Finite-Time Synchronization Analysis for BAM Neural Networks with Time-Varying Delays by Applying the Maximum-Value Approach with New Inequalities

Abstract

In this paper, we consider the finite-time synchronization for drive-response BAM neural networks with time-varying delays. Instead of using the finite-time stability theorem and integral inequality method, by using the maximum-value method, two new criteria are obtained to ensure the finite-time synchronization for the considered drive-response systems. The inequalities in our paper, applied to obtaining the maximum-valued and designing the novel controllers, are different from those in existing papers.

1. Introduction

In 1987, Kosko firstly proposed a two-way associative search for stored bipolar vector pairs and generalized the single-layer auto-associative Hebbian correlation to a two-layer pattern-matched hetero-associative circuit. These are a class of important neural networks, called bidirectional associative memory (BAM) neural networks [1,2]. The dynamic behaviors of BAM neural networks are of significant application prospects in various fields, such as automatic control, signal processing, pattern recognition and associative memory, which has arisen the great interest of researchers. To date, many researchers have analyzed various dynamical behaviors of BAM neural networks, and obtained many dynamical analysis results of BAM neural networks [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. In recent years, the global asymptotic/exponential synchronization of BAM neural networks has been widely investigated, for example, see [3,4,5,6,7,8,9,10,11,12,13] and references therein. Meanwhile, many scholars have devoted themselves to the study of the finite-time synchronization of BAM neural networks, for example, see [14,15,16,17,18,19,20,21,22,42] and references therein.

In [14], the authors developed a finite-time synchronization for the drive–response fuzzy inertial BAM neural networks by employing integral inequality techniques and the figure analysis approach. Furthermore, the finite-time stochastic synchronization for memristor-based BAM neural networks with time-varying delays and stochastic disturbances was considered in [15]. In addition, the authors built more reasonable switching conditions of the finite-time synchronization by using a stochastic analysis technique. In [16], researchers introduced a different inequality, the fractional-order Gronwall inequality with time delays, and then dealt with the finite-time synchronization problem of fractional-order memristor-based neural networks with time delays. Zhang and Yang [17] studied the finite-time impulsive synchronization issue of fractional-order memristive BAM neural networks involving switch jumps mismatch by designing two impulsive controllers and using the properties of Gamma functions. In [18], the finite-time synchronization of drive–response BAM fuzzy neural networks with time delays and impulsive effects was investigated. The authors’ goal was to illustrate the effects of both impulse and time delay in a finite-time control area in terms of novel Lyapunov functionals and special analytical techniques. Motivated by security applications in the image transmission, Wang et al. [19] provided some sufficient conditions to guarantee the finite-time projective synchronization for memristor-based BAM neural networks.

So far, the investigated results about the finite-time synchronization of BAM neural networks and non-BAM neural networks have been acquired mainly by applying Lyapunov functionals [15,18], analysis approaches [15,17,18], inequality approaches [14,16,19,21,22], the finite-time stability theorem [20,21] and the integral inequality method [48,49,50]. Recently, two novel sufficient conditions were obtained to guarantee the finite-time synchronization for the considered drive–response fuzzy inertial neural networks by using the maximum-value approach [42].

Inspired by [42], we studied the finite-time synchronization of BAM neural networks by applying the maximum-value approach. In this paper, by using different inequality techniques from those in [42] and designing controllers different from those in [14,15,16,17,18,19,20,21,22,42], the finite-time synchronization criteria for drive–response BAM neural networks with time-varying delays are obtained. In proving the main results, one of the two difficulties in our paper was how to construct novel inequalities in the controllers to get $V^{\prime }(t)<\psi (t)$ and the other one was how to establish the new inequalities in the controllers to achieve a finite-time synchronization. Thus, the novelty of this paper is to construct some novel inequalities to obtain the maximum value of the considered functions and to get the finite time $t_1$ needed to achieve synchronization. As a result, the main contribution of this paper lies in the following three aspects:

(1) Some new inequalities which are different from those in [42] are constructed in the process of obtaining the maximum-value of the considered functions; (2) new inequalities described with fractional and integral functions are introduced to design the novel controllers; (3) using the maximum-value approach and designing novel controllers, two different conditions are attained to assure the finite-time synchronization between the drive system and the response system.

