Global Exponential Stability of Impulsive Delayed Neural Networks with Parameter Uncertainties and Reaction–Diffusion Terms

Abstract

In this paper, we present a method to achieve exponential stability in a class of impulsive delayed neural networks containing parameter uncertainties, time-varying delays, and impulsive effect and reaction–diffusion terms. By using an integro-differential inequality with impulsive initial conditions and employing the M-matrix theory and the nonlinear measure approach, some new sufficient conditions ensuring the global exponential stability and global robust exponential stability of the considered system are derived. In particular, the results obtained are presented by simple algebraic inequalities, which are certainly more concise than the previous methods. By comparisons and examples, it is shown that the results obtained are effective and useful.

1. Introduction

It is well-known that cellular neural networks (CNNs), proposed by Chuan and Yang in 1998 [1], include many models from different research fields, such as signal recognition, image processing, pattern classification, and other fields. As a dynamical behavior of CNNs, stabilization plays an important role in various fields such as engineering and biology. Depending on the content of the investigation, there are various stabilization types, such as asymptotic stabilization, exponential stabilization, finite-time stabilization, fixed-time stabilization, input-to-state stabilization, and so on. Recently, the studies on the stability of neural networks have received much attention [2,3,4,5].

Aswe all know, time delays are unavoidable due to the limitations of signal transmission and the limited switching speeds of amplifiers and the communication time. In addition, time delays are applied extensively in many aspects, such as biology and engineering [6,7,8]. In engineering applications, the system is suddenly affected by external factors, and the state of the system will suddenly change, involving impulsive effects. Although the duration of the impulses is very short, it can cause a sudden change for the state of the system, so it is harmful. At present, a quite hot topic is to stabilize the system with impulses. Therefore, it is particularly important to ensure the stability and good performance of systems in the presence of time delays and impulses. Research on the stability of neural networks with time delays and impulses has been conducted extensively; for example, see [9,10,11,12,13,14,15,16,17,18]. Among the studies, a novel exponential stability criterion in the form of linear matrix inequalities was first established in [15]. By applying the properties of M-matrices, Li et al. [16] investigated the global exponential stability conditions for discrete-time BAM neural networks affected by impulses and time-varying delays. In [17], Rajchakit et al. investigated the global asymptotic stability problem for Clifford-valued neural network models with time-varying delays and impulsive effects. An auxiliary model-based nonsingular M-matrix approach was used to study the global exponential stability of neural networks with time-varying delays, connection weights, and impulses in [18]. In [19], Sergey et al. considered the input-to-state stability (ISS) of impulsive control systems with and without time delays and proved that, if the time-delay system possesses an exponential Lyapunov–Razumikhin function or an exponential Lyapunov–Krasovskii functional, then the system is uniformly ISS provided that the averaged well-time condition is satisfied.

In particular, the neural networks generate a diffusion phenomenon when electrons move in an asymmetric electromagnetic field, which means that the state of the neurons changes not only with time but also with space. This phenomenon can be modeled by introducing the diffusion term into the traditional neural network models. In order to better reflect the actual states of the neural networks, it is necessary to describe the reaction–diffusion of neurons in space by partial differential equations. The relevant investigations on the stability of neural networks with diffusion terms expressed by partial differential equations are presented in [12,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] and the references therein. Among them, the global exponential stability of reaction–diffusion neural networks with time-varying was investigated in [12,20,21,23,28,29,30]. In [22], the stability of delay fuzzy cellular neural networks with reaction–diffusion was investigated by some new conditions. These conditions do not depend on the delay. Lu [25] studied a model with delayed reaction–diffusion and distributed delays and further discussed the robust global exponential stability and periodicity. Gokulakrishnan and Srinivasan [33] presented new criteria to investigate the exponential input-to-state stabilization of stochastic reaction–diffusion Cohen–Grossberg neural networks by designing an event-triggered controller. Here, the Zeno phenomenon of the event-triggered controller is avoided. By applying a new adaptive intermittent control approach, the finite-time synchronization of complex-valued neural networks with reaction–diffusion terms was studied in [34].

Moreover, parameter uncertainties are often encountered in engineering and biology systems due to changes in the environment. In recent years, in order to solve the problems caused by parameter uncertainties, many scholars have conducted a great deal of work on different uncertainty problems in this area and have achieved great progress; for example, see [9,31,35,36,37,38,39]. In [35,36], the authors investigated the global exponential stability of neural networks by using the linear matrix inequality (LMI) methods. Here, the time-delay correlation criterion was proposed to ensure the robust stability of uncertain time-delay cellular neural networks. Hua et al. [37] investigated an uncertain stochastic neural network model and obtained new results on robust exponential stability. The robust exponential stability of uncertain neural networks with time-varying was investigated by Balasubramaniam and Syed Ali in [38] and Zhang in [9]. Uncertain fuzzy cellular neural networks with time-varying delays and reaction–diffusion terms were discussed in [31,32].

Despite the widespread works on the stability analysis of neural networks, little attention has been paid to the cases that the uncertain systems contain many complex terms, for example, impulsive effects or time-varying and reaction terms. Motivated by the works above and the need for practical problems, we aim to study the global exponential stability of impulsive delayed neural networks with parameter uncertainties and reaction–diffusion terms.

To the best of our knowledge, most authors have studied the global asymptotic stability of different types of neural networks by using the linear matrix inequality (LMI) technique; for example, see [31,34,35,36,37]. In their works, the obtained conditions of stability are quite complex and cumbersome. In addition, some conditions obtained cannot achieve global robust exponential stability. In this paper, a new differential inequality simultaneously ensuring the global exponential stability and global robust exponential stability of the considered system is established by using the properties of an M-matrix and the technique of simple inequalities. In addition, the time delay is not the key factor to determine whether the system is stable but only determines the rate of exponential convergence of the system.

Our presentation is organized as follows. In Section 2, we describe the mathematical model of the system and provide some preliminaries. The stability analysis of the given system is discussed in Section 3. In Section 4, two examples are provided to verify the validity of the obtained results.

