)-Dx(\sigma t)|}^2\le \frac{{\alpha }_2}{{\alpha }_1}{(1+k)}^2{(1+t_0)}^{\eta }\underset{\sigma t_0\le s\le t_0}{sup}{\left|x(s)\right|}^2,$$

we can calculate that

$$\begin{array}{cc}\hfill \underset{t_0\le s\le t}{sup}{((1+t)}^{\eta }{\left|x(s)\right|}^2)\le & \frac{1}{1-ϵ}\underset{t_0\le s\le t}{sup}{((1+t)}^{\eta }{|x(s)-Dx(\sigma s)|}^2)\hfill \\ \hfill & +\frac{k^2}{ϵ}\underset{t_0\le s\le t}{sup}{((1+t)}^{\eta }\underset{\sigma t_0\le \theta \le t}{sup}{\left|x(\theta )\right|}^2)\hfill \\ \hfill \le & \frac{{\alpha }_2}{{\alpha }_1}{(1+t)}^{\eta }\frac{{(1+k)}^2{(1+t_0)}^{\eta }\underset{\sigma t_0\le s\le t_0}{sup}{\left|x(s)\right|}^2}{1-ϵ}\hfill \\ \hfill & +\frac{k^2}{ϵ}{(\frac{1+t}{1+\sigma t})}^{\eta }\underset{\sigma t_0\le s\le t_0}{sup}{(1+s)}^{\eta }{\left|x(s)\right|}^2,\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill \underset{\sigma t_0\le s\le t}{sup}{((1+t)}^{\eta }{\left|x(s)\right|}^2)\le & \frac{{\alpha }_2}{{\alpha }_1}{(1+t)}^{\eta }\frac{{(1+k)}^2{(1+t_0)}^{\eta }\underset{\sigma t_0\le s\le t_0}{sup}{\left|x(s)\right|}^2}{1-ϵ}\hfill \\ \hfill & +\frac{k^2}{ϵ}{(\frac{1+t}{1+\sigma t})}^{\eta }\underset{\sigma t_0\le s\le t_0}{sup}{(1+s)}^{\eta }{\left|x(s)\right|}^2,t\ge t_0.\hfill \end{array}$$

By taking $ϵ=k{(\frac{1}{\sigma })}^{\frac{\eta }{2}},$ we immediately acquire that

$$\begin{array}{cc}\hfill \underset{\sigma t_0\le s\le t}{sup}{((1+t)}^{\eta }{\left|x(s)\right|}^2)\le & \frac{{\alpha }_2{(1+k)}^2{(1+t_0)}^{\eta }}{{\alpha }_1{[1-k{(\frac{1}{\sigma })}^{\frac{\eta }{2}}]}^2}\underset{\sigma t_0\le \theta \le t_0}{sup}{\left|x(\theta )\right|}^2.\hfill \end{array}$$

□

Lemma 4.

Let Assumption 1 hold. Suppose that functions $H_1$, $H_2$ and V satisfy the same conditions as Lemma 2. $x(t)$ is the solution of the equation

$$\frac{d[x-Dx(\sigma t)]}{dt}=f(x(t),x(\sigma t),t).$$

(24)

Let $\varpi (t)=x(t)-Dx(\sigma t),\omega (t)=V(\varpi (t))$ and $u(t)=V(x(t)).$ Functions $\omega (t),u(t)$ and $\tilde{u}(t)$ satisfy the following differentiable inequality

$$\dot{\omega }(t)\le [-H_1(u(t))+H_2(\tilde{u}(t))]\omega (t),t\ge t_0,$$

(25)

when $u(t+s)<q\omega (t),s\in [(\sigma -1)t,0]$, where $\tilde{u}(t)={\Vert u\Vert }_{t,(1-\sigma )t}$ and $\overline{q}>{(1-k)}^{-2}$, $\overline{q}=\frac{{\alpha }_{1}q}{{\alpha }_2}$. If $H_1(0)>H_2(0)$ and $M={\Vert u\Vert }_{t_0,(1-\sigma )t_0}<\frac{r_0}{M_0}$, then there exists a sufficiently small positive constant ${\eta }_1$ such that

$$u(t)\le r_0{(1+t_0)}^{{\eta }_1}{(1+t)}^{-{\eta }_1},t\ge t_0,$$

(26)

where a series of parameters satisfy $r_0=sup\left\{r\right|H_1(r)-H_2(r)>0\}$, $M_0=\frac{{\alpha }_2^2{(1+k)}^2}{{\alpha }_1^2{(1-k)}^2}{(1+t_0)}^{{\eta }_1}$, $r_1=M_{0}M$, $\lambda =H_1(r_1)-H_2(r_1)$ and $0<{\eta }_1<min\left\{\lambda ,\frac{1}{ln\frac{1}{\sigma }}ln\left(\frac{\overline{q}}{{(1+k\sqrt{\overline{q}})}^2}\right)\right\}$.

Proof. 

In light of $r_1=M_{0}M<r_0$, we can also choose two small enough constants ${\epsilon }_0>0$ and

$${\eta }_1\in \left(0,min\left\{\lambda ,\frac{1}{ln\frac{1}{\sigma }}ln\left(\frac{q}{{(1+k\sqrt{q})}^2}\right)\right\}\right)$$

such that

$$H_1(r_2)-H_2(r_2)>{\eta }_1,$$

(27)

and

$$\frac{{\alpha }_2^2{(1+k)}^2{(1+t_0)}^{{\eta }_1}}{{\alpha }_1^2{[1-k{(\frac{1}{\sigma })}^{\frac{{\eta }_1}{2}}]}^2}\le M_0+{\epsilon }_0,$$

(28)

where $r_2=(M_0+{\epsilon }_0)M<r_0$. In what follows, we will prove that

$${(1+t)}^{{\eta }_1}\omega (t)\le \frac{{\alpha }_2{(1+k)}^2{(1+t_0)}^{{\eta }_1}}{{\alpha }_1}M.$$

(29)

If the above inequality is not true, then there exists one constant $\rho >t_0\ge 0$ such that

$${(1+t)}^{{\eta }_1}\omega (t)\le {(1+\rho )}^{{\eta }_1}\omega (\rho )=\frac{{\alpha }_2{(1+k)}^2{(1+t_0)}^{{\eta }_1}}{{\alpha }_1}M,t\in [\sigma t_0,\rho ].$$