The rest of this paper is arranged as follows. In Section 2, some necessary preliminaries are given. In Section 3, two novel criteria to realize the finite-time synchronization are put forward for the drive system (1) and the response system (2). In Section 4, we exhibit two examples to validate the effectiveness and feasibility of the derived consequences.

2. Preliminaries

In this paper, we consider a class of BAM neural networks with time-varying delays described by

$$\left\{\begin{array}{c}\frac{d{x}_i(t)}{dt}=-a_i{x}_i(t)+{\displaystyle \sum _{j=1}^m}{c}_{ij}{f}_j(y_j(t))+{\displaystyle \sum _{j=1}^m}{d}_{ij}{f}_j(y_j(t-{\tau }_{ji}(t)))+I_i,\\ \frac{d{y}_j(t)}{dt}=-b_j{y}_j(t)+{\displaystyle \sum _{i=1}^n}{p}_{ji}{g}_i(x_i(t))+{\displaystyle \sum _{i=1}^n}{q}_{ji}{g}_i(x_i(t-{\sigma }_{ij}(t)))+J_j,\end{array}\right.$$

(1)

where $i=1,2,\cdots ,n,j=1,2,\cdots ,m,a_i>0$ and $b_j>0$ are constants; $x_i(t),y_j(t)$ are the states of the ith neuron and jth neuron; constants $c_{ij},d_{ij},p_{ji},q_{ji}$ denote the connection strengths; ${\tau }_{ji}(t),{\sigma }_{ij}(t)$ are time delays, $0<{\sigma }_{ij}^{\prime }(t)<{\sigma }^{*},0<{\tau }_{ji}^{\prime }(t)<{\tau }^{*}$; functions $f_j,g_i$ denote the activation functions; constants $I_i,J_j$ denote the external inputs.

The initial values of system (1) are given as follows:

$$x_i(s)={\varphi }_{xi}(s),s\in [-\sigma ,0],y_j(s)={\psi }_{yj}(s),s\in [-\tau ,0],$$

where $\sigma =\underset{1\le i\le n,1\le j\le m}{max}\{{\sigma }_{ij}(t)\},\tau =\underset{1\le i\le n,1\le j\le m}{max}\{{\tau }_{ji}(t)\},{\varphi }_{xi}(s),{\psi }_{yj}(s)$ are bounded continuous functions.

Throughout this paper, for system (1), we always assume that

Zhen: have modified. $(H_1)$ There exist constants $F_j>0,G_i>0$ such that

$$|f_j(x)-f_j(y)|\le F_j|x-y|,|g_i(x)-g_i(y)|\le G_i|x-y|,$$

for all $x,y\in R,i=1,2,\cdots ,n,j=1,2,\cdots ,m,|.|$ is the norm of the Euclidean space $R.$

For simplicity, we refer to (1) as the drive system, and consider the response BAM neural networks with time-varying delays described as follows:

$$\left\{\begin{array}{c}\frac{d{u}_i(t)}{dt}=-a_i{u}_i(t)+{\displaystyle \sum _{j=1}^m}{c}_{ij}{f}_j(v_j(t))+{\displaystyle \sum _{j=1}^m}{d}_{ij}{f}_j(v_j(t-{\tau }_{ji}(t)))+I_i+P_i(t),\\ \frac{d{v}_j(t)}{dt}=-b_j{v}_j(t)+{\displaystyle \sum _{i=1}^n}{p}_{ji}{g}_i(u_i(t))+{\displaystyle \sum _{i=1}^n}{q}_{ji}{g}_i(u_i(t-{\sigma }_{ij}(t)))+J_j+Q_j(t),\end{array}\right.$$

(2)

where $u_i(t),v_j(t)$ denote the states, the parameters are the same as those in system (1) and $P_i(t),Q_j(t)$ are the controllers to realize finite-time synchronization between the drive system (1) and the response system (2).

The initial values of system (2) are given as follows:

$$u_i(s)={\varphi }_{ui}(s),s\in [-\sigma ,0],v_j(s)={\psi }_{vj}(s),s\in [-\tau ,0],$$

where $\sigma =\underset{1\le i\le n,1\le j\le m}{max}\{{\sigma }_{ij}(t)\},\tau =\underset{1\le i\le n,1\le j\le m}{max}\{{\tau }_{ji}(t)\},{\varphi }_{ui}(s),{\psi }_{vj}(s)$ are bounded continuous functions.