2. Model Description and Some Preliminaries

In this paper, we consider the following impulsive delayed neural network model with reaction–diffusion terms and parameter uncertainties:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\displaystyle \frac{\partial u_p(t,x)}{\partial t}}=& {\displaystyle \sum _{k=1}^m}{\displaystyle \frac{\partial }{\partial x_k}}\left(D_{pk}{\displaystyle \frac{\partial u_p(t,x)}{\partial x_k}}\right)-(a_p+\Delta a_p\left(t\right))u_p(t,x)+{\displaystyle \sum _{q=1}^n}(b_{pq}+\Delta b_{pq}\left(t\right))f_q\left(u_q(t,x)\right)\hfill \\ & +{\displaystyle \sum _{q=1}^n}(c_{pq}+\Delta c_{pq}\left(t\right))g_q\left(u_q\left(t-{\tau }_{pq}\left(t\right),x\right)\right)\hfill \\ & +J_p,t\ne t_l,p=1,2,\dots ,n,x\in \Xi ,\hfill \\ \hfill u_p(t_l^{+},x)=& I_{pl}\left(u_p(t_l^{-},x)\right),p=1,2,\dots ,n,x\in \Xi ,l\in N=\{1,2,\dots \},\hfill \\ \hfill u_p(t,x)=& 0,t⩾0,x\in \partial \Xi ,\hfill \\ \hfill u_p(s,x)=& {\mu }_p(s,x),s\in [-\tau ,0],p=1,2,\dots ,n,\hfill \end{array}\right.\end{array}$$

(1)

where $u_p(t,x)$ is the state of the pth neuron in space x and time t. $x={\left(x_1,x_2,\dots ,x_m\right)}^{\mathrm{T}}\in \Xi \subset R^m$. $\Xi =\{x={\left(x_1,x_2,\dots ,x_m\right)}^{\mathrm{T}}||x_k\mid <L_k,k=1,2,\dots ,m\}$ is a bounded compact set with boundary $\partial \Xi$ and mes $\Xi >0$ in space $R^m\left(L_k\right.$ is a positive constant). $u(t,x)=\left(u_1(t,x),\right.$${\left.u_2(t,x),\dots ,u_n(t,x)\right)}^{\mathrm{T}}\in R^n$. $D_{pk}>0$ denotes diffusion coefficient. $a_p>0$ denotes the intensity of the pth element. $b_{pq}$ and $c_{pq}$ represent elements of delay and undelayed coupling gains. $\Delta a_p\left(t\right), \Delta b_{pq}\left(t\right), \Delta c_{pq}\left(t\right)$ represent bounded unstructured perturbations. ${\tau }_{pq}\left(t\right)\in [0,\tau ]$ represents the transmission delays from the pth element to the qth element; $f_q(·)\in C\left(R,R\right)$ and $g_q(·)\in C\left(R,R\right)$ represent the activation functions with no delays and delays, respectively. $J_p$ represents the external input. $I_{pk}$ shows impulsive perturbation of the pth neuron.

$PC(\Xi )\triangleq \left\{\omega :[-\tau ,0]\times \Xi \to R^n\mid \omega (s,x)\right.$ is bounded on $[-\tau ,0]\times \Xi$, $\omega \left(s^{+},x\right)=$$\omega (s,x)$ for $s\in [-\tau ,0],\omega \left(s^{-},x\right)$ exists for $s\in$ $[-\tau ,0]$, and $\omega \left(s^{-},x\right)=\omega (s,x)$ for all but a finite number of points $s\in [-\tau ,0]\}$.

$PC\triangleq \left\{\omega :[-\tau ,0]\to R^n\mid \omega \left(s\right)\right.$ is bounded on $[-\tau ,0]$, $\omega \left(s^{+}\right)=\omega \left(s\right)$ for $s\in [-\tau ,0]$, $\omega \left(s^{-}\right)$ exists for $s\in [-\tau ,0]$, and $\omega \left(s^{-}\right)=\omega \left(s\right)$ for all but a finite number of points $s\in [-\tau ,0]\}$.

For any $\omega (s,x)={({\omega }_1(s,x),{\omega }_2(s,x),\dots ,{\omega }_n(s,x))}^T\in PC(\Xi )$, the norm on $PC(\Xi )$ is defined by

$$\parallel \omega \parallel =\underset{-\tau ⩽s⩽0}{sup}{\displaystyle \sum _{p=1}^n}{∥{\omega }_p(s,x)∥}_2\mathrm{and}{∥u_p(t,x)∥}_2={\left[{\int }_{\Xi }{\left|u_p(t,x)\right|}^{2}dx\right]}^{{\displaystyle \frac{1}{2}}}, p=1,2,\dots ,n.$$

For $C,D\in R^{m\times n},$ we define the Schur product by $C\otimes D={\left(c_{pq}{d}_{pq}\right)}_{m\times n}$, in which $C⩾D(C>D)$ means that the inequality $c_{pq}⩾b_{pq}\left(c_{pq}>d_{pq}\right)$ holds. $\mathbf{e}={(1,1,\dots ,1)}^T\in R^n$, and E is an n-dimensional identity matrix.

To complete the goal of this paper, we make the following assumptions:

Hypothesis 1.

There exist positive diagonal matrices F = diag$(F_1,F_2,\dots ,F_n)$ and G = diag$(G_1,G_2,\dots ,G_n)$ such that

$$F_p=\underset{u\ne w}{sup}\left|{\displaystyle \frac{f_p\left(u\right)-f_p\left(w\right)}{u-w}}\right|,G_p=\underset{u\ne w}{sup}\left|{\displaystyle \frac{g_p\left(u\right)-g_p\left(w\right)}{u-w}}\right|,$$

for all $u, w\in R(u\ne w), p=1,2\dots ,n.$

Hypothesis 2.

There exists a positive diagonal matrix ${\mathfrak{I}}_l=diag\left({\mathcal{I}}_{1l},\dots ,{\mathcal{I}}_{nl}\right)$ such that

$$\left|I_{pl}\left(u\right)-I_{pl}\left(v\right)\right|⩽{\mathcal{I}}_{pl}|u-v|$$

for all $u, v\in R(u\ne v),p\in \{1,2,\dots ,n\},l\in N.$

Hypothesis 3.

There exist some positive numbers ${\delta }_{p1},{\delta }_{p2}$, and ${\delta }_{p3}$, $p=1,2,\dots ,n$ such that $max|\Delta a_p\left(t\right)|⩽{\delta }_{p1},max|\Delta b_{pq}\left(t\right)|⩽{\delta }_{p2}$, and $max|\Delta c_{pq}\left(t\right)|⩽{\delta }_{p3}$, $p,q=1,2,\dots ,n$, $\forall t⩾0$.

Remark 1.

It is worth noting that the nonlinear functions $f_q$ and $g_p$ in the system (1) are subjected to H1 and the impulsive map $I_{pl}$ is assumed to be globally Lipschitz. Hence, the whole system (1) is closed to a linear one.

Remark 2.

In [9,35,36], the unstructured perturbations were dealt with through a relatively complex process. In real life, many perturbations are micro-perturbations, so we only assume that the unstructured perturbations are bounded as shown in assumption (H3).

Definition 1

([40,41]). If a function $u(t,x):R\times \Xi \to R^n$ satisfies the following conditions:

(1) 

$u(t,x)$ is a piecewise continuous function with the first kind of discontinuity point at $t_l$ for all $l\in N$ and $u(t,x)$ is right-continuous at each discontinuity point $t_l$;

(2) 

$u(t,x)$ satisfies system (1) for all $t⩾0$, and $u(t,x)=\mu (t,x)(t\in [-\tau ,0\left]\right)$;

then $u(t,x)$ is a particular solution that satisfies the initial conditions for system (1), and $u(t,\mu ,x)$ represents the special solution of system (1) under initial condition $\mu \in PC(\Xi )$.