(30)

Correspondingly, there exists a time sequence of ${\left\{t_k\right\}}_{k\ge 1}$ satisfying $t_k\downarrow \rho$ and

$${(1+t_k)}^{{\eta }_1}\omega (t_k)-{(1+\rho )}^{{\eta }_1}\omega (\rho )>0.$$

(31)

By utilizing Lemma 3, we obtain that

$${(1+t)}^{{\eta }_1}u(t)\le {(1+t_0)}^{{\eta }_1}\frac{{\alpha }_2^2{(1+k)}^{2}M}{{\alpha }_1^2{[1-k{(\frac{1}{\sigma })}^{\frac{{\eta }_1}{2}}]}^2}=\frac{{\alpha }_2{(1+\rho )}^{{\eta }_1}\omega (\rho )}{{\alpha }_1{[1-k{(\frac{1}{\sigma })}^{\frac{{\eta }_1}{2}}]}^2}.$$

(32)

When $\theta \in [(\sigma -1)\rho ,0]$, we have that

$$\begin{array}{cc}\hfill u(\rho +\theta )\le & {(1+\rho +\theta )}^{-{\eta }_1}\frac{{\alpha }_2{(1+\rho )}^{{\eta }_1}\omega (\rho )}{{\alpha }_1{[1-k{(\frac{1}{\delta })}^{\frac{{\eta }_1}{2}}]}^2}\hfill \\ \hfill \le & {\left(\frac{1+\rho }{1+\rho +\theta }\right)}^{{\eta }_1}\frac{{\alpha }_2\omega (\rho )}{{\alpha }_1{[1-k{(\frac{1}{\delta })}^{\frac{{\eta }_1}{2}}]}^2}\hfill \\ \hfill \le & {(\frac{1}{\sigma })}^{{\eta }_1}\frac{{\alpha }_2}{{\alpha }_1{[1-k{(\frac{1}{\delta })}^{\frac{{\eta }_1}{2}}]}^2}\omega (\rho )\hfill \\ \hfill <& q\omega (\rho ).\hfill \end{array}$$

(33)

Thus, we have $\dot{\omega }(t)\le [-H_1(u(t))+H_2(\tilde{u}(t))]\omega (t)$, $t=\rho$. When $t\in [\sigma t_0,\rho ]$, it follows that

$$u(t)\le \frac{{\alpha }_2^2{(1+k)}^2{(1+t_0)}^{{\eta }_1}}{{\alpha }_1^2{[1-k{(\frac{1}{\sigma })}^{\frac{{\eta }_1}{2}}]}^2}M{(1+t)}^{-{\eta }_1}\le (M_0+{\epsilon }_0)M=r_2.$$

(34)

When $t=t_k$, by using the Newton–Leibniz formula, we can deduce that

$$\begin{array}{cc}\hfill & {(1+t_k)}^{{\eta }_1}\omega (t_k)-{(1+\rho )}^{{\eta }_1}\omega (\rho )\hfill \\ \hfill =& {\int }_{\rho }^{t_k}{\left[{(1+s)}^{{\eta }_1}\omega (s)\right]}^{\prime }ds\hfill \\ \hfill \le & {\int }_{\rho }^{t_k}{(1+s)}^{{\eta }_1}\left[\frac{{\eta }_1}{1+s}\omega (s)+\left(-H_1(u(s))+H_2(\tilde{u}(s))\right)\omega (s)\right]ds\hfill \\ \hfill \le & {\int }_{\rho }^{t_k}{(1+s)}^{{\eta }_1}\left[\frac{{\eta }_1}{1+s}\omega (s)-\left(H_1(r_2)-H_2(r_2)\right)\omega (s)\right]ds\hfill \\ \hfill \le & {\int }_{\rho }^{t_k}{(1+s)}^{{\eta }_1}\left(\frac{{\eta }_1}{1+s}-{\eta }_1\right)\omega (s)ds\hfill \\ \hfill \le & 0,\hfill \end{array}$$

which yields a contradiction, and inequality (29) holds. Therefore, one obtains

$$\begin{array}{cc}\hfill u(t)\le & \frac{{\alpha }_2^2{(1+k)}^2{(1+t_0)}^{{\eta }_1}{(1+t)}^{-{\eta }_1}M}{{\alpha }_1^2{[1-k{(\frac{1}{\sigma })}^{\frac{{\eta }_1}{2}}]}^2}\hfill \\ \hfill \le & r_0{(1+t_0)}^{{\eta }_1}{(1+t)}^{-{\eta }_1}.\hfill \end{array}$$

(35)

□

Remark 3.

Lemma 2 and Lemma 4 give the estimations of DOA of neutral differential systems with bounded time-varying delays or unbounded proportional delays in the light of the Razumikhin approach. Compared with the results proposed in [31], our findings are not required to choose the complicated Lyapunov–Krasovskii functional, and the acquired conditions are easier to verify. Meanwhile, since the neutral term is incorporated, Lemma 2 in [38] cannot be directly applied to this circumstance. We further develop the theory work [37,38] to neutral differential systems.

Lemma 5.

([45]). Let $P_1,P_2\in {\mathbb{R}}^n$ be n-dimensional vectors. The following inequality holds.

$$2P_1^T{P}_2\le P_1^T\Lambda P_1+P_2^T{\Lambda }^{-1}{P}_2,$$

where $\Lambda \in {\mathbb{R}}^{n\times n}$ denotes arbitrary positive definite matrix.

3. Main Results

In this section, based on the established lemmas, we will estimate the DOA of HNNCNS (1). The cases of bounded delays and unbounded proportional delays are discussed, respectively. When the initial values belong to the DOA, the states of system locally exponentially or locally polynomially converge to the equilibrium point, which means that local synchronization is achieved.

3.1. Estimation of DOA on HNNCNs with Bounded Delays

In this subsection, we consider the case of the bounded delay $\delta (t)$. Accordingly, we can derive the following theorem.

Theorem 1.