Definition 1.

Drive–response systems (1) and (2) are said to be finite-time synchronized if for arbitrary solutions of system (1) and system (2) denoted by $[x_1(t),x_2(t),\cdots ,x_n(t),y_1(t),y_2(t),$$\cdots ,y_m{(t)]}^T$ and ${[u_1(t),u_2(t),\cdots ,u_n(t),v_1(t),v_2(t),\cdots ,v_m(t)]}^T$, under a suitable designed controller, there exists a time $T>0$ which is related to the initial condition, such that for $i=1,2,\cdots ,n;j=1,2,\cdots ,m$,

$$\underset{t\to T}{lim}|u_i(t)-x_i(t)|=0;\underset{t\to T}{lim}|v_j(t)-y_j(t)|=0;$$

$$|u_i(t)-x_i(t)|=0,t>T;|v_j(t)-y_j(t)|=0,t>T.$$

Lemma 1

([51]). If $x>0,y>0,p>1,\frac{1}{p}+\frac{1}{k}=1,k$ is a constant, then

$$xy\le \frac{x^p}{p}+\frac{y^k}{k}.$$

Lemma 2.

If $a>1,b>1,$ and $\frac{1}{a}+\frac{1}{b}=1,$ when $t>\frac{3}{4}$, the following inequality holds:

$$ln(4t+1)+1-2ln2<\frac{1}{a}{(t+\frac{1}{4})}^a+\frac{1}{b}.$$

Proof. 

Let $f(t)=\frac{1}{a}{(t+\frac{1}{4})}^a+\frac{1}{b}-ln(4t+1)-1+2ln2,t>\frac{3}{4}.$ Then $f^{\prime }(t)={(t+\frac{1}{4})}^{a-1}-\frac{4}{4t+1},t>\frac{3}{4}.$ Thus, $f^{″}(t)=\frac{16}{{(4t+1)}^2}\left[(a-1){(t+\frac{1}{4})}^{a+1}\right]>0,t>\frac{3}{4}.$ So $f^{\prime }(t)$ is a monotonically increasing function on $(\frac{3}{4},+\infty )$ and then $f^{\prime }(t)>f^{\prime }(\frac{3}{4})=0$. Therefore, when $t>\frac{3}{4}$, $f(t)$ is also a monotonically increasing function and $f(t)>f(\frac{3}{4})=0.$ Finally, $ln(4t+1)+1-2ln2<\frac{1}{a}{(t+\frac{1}{4})}^a+\frac{1}{b}$ holds. □

Lemma 3.

If $t\ge \frac{1}{e},a^{*}>\frac{1+e}{e^{\frac{1}{e}}-1},$ then $t+1+a^{*}t-a^{*}t{e}^t<0.$

Proof. 

We only need to prove $max\left\{\frac{1+t}{t(e^t-1)}\right\}\le \frac{1+e}{e^{\frac{1}{e}}-1}$.

Denote $F(t)=\frac{1+t}{t(e^t-1)}$. Then, we have $F^{\prime }(t)=\frac{1-e^t-t^2{e}^t-t{e}^t}{t^2{(e^t-1)}^2}<0,t\ge \frac{1}{e}$.

Hence, $F(t)$ is a monotonically decreasing function on $[\frac{1}{e},+\infty )$.

Furthermore, $F(t)\le F(\frac{1}{e})=\frac{1+e}{e^{\frac{1}{e}}-1}<a^{*}$. □

Lemma 4

([42]). Assume that z = f(x) defined on $(-\infty ,+\infty )$, and $x_0$ is a unique local maximum value point. Then, $\underset{x\in (-\infty ,+\infty )}{max}f(x)=max\{f(x_0);f(-\infty );f(+\infty )\}$.

Proof. 