Definition 2.

$u(t,\omega ,x)$ is a particular solution of the system (1) and satisfies the initial conditions $\omega \in PC(\Xi )$. In the initial condition of $\varpi \in PC(\Xi )$, $u(t,\varpi ,x)$ is any solution of the system (1). If there is a positive number $\lambda >0$ and $M⩾1$ such that

$$∥u(t,\omega ,x)-u(t,\varpi ,x)∥⩽M∥\omega -\varpi ∥e^{-\lambda t}\mathrm{for}\mathrm{all}t⩾0$$

then the system (1) is said to be globally exponentially stable.

Definition 3

([42]). A real matrix $D={\left(d_{pq}\right)}_{n\times n}$ is said to be a nonsingular M-matrix if $d_{pq}⩽0,p,q=1,2,\dots ,n, p\ne q,$ and all successive principal minors of D are positive.

Lemma 1

([42]). Let $D=$${\left(d_{pq}\right)}_{n\times n}$ with $d_{pq}⩽0(p\ne q),$ and then D is a nonsingular M-matrix if and only if the diagonal elements of D are all positive and there exists a positive vector d such that $Dd>0$ or $D^{T}d>0.$

Lemma 2

([26]). Let $\left|x_k\right|<L_k$$(k=1,2,\dots ,m)$ and $h\left(x\right)$ be a real-valued function belonging to $C^1(\Xi )$, which vanishes on the boundary an of $\Xi ,$ i.e., ${\left.h\left(x\right)\right|}_{\partial \Xi }=0$, and then

$${\int }_{\Xi }{h}^2\left(x\right)dx⩽L_k^2{\int }_{\Xi }{\left|{\displaystyle \frac{\partial h}{\partial x_k}}\right|}^{2}dx.$$

In the following, we establish an impulsive delay inequality. It can be regarded as a generalization of the Halanay inequality and Lemma 2.1 in [43] and Lemma 4.1 in [40]. Later, we apply this inequality to analyze exponential stability of impulsive delayed neural networks with parameter uncertainties and reaction–diffusion terms.

Lemma 3.

Let $\tau >0, a<b⩽+\infty$, and suppose that $U\left(t\right)=$${\left(U_1\left(t\right),U_2\left(t\right),\dots ,U_n\left(t\right)\right)}^{\mathrm{T}}\in C\left[[a,b),R^n\right]$ satisfies the following differential inequality:

$$\left\{\begin{array}{cc}{D}^{+}U\left(t\right)⩽PU\left(t\right)+(Q\otimes \overline{U}\left(t\right))\mathbf{e},\hfill & a⩽t<b,\hfill \\ U(a+s)\in PC,\hfill & -\tau ⩽s<0,\hfill \end{array}\right.$$

where $P={\left({\tilde{p}}_{pq}\right)}_{n\times n}$, ${\tilde{p}}_{pq}⩾0$$(p\ne q), Q={\left({\tilde{q}}_{pq}\right)}_{n\times n}$, ${\tilde{q}}_{pq}⩾0$, $\overline{U}\left(t\right)={\left(U\left(t-{\tau }_{pq}\left(t\right)\right)\right)}_{n\times n}$. If the initial condition satisfies

$$U\left(t\right)⩽\kappa \xi {\mathrm{e}}^{-\lambda (t-a)},\kappa ⩾0,\xi ={\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)}^{\mathrm{T}}>0,t\in [a-\tau ,a],$$

where the value of λ is determined by the following inequality:

$$[\lambda E+P+Q\otimes \zeta \left(\lambda \right)]\xi <0,\zeta \left(\lambda \right)={\left({\mathrm{e}}^{\lambda {\tau }_{pq}\left(t\right)}\right)}_{n\times n},$$

then $U\left(t\right)⩽\kappa \xi e^{-\lambda (t-a)}\mathrm{for}t\in [a,b)$.

Proof. 

For $p\in \{1,2,\dots ,n\}$ and arbitrary $\epsilon >0,$ define $w_p\left(t\right)\triangleq$$(\kappa +\epsilon ){\xi }_p{\mathrm{e}}^{-\lambda (t-a)}$. We prove that

$$\begin{array}{c}\hfill U_p\left(t\right)⩽w_p\left(t\right)=(\kappa +\epsilon ){\xi }_p{\mathrm{e}}^{-\lambda (t-a)},t\in [a,b),p=1,2,\dots ,n.\end{array}$$

(2)

If this is not true, then there exist a number $t^{*}\in [a,b)$ and some integer r such that

$$\begin{array}{c}\hfill U_r\left(t^{*}\right)=w_r\left(t^{*}\right),D^{+}{U}_r\left(t^{*}\right)⩾{\dot{w}}_r\left(t^{*}\right),U_p\left(t\right)⩽w_p\left(t\right),t\in \left[a,t^{*}\right],p=1,2,\dots ,n.\end{array}$$

(3)

From Lemma 3 and (2), we have

$$\begin{array}{ccc}\hfill D^{+}{U}_r\left(t^{*}\right)& ⩽& {\displaystyle \sum _{q=1}^n}\left[{\tilde{p}}_{rq}{U}_q\left(t^{*}\right)+{\tilde{q}}_{rq}{U}_q\left(t^{*}-{\tau }_{rq}\left(t^{*}\right)\right)\right]\hfill \\ & ⩽& {\displaystyle \sum _{q=1}^n}\left[{\tilde{p}}_{rq}(\kappa +\epsilon ){\xi }_q{\mathrm{e}}^{-\lambda \left(t^{*}-a\right)}+{\tilde{q}}_{rq}(\kappa +\epsilon ){\xi }_q{\mathrm{e}}^{-\lambda \left(t^{*}-{\tau }_{rq}\left(t^{*}\right)-a\right)}\right]\hfill \\ & =& {\displaystyle \sum _{q=1}^n}\left[{\tilde{p}}_{rq}(\kappa +\epsilon ){\xi }_q{\mathrm{e}}^{-\lambda \left(t^{*}-a\right)}+{\tilde{q}}_{rq}{\mathrm{e}}^{\lambda {\tau }_{rq}\left(t^{*}\right)}(\kappa +\epsilon ){\xi }_q{\mathrm{e}}^{-\lambda \left(t^{*}-a\right)}\right]\hfill \\ & ⩽& {\displaystyle \sum _{q=1}^n}\left[{\tilde{p}}_{rq}+{\tilde{q}}_{rq}{\mathrm{e}}^{\lambda {\tau }_{rq}\left(t^{*}\right)}\right](\kappa +\epsilon ){\xi }_q{\mathrm{e}}^{-\lambda \left(t^{*}-a\right)}.\hfill \end{array}$$