Let Assumption 1 hold. If control gain $c>0$ satisfies

$$\begin{array}{cc}\hfill \beta =& -{\lambda }_{\max }\left(2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)\right)\hfill \\ \hfill & -2\sqrt{q}\Vert I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\Vert >0,\hfill \end{array}$$

(36)

and $H_1(0)>H_2(0)$, then system (1) realizes local exponential synchronization toward the equilibrium Θ, and the DOA is estimated as

$$R_0(r_0)=\left\{\zeta \right|\Vert z(t){\Vert }_{t_0,{\delta }_0}<r_0{\}=\{\zeta |\Vert \zeta -\mathrm{\Theta }\Vert }_{t_0,{\delta }_0}<r_0\},$$

i.e.,

$$\left|\right|z(t)\left|\right|\le \sqrt{{\overline{r}}_0}{e}^{-\frac{{\eta }_1}{2}(t-t_0)},\zeta \in R_0(r_0),t\ge t_0,$$

where functions $H_1(r),H_2(r)$ satisfy $H_1(r)=\beta -q{\psi }_1(\sqrt{r})$, $H_2(r)=q{\psi }_2(\sqrt{r})+q{\psi }_3(\sqrt{r})$, and a series of parameters satisfy $q>{(1-k)}^{-2}$, ${\overline{r}}_0=sup\left\{r\right|-H_1(r)+H_2(r)<0\}$, $M_0=\frac{{(1+k)}^2}{{(1-k)}^2}$, $r_0=\sqrt{\frac{{\overline{r}}_0}{M_0}}$, $M={\left|\right|z(t)\left|\right|}_{t_0,{\delta }_0}^2$, $r_1=M_{0}M$, $\lambda =H_1(r_1)-H_2(r_1)>0$, ${\eta }_1\in \left(0,min\left\{\lambda ,\frac{1}{{\delta }_0}ln\left[\frac{q}{{(1+\sqrt{q}k)}^2}\right]\right\}\right)$.

Proof. 

To begin with, we can choose a candidate Lyapunov function

$$W(t)=\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T[z_i(t)-D{z}_i(t-\delta (t))].$$

Calculating the derivative of $W(t)$ along system (5), it can be deduced that

$$\begin{array}{cc}\hfill W^{\prime }(t)=& 2\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T(J_1-c{I}_n)z_i(t)\hfill \\ \hfill & +2\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T{J}_2{z}_i(t-\delta (t))\hfill \\ \hfill & +2\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T{\mathrm{\Phi }}_{(1)i}(t)\hfill \\ \hfill & +2l\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }{J}_3{z}_j(t-\delta (t))\hfill \\ \hfill & +2l\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }{\mathrm{\Phi }}_{(2)j}(t-\delta (t))\hfill \\ \hfill =& 2\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T(J_1-c{I}_n)[z_i(t)-D{z}_i(t-\delta (t))]\hfill \\ \hfill & +2\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T\left(J_2+(J_1-c{I}_n)D\right)z_i(t-\delta (t))\hfill \\ \hfill & +2\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T{\mathrm{\Phi }}_{(1)i}(t)\hfill \\ \hfill & +2l\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }{J}_3{z}_j(t-\delta (t))\hfill \\ \hfill & +2l\sum _{i=1}^N{[z_i(t)-D{z}_i(t-\delta (t))]}^T\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }{\mathrm{\Phi }}_{(2)j}(t-\delta (t))\hfill \\ \hfill =& 2{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(I_N\otimes (J_1-c{I}_n)\right)[z(t)-\overline{D}z(t-\delta (t))]\hfill \\ \hfill & +2{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(I_N\otimes (J_2+(J_1-c{I}_n)D)\right)z(t-\delta (t))\hfill \\ \hfill & +2{[z(t)-\overline{D}z(t-\delta (t))]}^T{\mathrm{\Phi }}_{(1)}(t)+2l{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(Q\otimes \mathrm{\Gamma }{J}_3\right)z(t-\delta (t))\hfill \\ \hfill & +2l{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(Q\otimes \mathrm{\Gamma }\right){\mathrm{\Phi }}_{(2)}(t-\delta (t))\hfill \\ \hfill =& 2{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(I_N\otimes (J_1-c{I}_n)\right)[z(t)-\overline{D}z(t-\delta (t))]\hfill \\ \hfill & +2{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\right)z(t-\delta (t))\hfill \\ \hfill & +2{[z(t)-\overline{D}z(t-\delta (t))]}^T{\mathrm{\Phi }}_{(1)}(t)+2l{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(Q\otimes \mathrm{\Gamma }\right){\mathrm{\Phi }}_{(2)}(t-\delta (t)),\hfill \end{array}$$

(37)

where $z(t)={[z_1^T(t),z_2^T(t),\dots ,z_N^T(t)]}^T\in {\mathbb{R}}^{nN}$, $\overline{D}=I_N\otimes D\in {\mathbb{R}}^{nN\times nN}$. On the basis of inequalities (6) and (7) and Lemma 5, we have that

$$\begin{array}{cc}\hfill & 2{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\right)z(t-\delta (t))\hfill \\ \hfill \le & 2\left|\right|z(t)-\overline{D}z(t-\delta (t))\left|\right|\left|\right|I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\left|\right|\left|\right|z(t-\delta (t))\left|\right|,\hfill \end{array}$$

(38)

$$\begin{array}{c}2{[z(t)-\overline{D}z(t-\delta (t))]}^T{\mathrm{\Phi }}_{(1)}(t)\hfill \\ {\le [z(t)-\overline{D}z(t-\delta (t))]}^T[z(t)-\overline{D}z(t-\delta (t))]+{\psi }_1(\Vert z(t)\Vert )z^T(t)z(t)\hfill \\ +{\psi }_2{(\Vert z(t)\Vert }_{t,\delta (t)})z^T(t-\delta (t))z(t-\delta (t)),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & 2l{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(Q\otimes \mathrm{\Gamma }\right){\mathrm{\Phi }}_{(2)}(t-\delta (t))\hfill \\ \hfill \le & l^2{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T\right)[z(t)-\overline{D}z(t-\delta (t))]\hfill \\ \hfill & +{\psi }_3{(\Vert z(t)\Vert }_{t,\delta (t)})z^T(t-\delta (t))z(t-\delta (t)).\hfill \end{array}$$