The proof is well-known and it is omitted. □

3. Main Results

Let $e_i(t)=u_i(t)-x_i(t),r_j(t)=v_j(t)-y_j(t)$, then the error system can be described as follows:

$$\left\{\begin{array}{c}\frac{d{e}_i(t)}{dt}=-a_i{e}_i(t)+{\displaystyle \sum _{j=1}^m}{c}_{ij}\left[f_j(v_j(t))-f_j(y_j(t))\right]+{\displaystyle \sum _{j=1}^m}{d}_{ij}[f_j(v_j(t-{\tau }_{ji}(t)))\\ -f_j(y_j(t-{\tau }_{ji}(t)))]+P_i(t),\\ \frac{d{r}_j(t)}{dt}=-b_j{r}_j(t)+{\displaystyle \sum _{i=1}^n}{p}_{ji}\left[g_i(u_i(t))-g_i(x_i(t))\right]+{\displaystyle \sum _{i=1}^n}{q}_{ji}[g_i(u_i(t-{\sigma }_{ij}(t)))\\ -g_i(x_i(t-{\sigma }_{ij}(t)))]+Q_j(t).\end{array}\right.$$

(3)

The controllers in system (3) are designed as follows for $i=1,2,\cdots ,n,j=1,2,\cdots ,m$:

$$\left\{\begin{array}{c}{P}_i\left(t\right)=sign\left[e_i\left(t\right)\right]k_1+k_2{e}_i\left(t\right)+sign\left[e_i\left(t\right)\right]{\left|e_i\left(t\right)\right|}^{1-p}{k}_3,e_i\left(t\right)\ne 0,\\ Q_j\left(t\right)=sign\left[r_j\left(t\right)\right]l_1+l_2{r}_j\left(t\right)+sign\left[r_j\left(t\right)\right]\times {\left|r_j\left(t\right)\right|}^{1-p}\frac{\varphi \left(t\right)}{p}+\\ sign\left[r_j\left(t\right)\right]{\left|r_j\left(t\right)\right|}^{1-p}{l}_3,r_j\left(t\right)\ne 0\end{array}\right.$$

(4)

or

$$\left\{\begin{array}{c}{P}_i\left(t\right)={\left[e_i\left(t\right)\right]}^{-1}{w}_1+w_2{e}_i\left(t\right)+{\left[e_i\left(t\right)\right]}^{1-2q}{w}_3,e_i\left(t\right)\ne 0,\\ Q_j\left(t\right)={\left[r_j\left(t\right)\right]}^{-1}{z}_1+z_2{r}_j\left(t\right)+{\left[r_j\left(t\right)\right]}^{1-2q}\frac{\psi \left(t\right)}{2q}+\\ {\left[r_j\left(t\right)\right]}^{1-2q}{z}_3,r_j\left(t\right)\ne 0\end{array}\right.$$

(5)

where $p\ge 1,\varphi \left(t\right)=\frac{4}{4t+1}-{(t+\frac{1}{4})}^{a-1}-1,t\ge \frac{3}{4},a>1,b>1,$ and $\frac{1}{a}+\frac{1}{b}=1,$ $p{k}_3+k_1^p{\left(\frac{p-1}{-A_i}\right)}^{p-1}<0,$ $p{l}_3+l_1^p{\left(\frac{p-1}{-B_j}\right)}^{p-1}<0,$ $k_1>0,$ $k_2<0,$ $k_3<0,$ $l_1>0,$ $l_2<0,$ $l_3<0;$ $q\ge 1,$ $\psi \left(t\right)=1+a^{*}-a^{*}(e^t+t{e}^t)-e^t-3M\left(0\right),$ $t\ge \frac{1}{e},$ $a^{*}\ge \frac{1+e}{e^{\frac{1}{e}}-1},$ $2q{w}_3+w_1^q{2}^q{\left(\frac{q-1}{-C_i}\right)}^{2q-1}<0,$ $2q{z}_3+z_1^q{2}^q{\left(\frac{q-1}{-D_i}\right)}^{q-1}<0,$ $w_1>0,$ $w_2<0,$ $w_3<0,$ $z_1>0,$ $z_2<0,$ $z_3<0,$ $A_i,B_j$ are defined in Theorem 1, $C_i,D_j,M\left(t\right)$ are defined in Theorem 2 and $M\left(0\right)$ is the value when $t=0$ in $M\left(t\right)$.

Notation 1.