(4)

Since $[\lambda E+P+Q\otimes \zeta \left(\lambda \right)]\xi <0,{\tilde{p}}_{pq}⩾0,$${\tilde{q}}_{pq}⩾0, (p\ne q)$, and $e^{\lambda {\tau }_{pq}}>0,$ it follows that

$${\displaystyle \sum _{q=1}^n}\left[{\tilde{p}}_{rq}+{\tilde{q}}_{rq}{e}^{\lambda {\tau }_{rq}\left(t^{*}\right)}\right]{\xi }_q<-\lambda {\xi }_r<0,$$

From (4), one can obtain

$$D^{+}{U}_r\left(t^{*}\right)<-\lambda {\xi }_r(\kappa +\epsilon ){\mathrm{e}}^{-\lambda \left(t^{*}-l\right)}={\dot{w}}_r\left(t^{*}\right).$$

It is obvious that the result obtained is in contradiction with Equation (2), so inequality (2) holds. Let $\epsilon \to 0$ and one has

$$U_p\left(t\right)⩽\kappa {\xi }_p{\mathrm{e}}^{-\lambda (t-a)},p=1,2,\dots ,n,t\in [a,b).$$

The proof is completed. □

3. Global Exponential Stability Criteria

In this section, we obtain sufficient conditions for the global exponential stability of system (1).

Theorem 1.

Under assumptions (H1)–(H3), then system (1) is globally exponentially stable if the following two conditions (C1) and (C2) hold:

(C1) 

There exist a vector $\xi ={\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)}^{\mathrm{T}}>0$ and a constant $\lambda >0$ such that

$$[\lambda E-D+BF+CG\otimes \zeta \left(\lambda \right)]\xi <0,t⩾0,\zeta \left(\lambda \right)=e^{\lambda {\tau }_{pq}\left(t\right)},$$

where $D=diag\left(d_1,d_2,\dots ,d_n\right)$ with $d_p=a_p-{\delta }_{p1}+{\sum }_{k=1}^m{\displaystyle \frac{D_{pk}}{L_k^2}}$, $\zeta \left(\lambda \right)={\left({\mathrm{e}}^{\lambda {\tau }_{pq}\left(t\right)}\right)}_{n\times n}$, $B={\left(|b_{pq}+{\delta }_{p2}|\right)}_{n\times n},C={\left(|c_{pq}+{\delta }_{p3}|\right)}_{n\times n}$, $F=$ diag$(F_1,F_2,\dots ,F_n)$, $G=$ diag$(G_1,G_2,$$\dots ,G_n)$, and E is a n-dimensional identity matrix.

(C2) 

There is a positive number $\gamma >0$ such that

$$\eta =\underset{l\in N}{sup}\left\{{\displaystyle \frac{ln{\gamma }_l}{t_l-t_{l-1}}}\right\}⩽\gamma <\lambda ,$$

where ${\gamma }_l=\underset{1⩽p⩽n}{max}\left\{1,{\mathcal{I}}_{pl}\right\},l\in N$.

Proof. 

For $\omega ,\varpi \in PC(\Xi )$, the solutions of system (1) through $(0,\omega )$ and $(0,\varpi )$ are denoted, respectively, as follows:

$$\begin{array}{ccc}\hfill u(t,\omega ,x)& =& {\left(u_1(t,\omega ,x),u_2(t,\omega ,x),\dots ,u_n(t,\omega ,x)\right)}^T,\hfill \\ \hfill u(t,\varpi ,x)& =& {\left(u_1(t,\varpi ,x),u_2(t,\varpi ,x),\dots ,u_n(t,\varpi ,x)\right)}^T.\hfill \end{array}$$

Define $u_t(\omega ,x)=u(t+s,\omega ,x),u_t(\varpi ,x)=u(t+s,\varpi ,x),-\tau <s⩽0,t⩾0,$ and then $u_t(\omega ,x)$, $u_t(\varpi ,x)\in PC(\Xi )$ for $t⩾0.$ Let $U_p(t,\psi ,x)=$ $u_p(t,\omega ,x)-u_p(t,\varpi ,x),p=1,2,\dots ,n,\psi =\omega -\varpi$, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial U_p(t,\psi ,x)}{\partial t}}& =& {\displaystyle \sum _{k=1}^m}{\displaystyle \frac{\partial }{\partial x_k}}\left(D_{pk}{\displaystyle \frac{\partial U_p(t,\psi ,x)}{\partial x_k}}\right)-(a_p+\Delta a_p\left(t\right))U_p(t,\psi ,x)\hfill \\ & & -{\displaystyle \sum _{q=1}^n}(b_{pq}+\Delta b_{pq}\left(t\right))\left[f_q\left(u_q(t,\omega ,x)\right)-f_q\left(u_q(t,\varpi ,x)\right)\right]\hfill \\ & & -{\displaystyle \sum _{q=1}^n}(c_{pq}+\Delta c_{pq}\left(t\right))\left[g_q\left(u_q(t-{\tau }_{pq}\left(t\right),\omega ,x)\right)-g_q\left(u_q(t-{\tau }_{pq}\left(t\right),\varpi ,x)\right)\right]\hfill \end{array}$$

(5)

for $t\ne t_l,x\in \Xi ,p=1,2,\dots ,n$.

Multiplying both sides of (5) by $U_p(t,\psi ,x)$ and integrating them, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}{\int }_{\Xi }{\left(U_p(t,\psi ,x)\right)}^{2}dx& =& 2{\int }_{\Xi }{U}_p(t,\psi ,x){\displaystyle \sum _{k=1}^m}{\displaystyle \frac{\partial }{\partial x_k}}\left(D_{pk}{\displaystyle \frac{\partial U_p(t,\psi ,x)}{\partial x_k}}\right)dx-g_q\left(u_q(t-{\tau }_{pq}\left(t\right),\varpi ,x)\right)]dx\hfill \\ & & +2{\displaystyle \sum _{q=1}^n}(b_{pq}+\Delta b_{pq}\left(t\right)){\int }_{\Xi }{U}_p(t,\psi ,x)\left[f_q\left(u_q(t,\omega ,x)\right)-f_q\left(u_q(t,\varpi ,x)\right)\right]dx\hfill \\ & & +2{\displaystyle \sum _{q=1}^n}(c_{pq}+\Delta c_{pq}\left(t\right)){\int }_{\Xi }{U}_p(t,\psi ,x)[g_q\left(u_q(t-{\tau }_{pq}\left(t\right),\omega ,x)\right)\hfill \\ & & -2{\int }_{\Xi }(a_p+\Delta a_p\left(t\right))U_p^2(t,\psi ,x)dx.\hfill \end{array}$$

(6)