(40)

Substituting inequalities (38)–(40) to inequality (37) yields that

$$\begin{array}{cc}\hfill W^{\prime }(t)\le & 2{[z(t)-\overline{D}z(t-\delta (t))]}^T\left(I_N\otimes (J_1-c{I}_n)\right)[z(t)-\overline{D}z(t-\delta (t))]\hfill \\ \hfill & +2\left|\right|z(t)-\overline{D}z(t-\delta (t))\left|\right|\left|\right|I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\left|\right|\left|\right|z(t-\delta (t))\left|\right|\hfill \\ \hfill & +{[z(t)-\overline{D}z(t-\delta (t))]}^T[z(t)-\overline{D}z(t-\delta (t))]+{\psi }_1(\Vert z(t)\Vert )z^T(t)z(t)\hfill \\ \hfill & +{\psi }_2{(\Vert z(t)\Vert }_{t,\delta (t)})z^T(t-\delta (t))z(t-\delta (t))\hfill \\ \hfill & +l^2{[z(t)-\overline{D}z(t-\delta (t))]}^T(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)[z(t)-\overline{D}z(t-\delta (t))]\hfill \\ \hfill & +{\psi }_3{(\Vert z(t)\Vert }_{t,\delta (t)})z^T(t-\delta (t))z(t-\delta (t))\hfill \\ \hfill \le & {[z(t)-\overline{D}z(t-\delta (t))]}^T\left(2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)\right)[z(t)-\overline{D}z(t-\delta (t))]\hfill \\ \hfill & +2\left|\right|z(t)-\overline{D}z(t-\delta (t))\left|\right|\left|\right|I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\left|\right|\left|\right|z(t-\delta (t))\left|\right|\hfill \\ \hfill & +{\psi }_1(\Vert z(t)\Vert )z^T(t)z(t)+{\psi }_2{(\Vert z(t)\Vert }_{t,\delta (t)})z^T(t-\delta (t))z(t-\delta (t))\hfill \\ \hfill & +{\psi }_3{(\Vert z(t)\Vert }_{t,\delta (t)})z^T(t-\delta (t))z(t-\delta (t)).\hfill \end{array}$$

(41)

In light of Lemma 2, one has $u(t+s)<q\omega (t),s\in [-{\delta }_0,0],$ which further gives that

$$\begin{array}{cc}\hfill W^{\prime }(t)\le & {\lambda }_{\max }\left(2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)\right){\Vert z(t)-\overline{D}z(t-\delta (t))\Vert }^2\hfill \\ \hfill & +2\left|\right|z(t)-\overline{D}z(t-\delta (t))\left|\right|\left|\right|I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\left|\right|\left|\right|z(t-\delta (t))\left|\right|\hfill \\ \hfill & +{\psi }_1{(\Vert z(t)\Vert )\Vert z(t)\Vert }^2+{\psi }_2{(\Vert z(t)\Vert }_{t,\delta (t)}{)\Vert z(t-\delta (t))\Vert }^2+{\psi }_3{(\Vert z(t)\Vert }_{t,\delta (t)}{)\Vert z(t-\delta (t))\Vert }^2\hfill \\ \hfill \le & [{\lambda }_{\max }\left(2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)\right)\hfill \\ \hfill & +2\sqrt{q}\Vert I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\Vert ]{\Vert z(t)-\overline{D}z(t-\delta (t))\Vert }^2\hfill \\ \hfill & +q{\psi }_1(\Vert z(t)\Vert )\Vert z(t)-\overline{D}{z(t-\delta (t))\Vert }^2+q{\psi }_2{(\Vert z(t)\Vert }_{t,\delta (t)})\Vert z(t)-\overline{D}{z(t-\delta (t))\Vert }^2\hfill \\ \hfill & +q{\psi }_3{(\Vert z(t)\Vert }_{t,\delta (t)})\Vert z(t)-\overline{D}{z(t-\delta (t))\Vert }^2\hfill \\ \hfill \le & [-H_1(u(t))+H_2(\widehat{u}(t))]{\Vert z(t)-\overline{D}z(t-\delta (t))\Vert }^2=[-H_1(u(t))+H_2(\widehat{u}(t))]\omega (t),\hfill \end{array}$$

(42)

where parameter $\beta$ satisfies $\beta =-{\lambda }_{\max }\left(2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)\right)-2\sqrt{q}\Vert I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\Vert >0$, matrix $P=2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)$ is a symmetric matrix, and a series of functions satisfy $u(t)={\Vert z(t)\Vert }^2$, $\widehat{u}(t)={\Vert z(t)\Vert }_{t,\delta (t)}^2$, $\omega (t)=\Vert z(t)-\overline{D}{z(t-\delta (t))\Vert }^2$, $H_1(u)=\beta -q{\psi }_1(\sqrt{u})$, $H_2(\widehat{u})=q{\psi }_2(\sqrt{\widehat{u}})+q{\psi }_3(\sqrt{\widehat{u}})$. Noting that $H_1(0)>H_2(0)$, we can choose an appropriate parameter ${\overline{r}}_0$ such that

$$\begin{array}{cc}\hfill {\overline{r}}_0=& sup\left\{r\right|-H_1(r)+H_2(r)<0\}\hfill \\ \hfill =& sup\left\{r\right|-\beta +q{\psi }_1(\sqrt{r})+q{\psi }_2(\sqrt{r})+q{\psi }_3(\sqrt{r})<0\}.\hfill \end{array}$$

(43)

Here, let ${\alpha }_1={\alpha }_2=1$, $\overline{q}=q>{(1-k)}^{-2}$ and $M_0=\frac{{(1+k)}^2}{{(1-k)}^2}$. When initial data $\zeta \in R_0(r_0)=\left\{\zeta \right|\Vert z(t){\Vert }_{t_0,{\delta }_0}<r_0{\}=\{\zeta |\Vert \zeta -\mathrm{\Theta }\Vert }_{t_0,{\delta }_0}<r_0\}$, the states of system (1) exponentially converge to equilibrium $\mathrm{\Theta }$ with the convergence rate ${\eta }_1$ by Lemma 2, namely

$$\left|\right|z(t)\left|\right|\le \sqrt{{\overline{r}}_0}{e}^{-\frac{{\eta }_1}{2}(t-t_0)},t\ge t_0,$$

where $r_0=\sqrt{\frac{{\overline{r}}_0}{M_0}}$, $M={\left|\right|z(t)\left|\right|}_{t_0,{\delta }_0}^2$, $r_1=M_{0}M$, ${\eta }_1\in \left(0,min\left\{\lambda ,\frac{1}{{\delta }_0}ln\left[\frac{q}{{(1+\sqrt{q}k)}^2}\right]\right\}\right)$ and $\lambda =H_1(r_1)-H_2(r_1)>0$, which means that the network system (1) realizes local exponential synchronization towards the equilibrium $\mathrm{\Theta }$ with the convergence rate ${\eta }_1$ and the above DOA. □

Remark 4.