In (4), $\varphi \left(t\right)$ is independent of the initial values of the error systems, while in (5), $\psi \left(t\right)$ is dependent on the initial values of the error systems, so the controllers (4) and the controllers (5) are different. Under these two controllers, the finite-time synchronization for the drive system (1) and the response system (2) are achieved under some conditions.

Theorem 1.

Assume the condition $\left(H_1\right)$ holds. Then, the drive system (1) and the response system (2) are finite-time synchronized under the controllers (4) in a finite time $t_1$, where $t_1=max\left\{\frac{3}{4},\frac{V\left(0\right)}{m}+2ln2-1+\frac{1}{b}+\frac{1}{a}·\frac{1}{4^a}\right\}$ if there exists a positive constant $p\ge 1$ such that the following conditions hold:

$\left(H_2\right)$

$$A_i=(k_2-a_i)p+(p-1)\sum _{j=1}^m{F}_j(|c_{ij}|+|d_{ij}|)+G_i\sum _{j=1}^m(|p_{ji}|+\frac{1}{1-{\sigma }^{*}}|q_{ji}|)<0;$$

$(H_3)$

$$B_j=(l_2-b_j)p+(p-1)\sum _{i=1}^n{G}_i(|p_{ji}|+|q_{ji}|)+F_j\sum _{i=1}^n(|c_{ij}|+\frac{1}{1-{\tau }^{*}}|d_{ij}|)<0.$$

Proof. 

Without loss of generalization, we assume that $e_i(t)\ne 0,r_j(t)\ne 0$ (If $e_i(t)=0,r_j(t)=0,$ then the finite-time synchronization has been proved (if $e_i(t)=0$ or $r_j(t)=0,$ then the proof is a special case of the following proof).

We construct a Lyapunov function as follows:

$$V(t)=V_1(t)+V_2(t),$$

where,

$$V_1(t)=\sum _{i=1}^n|e_i{(t)|}^p+\sum _{j=1}^m{|r_j(t)|}^p,$$

$$V_2(t)=\frac{1}{1-{\tau }^{*}}\sum _{i=1}^n\sum _{j=1}^m|d_{ij}|F_j{\int }_{t-{\tau }_{ji}(t)}^t|r_j{(s)|}^{p}ds+\frac{1}{1-{\sigma }^{*}}\sum _{i=1}^n\sum _{j=1}^m|q_{ji}|G_i{\int }_{t-{\sigma }_{ij}(t)}^t{|e_i(s)|}^{p}ds.$$

From system (3), Lemma 1, assumption $(H_1)$ and the controller (4), we have

$$\begin{array}{ccc}\hfill & & \frac{d{V}_1(t)}{dt}\hfill \\ & =& \sum _{i=1}^n\frac{d({(sign\left[e_i(t)\right])}^p{e}_i^p(t))}{dt}+\sum _{j=1}^m\frac{d({(sign\left[r_j(t)\right])}^p{r}_j^p(t))}{dt}\hfill \\ & =& \sum _{i=1}^n{(sign\left[e_i(t)\right])}^{p}p{e}_i^{p-1}(t)\{-a_i{e}_i(t)+\sum _{j=1}^m{c}_{ij}\left[f_j(v_j(t))-f_j(y_j(t))\right]+\sum _{j=1}^m{d}_{ij}\times \hfill \\ & & \left[f_j(v_j(t-{\tau }_{ji}(t)))-f_j(y_j(t-{\tau }_{ji}(t)))\right]+P_i(t)\}+\sum _{j=1}^m{(sign\left[r_j(t)\right])}^{p}p{r}_j^{p-1}(t)\hfill \\ & & \{-b_j{r}_j(t)+\sum _{i=1}^n{p}_{ji}\left[g_i(u_i(t))-g_i(x_i(t))\right]+\sum _{i=1}^n{q}_{ji}[g_i(u_i(t-{\sigma }_{ij}(t)))\hfill \\ & & -g_i(x_i(t-{\sigma }_{ij}(t)))]+Q_j(t)\}\hfill \\ & \le & \sum _{i=1}^n\{(k_2-a_i)p|e_i(t){|}^p+k_{1}p|e_i(t){|}^{p-1}+p|e_i(t){|}^{p-1}(\sum _{j=1}^m|c_{ij}||f_j(v_j(t))-f_j(y_j(t))|\hfill \\ & & +\sum _{j=1}^m|d_{ij}||f_j(v_j(t-{\tau }_{ji}(t)))-f_j(y_j(t-{\tau }_{ji}(t)))|)+p{k}_3\}+\sum _{j=1}^m\{(l_2-b_j)p{|r_j(t)|}^p\hfill \\ & & +l_{1}p|r_j{(t)|}^{p-1}+p|r_j{(t)|}^{p-1}(\sum _{i=1}^n|p_{ji}||g_i(u_i(t))-g_i(x_i(t))|+\sum _{i=1}^n|q_{ji}|\times \hfill \\ & & |g_i(u_i(t-{\sigma }_{ij}(t)))-g_i(x_i(t-{\sigma }_{ij}(t)))|)+\varphi (t)+p{l}_3\}\hfill \\ & \le & \sum _{i=1}^n\{(k_2-a_i)p|e_i(t){|}^p+k_{1}p|e_i(t){|}^{p-1}+p|e_i(t){|}^{p-1}(\sum _{j=1}^m|c_{ij}|F_j|r_j(t)|+\sum _{j=1}^m|d_{ij}|F_j\times \hfill \\ & & |r_j(t-{\tau }_{ji}(t))|)+p{k}_3\}+\sum _{j=1}^m\{(l_2-b_j)p|r_j(t){|}^p+l_{1}p|r_j(t){|}^{p-1}+p{|r_j(t)|}^{p-1}\hfill \\ & & \left(\sum _{i=1}^n|p_{ji}|G_i|e_i(t)|+\sum _{i=1}^n|q_{ji}|G_i|e_i(t-{\sigma }_{ij}(t))|\right)+\varphi (t)+p{l}_3\}.\hfill \end{array}$$