Applying the Dirichlet boundary conditions and Green’s formula, we can obtain

$$\begin{array}{c}\hfill {\int }_{\Xi }{U}_p(t,\psi ,x){\displaystyle \sum _{k=1}^m}{\displaystyle \frac{\partial }{\partial x_k}}\left(D_{pk}{\displaystyle \frac{\partial U_p(t,\psi ,x)}{\partial x_k}}\right)dx=-{\displaystyle \sum _{k=1}^m}{\int }_{\Xi }{D}_{pk}{\left({\displaystyle \frac{\partial \left(U_p(t,\psi ,x)\right)}{\partial x_k}}\right)}^{2}dx.\end{array}$$

By Lemma 2, we have

$$\begin{array}{ccc}\hfill {\int }_{\Xi }{U}_p(t,\psi ,x){\displaystyle \sum _{k=1}^m}{\displaystyle \frac{\partial }{\partial x_k}}\left(D_{pk}{\displaystyle \frac{\partial U_p(t,\psi ,x)}{\partial x_k}}\right)dx& \le & -{\displaystyle \sum _{k=1}^m}{\displaystyle \frac{D_{pk}}{L_k^2}}{\int }_{\Xi }{\left(U_p(t,\psi ,x)\right)}^{2}dx\hfill \\ & =& -{\displaystyle \sum _{k=1}^m}{\displaystyle \frac{D_{pk}}{L_k^2}}{∥U_p(t,\psi ,x)∥}_2^2.\hfill \end{array}$$

(7)

Applying assumptions (H1)–(H3) and Holder inequality, it yields

$$\begin{array}{ccc}& & {\displaystyle \sum _{q=1}^n}(b_{pq}+\Delta b_{pq}\left(t\right)){\int }_{\Xi }{U}_p(t,\psi ,x)\left[f_q\left(u_q(t,\omega ,x)\right)-f_q\left(u_q(t,\varpi ,x)\right)\right]dx\hfill \\ & ⩽& {\displaystyle \sum _{q=1}^n}\left(b_{pq}+\Delta b_{pq}\left(t\right)\right){\int }_{\Xi }\left|U_p(t,\psi ,x)\right|\left|f_q\left(u_q(t,\omega ,x)\right)-f_q\left(u_q(t,\varpi ,x)\right)\right|dx\hfill \\ & ⩽& {\displaystyle \sum _{q=1}^n}\left(b_{pq}+{\delta }_{p2}\right){\int }_{\Xi }\left|U_p(t,\psi ,x)\right|\left|U_q(t,\psi ,x)\right|F_{q}dx\hfill \\ & ⩽& {\displaystyle \sum _{q=1}^n}\left|b_{pq}+{\delta }_{p2}\right|{∥U_p(t,\psi ,x)∥}_2{F}_q{∥U_q(t,\psi ,x)∥}_2.\hfill \end{array}$$

(8)

Similarly, we can obtain

$$\begin{array}{ccc}& & {\displaystyle \sum _{q=1}^n}(c_{pq}+\Delta c_{pq}\left(t\right)){\int }_{\Xi }{U}_p(t,\psi ,x)\left[g_q\left(u_q(t-{\tau }_{pq}\left(t\right),\mu ,x)\right)-g_q\left(u_q(t-{\tau }_{pq}\left(t\right),\varpi ,x)\right)\right]dx\hfill \\ & ⩽& {\displaystyle \sum _{q=1}^n}\left|c_{pq}+{\delta }_{p3}\right|{∥U_q(t-{\tau }_{pq}\left(t\right),\psi ,x)∥}_2{G}_q{∥U_p(t,\psi ,x)∥}_2.\hfill \end{array}$$

(9)

Applying (6)–(9) to (5), we can obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥U_p(t,\psi ,x)∥}_2^2& \le & -\left(a_p-{\delta }_{p1}+{\displaystyle \sum _{k=1}^m}{\displaystyle \frac{D_{pk}}{L_k^2}}\right){\parallel U_p(t,\psi ,x)\parallel }_2^2\hfill \\ & & +{\displaystyle \sum _{q=1}^n}\left|c_{pq}+{\delta }_{p3}\right|{∥U_q(t-{\tau }_{pq}\left(t\right),\psi ,x)∥}_2{G}_q{∥U_p(t,\psi ,x)∥}_2\hfill \\ & & +{\displaystyle \sum _{q=1}^n}|b_{pq}+{\delta }_{p2}|\parallel U_p{(t,\psi ,x)\parallel }_2{F}_q{\parallel U_q(t,\psi ,x)\parallel }_2,\hfill \end{array}$$

i.e.,

$$\begin{array}{ccc}\hfill D^{+}{∥U_p(t,\psi ,x)∥}_2& \le & -\left(a_p-{\delta }_{p1}+{\displaystyle \sum _{k=1}^m}{\displaystyle \frac{D_{pk}}{L_k^2}}\right){∥U_p(t,\psi ,x)∥}_2+{\displaystyle \sum _{q=1}^n}\left|b_{pq}+{\delta }_{p2}\right|F_q{∥U_q(t,\vartheta ,x)∥}_2\hfill \\ & & +{\displaystyle \sum _{q=1}^n}\left|c_{pq}+{\delta }_{p3}\right|{∥U_q(t-{\tau }_{pq}\left(t\right),\psi ,x)∥}_2{G}_q.\hfill \end{array}$$

(10)

Let $U_p\left(t\right)={∥U_p(t,\psi ,x)∥}_2,U\left(t\right)={(U_1\left(t\right),U_2\left(t\right),\dots ,U_n\left(t\right))}^T,\overline{U}\left(t\right)={\left(U_p\left(t-{\tau }_{pq}\left(t\right)\right)\right)}_{n\times n},$$d_p=a_p-{\delta }_{p1}+{\sum }_{k=1}^m\left(D_{pk}/L_k^2\right), p=1,2,\dots ,n,D=diag\left(d_1\right.\left.d_2,\dots ,d_n\right)$, $B={\left(|b_{pq}+{\delta }_{p2}|\right)}_{n\times n},$ $C={\left(|c_{pq}+{\delta }_{p3}|\right)}_{n\times n}$, $P=-D+BF, Q=CG$. Then, (10) can be reduced to the following form:

$$\begin{array}{c}\hfill D^{+}U\left(t\right)⩽PU\left(t\right)+(Q\otimes \overline{U}\left(t\right))\mathbf{e}.\end{array}$$

(11)

It means that there exist a vector $\xi ={\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)}^{\mathrm{T}}>0$ and a constant $\lambda >0$ such that

$$\begin{array}{c}\hfill [\lambda E-D+BF+CG\otimes \zeta (\lambda \left)\right]\xi <0.\end{array}$$

(12)