In Theorem 1, on the basis of Lemma 2, the problem of DOA estimation is solved for network system (1) with bounded delays. If control gain c is large enough, then network (1) achieves local exponential synchronization when the initial value $\zeta \in R_0(r_0)$. Namely, there exists a close relationship among the control gain c, initial value ζ, and the local synchronization of the network system. On the other hand, although the dynamic properties of neutral complex networks such as stability, periodic solution, and synchronization have been sufficiently analyzed in [39,40,41,42], the research into DOA for highly nonlinear neutral complex networks has not been examined, and the above result bridges the gap. Additionally, it can also be viewed as an extension of the existing achievements [31,37,38].

3.2. Estimation of DOA on HNNCNs with Unbounded Proportional Delay

In this subsection, we consider the case of the unbounded proportional delay $\delta (t)=(1-\sigma )t,0<\sigma <1$. In system (1), the initial data are transformed into $y_i(s)={\zeta }_i(s)$, $s\in [\sigma t_0,t_0],1\le i\le N$. Accordingly, we can derive the following theorem.

Theorem 2.

If all the conditions in Theorem 1 are satisfied, then system (1) realizes local polynomial synchronization toward the equilibrium Θ, and the DOA is estimated as

$$R_0(r_0)=\left\{\zeta \right|\Vert z(t){\Vert }_{t_0,(1-\sigma )t_0}<r_0{\}=\{\zeta |\Vert \zeta -\mathrm{\Theta }\Vert }_{t_0,(1-\sigma )t_0}<r_0\},$$

i.e.,

$$\left|\right|z(t)\left|\right|\le \sqrt{{\overline{r}}_0}{(1+t_0)}^{\frac{{\eta }_1}{2}}{(1+t)}^{-\frac{{\eta }_1}{2}},\zeta \in R_0(r_0),t\ge t_0,$$

where functions $H_1(r),H_2(r)$ satisfy $H_1(r)=\beta -q{\psi }_1(\sqrt{r})$, $H_2(r)=q{\psi }_2(\sqrt{r})+q{\psi }_3(\sqrt{r})$, and a series of parameters satisfy $q>{(1-k)}^{-2}$, ${\overline{r}}_0=sup\left\{r\right|-H_1(r)+H_2(r)<0\}$, $M_0=\frac{{(1+k)}^2}{{(1-k)}^2}{(1+t_0)}^{{\eta }_1}$, $r_0=\sqrt{\frac{{\overline{r}}_0}{M_0}}$, $M={\left|\right|z(t)\left|\right|}_{t_0,(1-\sigma )t_0}^2$, $r_1=M_{0}M$, $\lambda =H_1(r_1)-H_2(r_1)>0$, ${\eta }_1\in \left(0,min\left\{\lambda ,\frac{1}{ln\frac{1}{\sigma }}ln\left(\frac{q}{{(1+\sqrt{q}k)}^2}\right)\right\}\right).$

Proof. 

Let $W(t)$ be a candidate Lyapunov function

$$W(t)=\sum _{i=1}^N{[z_i(t)-D{z}_i(\sigma t)]}^T[z_i(t)-D{z}_i(\sigma t)].$$

By repeating the previous proof procedure of Theorem 1, we can obtain the derivative of $W(t)$ as follows

$$\begin{array}{cc}\hfill W^{\prime }(t)=& 2{[z(t)-\overline{D}z(\sigma t)]}^T\left(I_N\otimes (J_1-c{I}_n)\right)[z(t)-\overline{D}z(\sigma t)]\hfill \\ \hfill & +2{[z(t)-\overline{D}z(\sigma t)]}^T\left(I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\right)z(\sigma t)\hfill \\ \hfill & +2{[z(t)-\overline{D}z(\sigma t)]}^T{\mathrm{\Phi }}_{(1)}(t)+2l{[z(t)-\overline{D}z(\sigma t)]}^T\left(Q\otimes \mathrm{\Gamma }\right){\mathrm{\Phi }}_{(2)}(\sigma t).\hfill \end{array}$$

(44)

Subsequently, on the basis of Lemma 4, one obtains $u(t+s)<q\omega (t),s\in [(\sigma -1)t,0]$, which further yields that

$$\begin{array}{cc}\hfill W^{\prime }(t)\le & [{\lambda }_{\max }\left(2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)\right)\hfill \\ \hfill & +2\sqrt{q}\Vert I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\Vert ]{\Vert z(t)-\overline{D}z(\sigma t)\Vert }^2\hfill \\ \hfill & +q{\psi }_1(\Vert z(t)\Vert )\Vert z(t)-\overline{D}{z(\sigma t)\Vert }^2+q{\psi }_2{(\Vert z(t)\Vert }_{t,(1-\sigma )t})\Vert z(t)-\overline{D}{z(\sigma t)\Vert }^2\hfill \\ \hfill & +q{\psi }_3{(\Vert z(t)\Vert }_{t,(1-\sigma )t})\Vert z(t)-\overline{D}{z(\sigma t)\Vert }^2\hfill \\ \hfill \le & [-H_1(u(t))+H_2(\tilde{u}(t))]{\Vert z(t)-\overline{D}z(\sigma t)\Vert }^2=[-H_1(u(t))+H_2(\tilde{u}(t))]\omega (t),\hfill \end{array}$$