(6)

Since $p>1,$ applying Lemma 1 gives

$$\begin{array}{c}\hfill |e_i{(t)|}^{p-1}|r_j(t)|\le \frac{(p-1)|e_i{(t)|}^p}{p}+\frac{|r_j{(t)|}^p}{p},\end{array}$$

(7)

$$\begin{array}{c}\hfill |e_i{(t)|}^{p-1}|r_j(t-{\tau }_{ji}(t))|\le \frac{(p-1)|e_i{(t)|}^p}{p}+\frac{|r_j(t-{\tau }_{ji}(t)){|}^p}{p},\end{array}$$

(8)

$$\begin{array}{c}\hfill |r_j{(t)|}^{p-1}|e_i(t)|\le \frac{(p-1)|r_j{(t)|}^p}{p}+\frac{|e_i{(t)|}^p}{p},\end{array}$$

(9)

and

$$\begin{array}{c}\hfill |r_j{(t)|}^{p-1}|e_i(t-{\sigma }_{ij}(t))|\le \frac{(p-1)|r_j{(t)|}^p}{p}+\frac{|e_i(t-{\sigma }_{ij}(t)){|}^p}{p}.\end{array}$$

(10)

Substituting (7)–(10) into (6), it follows that

$$\begin{array}{ccc}\hfill & & V_1^{\prime }(t)\hfill \\ & \le & \sum _{i=1}^n\{(k_2-a_i)p|e_i(t){|}^p+k_{1}p|e_i(t){|}^{p-1}+\sum _{j=1}^m(|c_{ij}|[(p-1)|e_i(t){|}^p+|r_j(t){|}^p]+\hfill \\ & & |d_{ij}|[(p-1)|e_i{(t)|}^p+|r_j(t-{\tau }_{ji}(t)){|}^p])F_j+p{k}_3\}+\sum _{j=1}^m\{(l_2-b_j)p{|r_j(t)|}^p\hfill \\ & & +l_{1}p|r_j{(t)|}^{p-1}+\sum _{i=1}^n(|p_{ji}|[(p-1)|r_j{(t)|}^p+|e_i{(t)|}^p]+|q_{ji}|[(p-1)|r_j{(t)|}^p+\hfill \\ & & |e_i(t-{\sigma }_{ij}(t)){|}^p])G_i+\varphi (t)+p{l}_3\}.\hfill \end{array}$$

(11)