Take $\kappa =\parallel \psi \parallel /\underset{1⩽p⩽n}{min}\left\{{\xi }_p\right\}$; it is easy to obtain

$$\begin{array}{c}\hfill U\left(t\right)⩽\kappa \xi e^{-\lambda t},t\in [-\tau ,t_0],t_0=0.\end{array}$$

(13)

By Lemma 3, we have

$$U\left(t\right)⩽\kappa \xi e^{-\lambda t},t>0.$$

If $m⩽l$, the following inequality holds:

$$\begin{array}{c}\hfill U\left(t\right)⩽\kappa {\gamma }_0\dots {\gamma }_{m-1}\xi {\mathrm{e}}^{-\lambda t},t\in [t_{m-1},t_m),{\gamma }_0=1.\end{array}$$

(14)

When $m=l+1,$ we obtain

$$\begin{array}{ccc}\hfill U\left(t_l\right)& =& {∥u(t_l,\omega ,x)-u(t_l,\varpi ,x)∥}_2\hfill \\ & =& \parallel I_l\left(u(t_l^{-},\omega ,x)\right)-I_l\left(u(t_l^{-},\varpi ,x)\right){\parallel }_2\hfill \\ & ⩽& {\mathfrak{I}}_l{\parallel U(t_l^{-},\psi ,x)\parallel }_2\hfill \\ & =& {\mathfrak{I}}_{l}U\left(t_l^{-}\right)\hfill \\ & ⩽& \kappa {\gamma }_0\dots {\gamma }_{l-1}{\mathfrak{I}}_l\xi \underset{t\to t_l^{-}}{lim}{\mathrm{e}}^{-\lambda t_l}\hfill \\ & ⩽& \kappa {\gamma }_0\dots {\gamma }_{l-1}{\gamma }_l\xi {\mathrm{e}}^{-\lambda t_l}.\hfill \end{array}$$

(15)

Combining (14), (15), and ${\gamma }_l⩾1,$ we have

$$\begin{array}{c}\hfill U\left(t\right)⩽\kappa {\gamma }_0\dots {\gamma }_{l-1}{\gamma }_l\xi {\mathrm{e}}^{-\lambda t},t_l-\tau ⩽t⩽t_l.\end{array}$$

(16)

By (12), (13), (16), and Lemma 3, we can obtain

$$\begin{array}{c}\hfill U\left(t\right)⩽\kappa {\gamma }_0\dots {\gamma }_{l-1}{\gamma }_l\xi {\mathrm{e}}^{-\lambda t},t_l⩽t<t_{l+1}.\end{array}$$

(17)

By mathematical induction, we can obtain

$$\begin{array}{c}\hfill U\left(t\right)⩽\kappa {\gamma }_0\dots {\gamma }_{l-1}\xi {\mathrm{e}}^{-\lambda t},t_{l-1}⩽t<t_l,l\in N.\end{array}$$

(18)

From condition $\left(\mathrm{C}2\right)$ and (18), we have

$$\begin{array}{c}\hfill U\left(t\right)\le \kappa {\mathrm{e}}^{\gamma t_1}{\mathrm{e}}^{\gamma \left(t_2-t_1\right)}\dots {\mathrm{e}}^{\gamma \left(t_{l-1}-t_{l-2}\right)}\xi {\mathrm{e}}^{-\lambda t}\le \kappa \xi {\mathrm{e}}^{\gamma t}{\mathrm{e}}^{-\lambda t}=\kappa \xi {\mathrm{e}}^{-(\lambda -\gamma )t}\end{array}$$

(19)

for all $l\in N,t_{l-1}⩽t<t_l$. It shows

$$\begin{array}{ccc}\hfill \parallel u(t,\omega ,x)-u(t,\varpi ,x)\parallel & =& {\displaystyle \sum _{p=1}^n}{∥u_p(t,\omega ,x)-u_p(t,\varpi ,x)∥}_2\hfill \\ & =& {\displaystyle \sum _{p=1}^n}{U}_p\left(t\right)\hfill \\ & ⩽& {\displaystyle \sum _{p=1}^n}\kappa {\xi }_p{e}^{-\lambda t}\hfill \\ & =& {\displaystyle \frac{{\displaystyle \sum _{p=1}^n}{\xi }_p}{\underset{1⩽p⩽n}{min}\left\{{\xi }_p\right\}}}\parallel \omega -\varpi \parallel e^{-\lambda t}.\hfill \end{array}$$

That is,

$$∥u_t(\omega ,x)-u_t(\varpi ,x)∥⩽M\parallel \omega -\varpi \parallel e^{-\lambda t},t⩾0,$$

where $M=\left({\displaystyle \sum _{p=1}^n}{\xi }_p/\underset{1⩽p⩽n}{min}\left\{{\xi }_p\right\}\right)$. The proof is completed. □

Remark 3.

In Theorem 1, the condition (C1) ensures the global exponential stability of the uncertainty system (1). The parameter $\lambda -\eta$ is actually an estimate of exponential convergence rate of system (1), which is delay-dependent. Condition (C2) shows the fact that the exponential stability of system (1) still remains if the global impulsive perturbation intensity $\eta \in [0,\lambda )$. Thus, Theorem 1 actually characterizes the robustness of stability for the impulsive perturbed system (1).

Remark 4.

It is easy to see from conditions (C1) and (C2) in Theorem 1 that the external input $J_p$ only changes the position of the equilibrium point of the system (1) and has no effect on the stability of the system (1). Moreover, the exponential stability of system (1) still remains when the unstructured perturbations $\Delta a_p\left(t\right), \Delta b_p\left(t\right), \Delta c_p\left(t\right)$ are bounded.

Corollary 1.

If ${\tau }_{pq}\left(t\right)=\tau ,p,q=1,2,\dots ,n$, τ is a positive constant and system (1) satisfies assumptions (H1)–(H3) and conditions (C1) and (C2), and then system (1) is globally exponentially stable.

If impulsive operator $I_{pl}\left(u_p\right)=0,p=1,2,\dots ,n,l\in N$, system (1) will become a model of the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\displaystyle \frac{\partial u_p(t,x)}{\partial t}}=& {\displaystyle \sum _{k=1}^m}{\displaystyle \frac{\partial }{\partial x_k}}\left(D_{pk}{\displaystyle \frac{\partial u_p(t,x)}{\partial x_k}}\right)-(a_p+\Delta a_p\left(t\right))u_p(t,x)+\hfill \\ & {\displaystyle \sum _{q=1}^n}(b_{pq}+\Delta b_{pq}\left(t\right))f_q\left(u_q(t,x)\right)\hfill \\ & +{\displaystyle \sum _{q=1}^n}(c_{pq}+\Delta c_{pq}\left(t\right))g_q\left(u_q\left(t-{\tau }_{pq}\left(t\right),x\right)\right)\hfill \\ & +J_p,p=1,2,\dots ,n,x\in \Xi ,\hfill \\ \hfill u_p(t,x)=& 0,t⩾0,x\in \partial \Xi ,\hfill \\ \hfill u_p(s,x)=& {\mu }_p(s,x),s\in [-\tau ,0],p=1,2,\dots ,n,\hfill \end{array}\right.\end{array}$$

(20)

Theorem 2.