(45)

where parameter $\beta$ satisfies $\beta =-{\lambda }_{\max }\left(2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)\right)-2\sqrt{q}\Vert I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\Vert >0$, matrix $P=2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)$ is a symmetric matrix, and a series of functions satisfy $u(t)={\Vert z(t)\Vert }^2$, $\tilde{u}(t)={\Vert z(t)\Vert }_{t,(1-\sigma )t}^2$, $\omega (t)=\Vert z(t)-\overline{D}{z(\sigma t)\Vert }^2$, $H_1(u)=\beta -q{\psi }_1(\sqrt{u})$, $H_2(\tilde{u}(t))=q{\psi }_2(\sqrt{\tilde{u}(t)})+q{\psi }_3(\sqrt{\tilde{u}(t)})$. Since $H_1(0)>H_2(0)$, we can select an appropriate parameter ${\overline{r}}_0$ satisfying

$$\begin{array}{cc}\hfill {\overline{r}}_0=& sup\left\{r\right|-H_1(r)+H_2(r)<0\}\hfill \\ \hfill =& sup\left\{r\right|-\beta +q{\psi }_1(\sqrt{r})+q{\psi }_2(\sqrt{r})+q{\psi }_3(\sqrt{r})<0\}.\hfill \end{array}$$

(46)

Here, let ${\alpha }_1={\alpha }_2=1$, $\overline{q}=q>{(1-k)}^{-2}$ and $M_0=\frac{{(1+k)}^2}{{(1-k)}^2}{(1+t_0)}^{{\eta }_1}$. In virtue of Lemma 4, the states of system (1) polynomially converge to equilibrium $\mathrm{\Theta }$ with the convergence rate ${\eta }_1$ when the initial data $\zeta \in R_0(r_0)=\left\{\zeta \right|\Vert z(t){\Vert }_{t_0,(1-\sigma )t_0}<r_0{\}=\{\zeta |\Vert \zeta -\mathrm{\Theta }\Vert }_{t_0,(1-\sigma )t_0}<r_0\}$, i.e.,

$$\left|\right|z(t)\left|\right|\le \sqrt{{\overline{r}}_0}{(1+t_0)}^{\frac{{\eta }_1}{2}}{(1+t)}^{-\frac{{\eta }_1}{2}},t\ge t_0,$$

where $r_0=\sqrt{\frac{{\overline{r}}_0}{M_0}}$, $M={\left|\right|z(t)\left|\right|}_{t_0,(1-\sigma )t_0}^2$, $r_1=M_{0}M$, ${\eta }_1\in \left(0,min\left\{\lambda ,\frac{1}{ln\frac{1}{\sigma }}ln\left(\frac{q}{{(1+\sqrt{q}k)}^2}\right)\right\}\right)$ and $\lambda =H_1(r_1)-H_2(r_1)>0$, which implies that the network system (1) realizes local polynomial synchronization towards the equilibrium $\mathrm{\Theta }$ with the convergence rate ${\eta }_1$ and the above DOA. □

Remark 5.

Although the estimations of DOA of nonlinear systems are extensively investigated, the case of proportional unbounded delays has been discussed comparatively rarely. Here, the issue of DOA estimation is solved for the network system (1) with the unbounded proportional delay $\delta (t)=(1-\sigma )t,0<\sigma <1$ by Lemma 4.

Remark 6.

In Assumption 1, it is noted that parameter k satisfies $0<k<1$ which makes our results applicable to only one special class of neutral CNs. Additionally, in order to estimate the DOA of HNNCNs, the Razumikhin approach was adopted. Consequently, we can directly derive the DOA estimation rather than seek complicated Lyapunov–Krasovskii functionals. However, we do not design some parameter optimization algorithms to obtain the more accurate DOA estimation. In the future, we might further explore the DOA estimation based on various optimization algorithms.

4. Examples and Simulations

In this section, several examples are presented to verify the effectiveness of theoretical results.

Example 1.

Consider a class of HNNCNS composed of N nodes with the bounded delay $\delta (t)$ as following

$$\left\{\begin{array}{cc}\hfill \frac{d[y_i(t)-D{y}_i(t-\delta (t))]}{dt}=& F(y_i(t),y_i(t-\delta (t)))+l\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }g(y_j(t-\delta (t)))+U_i(t),t\ge t_0,\hfill \\ \hfill y_i(s)=& {\zeta }_i(s),s\in [t_0-{\delta }_0,t_0],1\le i\le N,\hfill \end{array}\right.$$

(47)

where the bounded delay $\delta (t)=0.2|cos(t)|$, $l=0.5$, $N=10$, $t_0=0$ and

$$\begin{array}{cc}\hfill & F_1(y_i(t),y_i(t-\delta (t)))=0.3y_{i1}(t)-0.6y_{i2}(t)-0.2y_{i1}(t-\delta (t))+0.5y_{i2}(t-\delta (t))\hfill \\ \hfill & +y_{i1}(t){(y_{i1}^2(t)+y_{i2}^2(t))}^{\frac{3}{2}},\hfill \\ \hfill & F_2(y_i(t),y_i(t-\delta (t)))=0.2y_{i1}(t)-0.7y_{i2}(t)+0.6y_{i1}(t-\delta (t))+0.4y_{i2}(t-\delta (t))\hfill \\ \hfill & +y_{i2}(t){(y_{i1}^2(t)+y_{i2}^2(t))}^{\frac{3}{2}},\hfill \\ \hfill & g_1(y_i(t-\delta (t)))=0.8y_{i1}(t-\delta (t))+2y_{i1}(t-\delta (t))(y_{i1}^2(t-\delta (t))+y_{i2}^2(t-\delta (t))),\hfill \\ \hfill & g_2(y_i(t-\delta (t)))=-0.4y_{i1}(t-\delta (t))+0.5y_{i2}(t-\delta (t))\hfill \\ \hfill & +2y_{i2}(t-\delta (t))(y_{i1}^2(t-\delta (t))+y_{i2}^2(t-\delta (t))),\hfill \\ \hfill & D=\left[\begin{array}{cc}0.1& 0\\ 0& 0.12\end{array}\right],\mathrm{\Gamma }=\left[\begin{array}{cc}0.2& 0\\ 0& 0.5\end{array}\right],\hfill \\ \hfill & Q=\left[\begin{array}{cccccccccc}-2& 1& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& -3& 1& 0& 1& 0& 0& 1& 0& 0\\ 0& 0& -3& 0& 0& 0& 2& 0& 0& 1\\ 0& 1& 0& -4& 1& 0& 0& 0& 1& 1\\ 0& 0& 1& 0& -2& 0& 0& 0& 0& 1\\ 0& 1& 0& 1& 0& -3& 0& 1& 0& 0\\ 1& 1& 1& 1& 0& 0& -5& 1& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& -2& 1& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& -2& 1\\ 0& 0& 0& 1& 0& 0& 0& 0& 1& -2\end{array}\right].\hfill \end{array}$$