On the other hand, we have

$$\begin{array}{ccc}\hfill & & \frac{d{V}_2(t)}{dt}\hfill \\ & =& \frac{1}{1-{\tau }^{*}}\sum _{i=1}^n\sum _{j=1}^m|d_{ij}|F_j\left(|r_j{(t)|}^p-(1-{\tau }_{ji}^{\prime }(t)){|r_j(t-{\tau }_{ji(t)})|}^p\right)\hfill \\ & & +\frac{1}{1-{\sigma }^{*}}\sum _{i=1}^n\sum _{j=1}^m|q_{ji}|G_i\left(|e_i{(t)|}^p-(1-{\sigma }_{ij}^{\prime }(t)){|e_i(t-{\sigma }_{ij(t)})|}^p\right)\hfill \\ & \le & \frac{1}{1-{\tau }^{*}}\sum _{i=1}^n\sum _{j=1}^m|d_{ij}|F_j|r_j{(t)|}^p+\frac{1}{1-{\sigma }^{*}}\sum _{i=1}^n\sum _{j=1}^m|q_{ji}|G_i{|e_i(t)|}^p\hfill \\ & & -\left(\sum _{i=1}^n\sum _{j=1}^m|d_{ij}|F_j|r_j(t-{\tau }_{ji(t)}){|}^p+\sum _{i=1}^n\sum _{j=1}^m|q_{ji}|G_i{|e_i(t-{\sigma }_{ij(t)})|}^p\right).\hfill \end{array}$$

(12)

From (11) and (12), we have

$$\begin{array}{ccc}& & V^{\prime }(t)=V_1^{\prime }(t)+V_2^{\prime }(t)\hfill \\ & \le & \sum _{i=1}^n\{(k_2-a_i)p|e_i(t){|}^p+k_{1}p|e_i(t){|}^{p-1}+\sum _{j=1}^m(|c_{ij}|[(p-1)|e_i(t){|}^p+|r_j(t){|}^p]\hfill \\ & & +|d_{ij}|[(p-1)|e_i{(t)|}^p+\frac{1}{1-{\tau }^{*}}|r_j{(t)|}^p])F_j+p{k}_3\}+\sum _{j=1}^m\{(l_2-b_j)p{|r_j(t)|}^p\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & +l_{1}p|r_j{(t)|}^{p-1}+\sum _{i=1}^n(|p_{ji}|[(p-1)|r_j{(t)|}^p+|e_i{(t)|}^p]+|q_{ji}|[(p-1)|r_j{(t)|}^p\hfill \\ & & +\frac{1}{1-{\sigma }^{*}}|e_i(t){|}^p])G_i+\varphi (t)+p{l}_3\}\hfill \\ & =& \sum _{i=1}^n\{\left[(k_2-a_i)p+(p-1)\sum _{j=1}^m{F}_j(|c_{ij}|+|d_{ij}|)+G_i\sum _{j=1}^m(|p_{ji}|+\frac{1}{1-{\sigma }^{*}}|q_{ji}|)\right]{|e_i(t)|}^p\hfill \\ & & +k_{1}p|e_i{(t)|}^{p-1}+p{k}_3\}+\sum _{j=1}^m\{[(l_2-b_j)p+(p-1)\sum _{i=1}^n{G}_i(|p_{ji}|+|q_{ji}|)+F_j\sum _{i=1}^n(|c_{ij}|\hfill \\ & & +\frac{1}{1-{\tau }^{*}}|d_{ij}|)]|r_j{(t)|}^p+l_{1}p{|r_j(t)|}^{p-1}+\varphi (t)+p{l}_3\}\hfill \\ & =& \sum _{i=1}^n(A_i|e_i{(t)|}^p+k_{1}p|e_i{(t)|}^{p-1}+p{k}_3)+\sum _{j=1}^m(B_j|r_j{(t)|}^p+l_{1}p|r_j(t){|}^{p-1}+p{l}_3)+m\varphi (t).\hfill \end{array}$$

(13)