System (20) is globally exponentially stable if it satisfies assumptions (H1) and (H3) and the following condition:

$$\begin{array}{c}\hfill \left(a_p-{\delta }_{p1}+{\displaystyle \sum _{k=1}^m}\left({\displaystyle \frac{D_{pk}}{L_k^2}}\right)\right)>{\displaystyle \sum _{q=1}^n}\left[\left|b_{pq}+{\delta }_{p2}\right|F_q+\left|c_{pq}+{\delta }_{p3}\right|G_q\right],p,q=1,2,\dots ,n.\end{array}$$

(21)

Proof. 

Let $\Theta =D-BF-CG$, and then we know from condition (21) that all successive principal minors of $\Theta$ are positive because of diagonally dominant principle. It shows $\Theta$ is a nonsingular M-matrix by Definition 3. From Lemma 1, there exists $\xi ={\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)}^{\mathrm{T}}>0$ such that

$$[D-BF-CG\otimes \zeta (0\left)\right]\xi >0,\zeta \left(0\right)=1.$$

Since the continuity, there exists $\lambda >0$, satisfying

$$\begin{array}{c}\hfill [\lambda E-D+BF+CG\otimes \zeta \left(\lambda \right)]\xi <0,\zeta \left(\lambda \right)=e^{\lambda {\tau }_{pq}\left(t\right)},\end{array}$$

(22)

where $D=diag\left(d_1,d_2,\dots ,d_n\right)$, $d_p=a_p-{\delta }_{p1}+{\displaystyle \sum _{k=1}^m}{\displaystyle \frac{D_{pk}}{L_k^2}}$, $\zeta \left(\lambda \right)={\left({\mathrm{e}}^{\lambda {\tau }_{pq}\left(t\right)}\right)}_{n\times n}$, $B={\left(|b_{pq}+{\delta }_{p2}|\right)}_{n\times n},C={\left(|c_{pq}+{\delta }_{p3}|\right)}_{n\times n}$, $F=$ diag$(F_1,F_2,\dots ,F_n)$, and $G=$ diag$(G_1,G_2,\dots ,G_n)$. This shows that condition (C1) holds. □

Remark 5.

Theorem 2 shows that λ has no relation to the stability of the system; it only affects the exponential convergence rate of the system (23).

Remark 6.

It is worth noting that the existing works on some uncertain neural networks become a special case of the system (1) constructed in this paper; for example, see [35,36]. In their works, the results obtained are independent of the impulse and reaction–diffusion. Compared to [35], it is easy to see that our standards have improved. However, it must be noted that, compared to LMI method in [36], none of our criteria are independent of the time delays when impulses are not considered.

Remark 7.

Although the model considered in this paper is different from one in [31], the criteria we adopt have certain commonalities. And it is easy to see that all our criteria are presented in the form of simple algebraic inequalities; the obtained results are more concise.

When $D_{pk}=0$ or the reaction–diffusion can be neglected, the system (1) is rewritten as the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\displaystyle \frac{\partial u_p(t,x)}{\partial t}}=& -(a_p+\Delta a_p\left(t\right))u_p(t,x)+{\displaystyle \sum _{q=1}^n}(b_{pq}+\Delta b_{pq}\left(t\right))f_q\left(u_q(t,x)\right)\hfill \\ & +{\displaystyle \sum _{q=1}^n}(c_{pq}+\Delta c_{pq}\left(t\right))g_q\left(u_q\left(t-{\tau }_{pq}\left(t\right),x\right)\right)\hfill \\ & +J_p,t\ne t_l,p=1,2,\dots ,n,x\in \Xi ,\hfill \\ \hfill u_p(t_l^{+},x)=& I_{pl}\left(u_p(t_l^{-},x)\right),p=1,2,\dots ,n,x\in \Xi ,l\in N=\{1,2,\dots \},\hfill \\ \hfill u_p(t,x)=& 0,t⩾0,x\in \partial \Xi ,\hfill \\ \hfill u_p(s,x)=& {\mu }_p(s,x),s\in [-\tau ,0],p=1,2,\dots ,n,\hfill \end{array}\right.\end{array}$$

(23)

Corollary 2.

Under assumptions (H1)–(H3), then system (1) is globally exponentially stable if the following two conditions (D1) and (D2) hold:

(D1) 

There exist a vector $\xi ={\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)}^{\mathrm{T}}>0$ and a constant $\lambda >0$ such that

$$[\lambda E-D^{*}+BF+CG\otimes \zeta \left(\lambda \right)]\xi <0,t⩾0,\zeta \left(\lambda \right)=e^{\lambda {\tau }_{pq}\left(t\right)},$$

where $D^{*}=diag\left(d_1,d_2,\dots ,d_n\right)$, $d_p=a_p-{\delta }_{p1}$, $\zeta \left(\lambda \right)={\left({\mathrm{e}}^{\lambda {\tau }_{pq}\left(t\right)}\right)}_{n\times n}$, $B={\left(|b_{pq}+{\delta }_{p2}|\right)}_{n\times n}$, $C={\left(|c_{pq}+{\delta }_{p3}|\right)}_{n\times n}$, $F=$ diag$(F_1,F_2,\dots ,F_n)$, and $G=$ diag$(G_1,G_2,\dots ,G_n)$;

(D2) 

There is a positive number $\gamma >0$ such that

$$\underset{l\in N}{sup}\left\{{\displaystyle \frac{ln{\gamma }_l}{t_l-t_{l-1}}}\right\}⩽\gamma <\lambda ,$$

where ${\gamma }_l=\underset{1⩽p⩽n}{max}\left\{1,{\mathcal{I}}_{pl}\right\},l\in N$.

Corollary 3.

If $I_{pl}\left(u_p\right)=0$ and $D_{pk}=0$, system (1) is globally exponentially stable if it satisfies assumptions (H1) and (H3) and the following condition:

$$\begin{array}{c}\hfill \left(a_p-{\delta }_{p1}\right)>{\displaystyle \sum _{q=1}^n}\left[\left|b_{pq}+{\delta }_{p2}\right|F_q+\left|c_{pq}+{\delta }_{p3}\right|G_q\right],p,q=1,2,\dots ,n.\end{array}$$

(24)

4. Illustrative Examples

In this section, we will verify the validity of the results obtained by two examples.

Example 1.