By Equation (3), we can obviously have an equilibrium point $\vartheta ={[0,0]}^T$, which is locally exponentially stable. By means of computation, we have that $\Vert D\Vert =0.12$, ${\delta }_0=0.2$, $J_1=\left[\begin{array}{cc}0.3& -0.6\\ 0.2& -0.7\end{array}\right],J_2=\left[\begin{array}{cc}-0.2& 0.5\\ 0.6& 0.4\end{array}\right],J_3=\left[\begin{array}{cc}0.8& 0\\ -0.4& 0.5\end{array}\right],$ and ${\psi }_1(y)=y^6$, ${\psi }_2(y)=0$, ${\psi }_3(y)=4y^4.$ By Assumption 1, we obtain that $k=1.2$. Taking $q=1.3>{(1-k)}^{-2}=1.2913$ and $c=12$, by means of computation, we can have that $\beta =-{\lambda }_{\max }\left(2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)\right)-2\sqrt{q}\Vert I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\Vert =17.8354>0$, $M_0=\frac{{(1+k)}^2}{{(1-k)}^2}=1.6198>1$, ${\overline{r}}_0=sup\left\{r\right|-\beta +q{\psi }_1(\sqrt{r})+q{\psi }_2(\sqrt{r})+q{\psi }_3(\sqrt{r})<0\}=1.8803$ and $r_0=\sqrt{\frac{\overline{r}}{M_0}}=\sqrt{1.1608}=1.0774$. Let $r_1=M_{0}M=1.8<{\overline{r}}_0$, we obtain that

$${\eta }_1\in (0,min\{1.8298,\frac{1}{0.2}ln(\frac{1.3}{{(1+0.12\sqrt{1.3})}^2})\})=(0,0.0295).$$

According to Theorem 1, network (47) realizes local exponential synchronization toward the equilibrium Θ with the convergence rate ${\eta }_1$ if we can find that a sufficiently large control gain c such that $\beta >0$ and the DOA is estimated as $R_0(r_0)$. Let the initial conditions be ${\zeta }_1={[-1,0.4]}^T$, ${\zeta }_2={[-0.9,0.5]}^T$, ${\zeta }_3={[0.4,-0.8]}^T,{\zeta }_4={[-0.2,-0.9]}^T,{\zeta }_5={[-0.7,-0.8]}^T,{\zeta }_6={[0.9,0.5]}^T$, ${\zeta }_7={[0.6,-0.8]}^T$, ${\zeta }_8={[-0.5,0.9]}^T,{\zeta }_9={[1,-0.3]}^T,$${\zeta }_{10}={[0.7,0.8]}^T.$ Noting that $\zeta \in R_0(r_0),$ which illustrates that the initial condition belongs to the DOA $R_0(r_0)$. Taking ${\eta }_1=0.02$, we obtain that

$$\Vert z(t)\Vert \le \sqrt{1.8803}{e}^{-\frac{0.02}{2}t}=1.3712e^{-0.01t},t\ge 0,$$

which implies that the states of system (47) exponentially converge to the equilibrium point ${[0,0]}^T$ and further verifies the feasibility of Theorem 1. Figure 1 shows that the synchronization errors $z_{ij}(i=1,2,\dots ,10;j=1,2)$ of Example 1 diverge to infinity, which indicates that exponential synchronization cannot be achieved without control input when $\zeta \in R_0(r_0)$. On the other hand, Figure 2 shows that synchronization errors $z_{ij}(i=1,2,\dots ,10;j=1,2)$ converge to equilibrium point, which indicates that the exponential synchronization can be realized with the control input when $\zeta \in R_0(r_0)$. This example illustrates the significance of initial values and control input.

Example 2.

Consider a class of HNNCNS composed of N nodes with the unbounded proportional delay $\delta (t)=(1-\sigma )t,0<\sigma <1$ as follows

$$\left\{\begin{array}{cc}\hfill \frac{d[y_i(t)-D{y}_i(\sigma t)]}{dt}=& F(y_i(t),y_i(\sigma t))+l\sum _{j=1}^N{q}_{ij}\mathrm{\Gamma }g(y_j(\sigma t))+U_i(t),t\ge t_0=0,\hfill \\ \hfill y_i(t_0)=& {\zeta }_i,1\le i\le N,\hfill \end{array}\right.$$

(48)

where parameters $l,\sigma ,N,t_0$, functions $F_1(y_i(t),y_i(\sigma t)),F_2(y_i(t),y_i(\sigma t)),g_1(y_i(\sigma t)),g_2(y_i(\sigma t))$, and matrices $D,\mathrm{\Gamma },Q$ are chosen as below