Consider the uncertain impulsive system (1) with $n=m=2,{\mathcal{I}}_{pl}=1.1, \Xi =[0,1],t_l=0.1\pi l$, $max|\Delta a_p\left(t\right)|⩽{\delta }_{p1}=0.5,max|\Delta b_{pq}\left(t\right)|⩽{\delta }_{p2}=0.2,$ and $max|\Delta c_{pq}\left(t\right)|⩽{\delta }_{p3}=0.1$, where $p,q=1,2$. Let

$$f_q\left(u\right)=g_q\left(u\right)={\displaystyle \frac{|u+1|-|u-1|}{2}}.$$

Based on the definition of the above parameters and functions, it is easy to see that assumptions (H1)–(H3) are satisfied. Let

$$\begin{array}{c}\hfill \left(D_{p1}\right)=\left(\begin{array}{c}1\\ 1\end{array}\right),A=\left(\begin{array}{cc}1-{\delta }_{11}& 0\\ 0& 1-{\delta }_{21}\end{array}\right),B=\left(\begin{array}{cc}0.01+{\delta }_{12}& |-0.2+{\delta }_{12}|\\ 0.3+{\delta }_{22}& |-0.01+{\delta }_{22}|\end{array}\right)\end{array}$$

$$\begin{array}{c}\hfill C=\left(\begin{array}{cc}0.05+{\delta }_{13}& |-0.1+{\delta }_{13}|\\ |-0.2+{\delta }_{23}|& 0.08+{\delta }_{23}\end{array}\right),{\left(\tau \left(t\right)\right)}_{2\times 2}=\left(\begin{array}{cc}\left|sint\right|& \left|sint\right|\\ \left|cost\right|& \left|cost\right|\end{array}\right).\end{array}$$

and then we obtain

$$\begin{array}{c}\hfill \zeta \left(\lambda \right)=\left(\begin{array}{cc}{e}^{\lambda \left|sint\right|}& e^{\lambda \left|sint\right|}\\ e^{\lambda \left|cost\right|}& e^{\lambda \left|cost\right|}\end{array}\right),D=\left(\begin{array}{cc}1.1& 0\\ 0& 1.1\end{array}\right),F=G=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill B=\left(\begin{array}{cc}0.21& 0\\ 0.5& 0.19\end{array}\right),C=\left(\begin{array}{cc}0.15& 0\\ 0.1& 0.18\end{array}\right),\end{array}$$

and ${\gamma }_l=max\{1,1.1\}=1.1⩾1,\gamma =\underset{l\in N}{sup}{\displaystyle \frac{ln{\gamma }_l}{t_l-t_{l-1}}}={\displaystyle \frac{ln1.1}{0.1\pi }}\approx 0.3033.$

Take $\lambda =0.3946$ and $\xi =(1,1)$, and then condition (C1) in Theorem 1 is satisfied. Combined with the above equation, Theorem 1 holds. Therefore, the system (1) is globally exponentially stable. Under the given parameter conditions and different initial values, the numerical simulation of system (1) is shown in Figure 1. When ${\mathcal{I}}_{pl}=1.5, t_l=0.05\pi l$, the other parameters and functions remain uncharged. Under the initial value $x\left(t\right)={(0,0.5)}^T$, the numerical simulation of system (1) is shown in Figure 2.

Example 2.

Consider the uncertain impulsive system (1) with $n=m=2,{\mathcal{I}}_{pl}=0,\Xi =[0,1]$, $max|\Delta a_p\left(t\right)|⩽{\delta }_{p1}=0.2,max|\Delta b_{pq}\left(t\right)|⩽{\delta }_{p2}=0.5, andmax|\Delta c_{pq}\left(t\right)|⩽{\delta }_{p3}=0.2$, where $p,q=1,2$. Let

$$f_q\left(u\right)=g_q\left(u\right)={\displaystyle \frac{|u+1|-|u-1|}{2}}.$$

By selecting the parameters and functions, it is easy to see that assumptions (H1)–(H3) are satisfied. Let

$$\begin{array}{c}\hfill \left(D_{p1}\right)=\left(\begin{array}{c}1\\ 1\end{array}\right),A=\left(\begin{array}{cc}1-{\delta }_{11}& 0\\ 0& 1-{\delta }_{21}\end{array}\right),B=\left(\begin{array}{cc}0.02+{\delta }_{12}& |-0.3+{\delta }_{12}|\\ 0.3+{\delta }_{22}& 0.02+{\delta }_{22}\end{array}\right)\end{array}$$

$$\begin{array}{c}\hfill C=\left(\begin{array}{cc}0.05+{\delta }_{13}& 0.2+{\delta }_{13}\\ |-0.04+{\delta }_{23}|& 0.1+{\delta }_{23}\end{array}\right),{\left(\tau \left(t\right)\right)}_{2\times 2}=\left(\begin{array}{cc}\left|sint\right|& \left|sint\right|\\ \left|cost\right|& \left|cost\right|\end{array}\right).\end{array}$$

and then we obtain $F_1=1,F_2=0,G_1=0,G_2=1$, and

$$1.5⩽\left(a_p-{\delta }_{p1}+{\displaystyle \sum _{k=1}^1}\left(D_{pk}/L_k^2\right)\right),$$

$$0⩽{\displaystyle \sum _{q=1}^2}\left[\left|b_{1q}+{\delta }_{12}\right|F_q+\left|c_{1q}+{\delta }_{13}\right|G_q\right]⩽0.77,$$

$$0⩽{\displaystyle \sum _{q=1}^2}\left[\left|b_{2q}+{\delta }_{22}\right|F_q+\left|c_{2q}+{\delta }_{23}\right|G_q\right]⩽0.82,$$

for $p,q=1,2$. It is obvious that condition (22) in Theorem 2 holds. Then, the solution of system (1) under the given parameter conditions and different initial values is globally exponentially stable, and the numerical simulation of system (1) is shown in Figure 3.

Remark 8.

From Examples 1 and 2, we find that the system (1) is stable in the absence of impulses. The system (1) also stabilizes rapidly when the impulsive disturbance intensity is greater than 1 and conditions (C1) and (C2) are satisfied. In addition, when the parameter conditions are determined, the different initial values have no great influence on the stability of the system (1).

5. Conclusions

This paper investigates a class of impulsive delayed neural networks with parameter uncertainties and reaction–diffusion terms. By establishing a new differential inequality and combining the properties of M-matrices and some inequality tricks, several sufficient conditions are presented to guarantee the global exponential stability of the impulsive systems. Moreover, all the results are provided in the form of simple algebraic inequalities, which are certainly more concise than the previous LMI methods. Finally, the validity and generality of our results are illustrated with two arithmetic examples.

To reduce the computational difficulties, we mainly investigate the stability of neural networks with impulse, in which the impulsive intensity is a constant number. However, in engineering applications, the form of the impulsive intensity is rather complicated. Thus, in our future work, we plan to focus on investigating the stability of neural networks with random impulsive and reaction–diffusion terms. Therefore, it is important and interesting, but much more difficult.