$$\begin{array}{cc}\hfill & l=0.1,\sigma =0.8,N=10,t_0=0,\hfill \\ \hfill & F_1(y_i(t),y_i(\sigma t))=-0.1y_{i1}(t)+0.2y_{i1}(\sigma t)+0.4y_{i2}(\sigma t)+0.8y_{i1}(\sigma t){(y_{i1}^2(\sigma t)+y_{i2}^2(\sigma t))}^2,\hfill \\ \hfill & F_2(y_i(t),y_i(\sigma t))=-0.4y_{i1}(t)-0.3y_{i2}(t)-0.4y_{i1}(\sigma t)+0.5y_{i2}(\sigma t)\hfill \\ \hfill & +0.8y_{i2}(\sigma t){(y_{i1}^2(\sigma t)+y_{i2}^2(\sigma t))}^2,\hfill \\ \hfill & g_1(y_i(\sigma t))=0.4y_{i1}(\sigma t)+0.1y_{i2}(\sigma t)+1.5y_{i1}(\sigma t){(y_{i1}^2(\sigma t)+y_{i2}^2(\sigma t))}^{\frac{1}{2}},\hfill \\ \hfill & g_2(y_i(\sigma t))=0.1y_{i1}(\sigma t)+0.3y_{i2}(\sigma t)+1.5y_{i2}(\sigma t){(y_{i1}^2(\sigma t)+y_{i2}^2(\sigma t))}^{\frac{1}{2}},\hfill \\ \hfill & D=\left[\begin{array}{cc}0.2& 0\\ 0& 0.18\end{array}\right],\mathrm{\Gamma }=\left[\begin{array}{cc}0.3& 0\\ 0& 0.4\end{array}\right],\hfill \\ \hfill & Q=\left[\begin{array}{cccccccccc}-3& 1& 0& 0& 0& 0& 2& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 1& 0& 0\\ 1& 0& -2& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& -5& 1& 2& 0& 0& 1& 1\\ 0& 0& 1& 0& -3& 0& 1& 0& 0& 1\\ 0& 1& 0& 1& 0& -2& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -1& 0& 1& 0\\ 0& 1& 0& 0& 1& 0& 0& -3& 1& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& -2& 1\\ 1& 0& 0& 1& 0& 0& 1& 0& 1& -4\end{array}\right].\hfill \end{array}$$

By Equation (3), we can obviously have an equilibrium point $\vartheta ={[0,0]}^T$, which is locally polynomially stable. By computing, we have that $\Vert D\Vert =0.2$, $J_1=\left[\begin{array}{cc}-0.1& 0\\ -0.4& -0.3\end{array}\right]$, $J_2=\left[\begin{array}{cc}0.2& 0.4\\ -0.4& 0.5\end{array}\right]$, $J_3=\left[\begin{array}{cc}0.4& 0.1\\ 0.1& 0.3\end{array}\right],$ and ${\psi }_1(y)=0$, ${\psi }_2(y)=0.64y^8$, ${\psi }_3(y)=2.25y^2.$

According to Assumption 1, we get $k=0.2$. Taking $q=1.6>{(1-k)}^{-2}=1.5625$ and $c=18$. By simply computational procedure, we obtain that $\beta =-{\lambda }_{\max }\left(2(I_N\otimes (J_1-c{I}_n))+I_{nN}+l^2(Q{Q}^T\otimes \mathrm{\Gamma }{\mathrm{\Gamma }}^T)\right)-2\sqrt{q}\Vert I_N\otimes (J_2+(J_1-c{I}_n)D)+l(Q\otimes \mathrm{\Gamma }{J}_3)\Vert =26.2354>0$, $M_0=\frac{{(1+k)}^2}{{(1-k)}^2}{(1+t_0)}^{{\eta }_1}=\frac{{(1+k)}^2}{{(1-k)}^2}=2.2500>1$, ${\overline{r}}_0=sup\left\{r\right|-\beta +q{\psi }_1(\sqrt{r})+q{\psi }_2(\sqrt{r})+q{\psi }_3(\sqrt{r})<0\}=2.0216$ and $r_0=\sqrt{\frac{{\overline{r}}_0}{M_0}}=\sqrt{0.8985}=0.9479$. Let $r_1=M_{0}M=2<{\overline{r}}_0$, we have that

$${\eta }_1\in (0,min\{0.9554,\frac{1}{ln(\frac{1}{0.8})}ln(\frac{1.6}{{(1+0.2\sqrt{1.6})}^2})\})=(0,0.0849).$$

Based on Theorem 2, network (48) realizes local polynomial synchronization toward the equilibrium Θ with the convergence rate ${\eta }_1$ if we can find that a sufficiently large control gain c such that $\beta >0$ and the DOA is estimated as $R_0(r_0)$. We can also choose the initial condition $\zeta \in R_0(r_0)$ as follows ${\zeta }_1={[0.7,0.6]}^T,{\zeta }_2={[0.5,0.3]}^T,{\zeta }_3={[-0.7,-0.4]}^T,{\zeta }_4={[-0.5,0.4]}^T$, ${\zeta }_5={[-0.3,0.6]}^T$, ${\zeta }_6={[0.8,-0.2]}^T,{\zeta }_7={[-0.1,0.3]}^T,{\zeta }_8={[0.3,-0.7]}^T,{\zeta }_9={[-0.4,-0.8]}^T,$${\zeta }_{10}={[0.5,-0.5]}^T,$ which shows that the initial conditions belong to the DOA $R_0(r_0)$. Taking ${\eta }_1=0.08$, we can obtain that

$$\Vert z(t)\Vert \le \sqrt{2.0216}{(1+t_0)}^{\frac{0.08}{2}}{(1+t)}^{-\frac{0.08}{2}}=1.4218{(1+t)}^{-0.04},t\ge 0,$$

which means that the states of system (48) polynomially converge to equilibrium point ${[0,0]}^T$ and validate the effectiveness of Theorem 2. Figure 3 shows that synchronization errors $z_{ij}$$(i=1,2,\dots ,10;j=1,2)$ of Example 2 diverge towards infinity, indicating that polynomial synchronization cannot be achieved without control input when $\zeta \in R_0(r_0)$. On the other hand, Figure 4 shows that synchronization errors $z_{ij}(i=1,2,\dots ,10;j=1,2)$ converge to an equilibrium point, which indicates that polynomial synchronization can be realized with control input when $\zeta \in R_0(r_0)$. In contrast to Example 1, proportional delays are considered here. Actually, when $\zeta \notin R_0(r_0)$ and control gain c is not sufficiently large, polynomial synchronization cannot be realized.

5. Conclusions

In this paper, to begin with, several novel Razumikhin Lemmas on the estimations of DOA for a class of highly nonlinear neutral differential systems have been established. Thereafter, by utilizing the presented lemmas, the estimation of DOA of HNNCNs with bounded delays or unbounded proportional delays is given, respectively. The theoretical results reveal that the local exponential or polynomial synchronization of HNNCNs can be realized when initial values starting from the DOA finally exponentially or polynomially converge to an equilibrium point. Finally, several numerical simulations illustrate the feasibility of theoretical results. In the future, we will further explore the DOA estimation for highly nonlinear discrete systems and switching systems.