Let

$$F(|e_i(t)|)=A_i|e_i{(t)|}^p+k_{1}p{|e_i(t)|}^{p-1}+p{k}_3,$$

$$G(|r_j(t)|)=B_j|r_j{(t)|}^p+l_{1}p{|r_j(t)|}^{p-1}+p{l}_3.$$

Then $\frac{d[F(|e_i(t)|)]}{d|e_i(t)|}=|e_i{(t)|}^{p-2}p[A_i|e_i(t)|+k_1(p-1)].$ From $\frac{d[F(|e_i(t)|)]}{d|e_i(t)|}=0,$ we have $|e_i(t)|=\frac{k_1(p-1)}{-A_i}=x_0.$ Since when $|e_i(t)|<x_0,\frac{d[F(|e_i(t)|)]}{d|e_i(t)|}>0$ and when $|e_i(t)|>x_0,\frac{d[F(|e_i(t)|)]}{d|e_i(t)|}<0$, then $x_0$ is the local maximum-value point. It is clear that $F(x_0)=\frac{k_1^p{(p-1)}^{p-1}}{{(-A_i)}^{p-1}}+p{k}_3<0$.

Because $A_i<0$, then we have $\underset{|e_i(t)|\to +\infty }{lim}F(|e_i(t)|)=-\infty <0.$ So, from Lemma 4, we get $max\{F(|e_i(t)|\}<0.$ Then, $F(|e_i(t)|)<0.$ Similarly, we can prove $G(|r_j(t)|)<0.$

Substituting $F(|e_i(t)|)<0$ and $G(|r_j(t)|)<0$ into (13) yields

$$\begin{array}{c}\hfill V^{\prime }(t)<m\varphi (t).\end{array}$$

(14)

Integrating (14) over $[0,t]$ gives

$$\begin{array}{ccc}\hfill V(t)& <& V(0)+m\left[ln(4t+1)-\frac{1}{a}{(t+\frac{1}{4})}^a+\frac{1}{a}·\frac{1}{4^a}-t\right](t>\frac{3}{4})\hfill \\ & =& V(0)+m(ln(4t+1)+1-2ln2-\frac{1}{a}{(t+\frac{1}{4})}^a-\frac{1}{b})\hfill \\ & & +m(2ln2-1+\frac{1}{b}+\frac{1}{a}·\frac{1}{4^a}-t).\hfill \end{array}$$

(15)

It is clear that when $t>\frac{V(0)}{m}+2ln2-1+\frac{1}{b}+\frac{1}{a}·\frac{1}{4^a},$

$$\begin{array}{c}\hfill V(0)-m\left[2ln2-1+\frac{1}{b}+\frac{1}{a}·\frac{1}{4^a}\right]<0.\end{array}$$

(16)

According to Lemma 2, we have

$$\begin{array}{c}\hfill ln(4t+1)+1-2ln2-\frac{1}{a}{(t+\frac{1}{4})}^a-\frac{1}{b}<0,t>\frac{3}{4}.\end{array}$$

(17)

Substituting (16) and (17) into (15), it follows that when $t>t_1=max\left\{\frac{3}{4},\frac{V(0)}{m}+2ln2-1\right.\left.+\frac{1}{b}+\frac{1}{a}·\frac{1}{4^a}\right\},$

$$0\le V(t)\le 0,t\ge t_1,$$

that is, ${lim}_{t\to t_1}|e_i(t)|=0,{lim}_{t\to t_1}|r_j(t)|=0,|e_i(t)|=0,|r_j(t)|=0,t\ge t_1.$

Namely,

$$\underset{t\to t_1}{lim}|u_i(t)-x_i(t)|=0,|u_i(t)-x_i(t)|=0,t\ge t_1;$$

$$\underset{t\to t_1}{lim}|v_j(t)-y_j(t)|=0,|v_j(t)-y_j(t)|=0,t\ge t_1.$$

The proof of Theorem 1 is finished. □

Theorem 2.

Assume that $(H_1)$ holds. Then, the drive system (1) and the response system (2) are finite-time synchronized under the controllers (5) in a finite time $t_2$, where $t_2=max\{\frac{1}{3m},\frac{1}{e}\}$ if there exists a positive constant $q\ge 1$ such that the following conditions hold:

$(H_4)$

$$C_i=(w_2-a_i)2q+(2q-1)\sum _{j=1}^m{F}_j(|c_{ij}|+|d_{ij}|)+G_i\sum _{j=1}^m(|p_{ji}|+\frac{1}{1-{\sigma }^{*}}|q_{ji}|)<0;$$

$(H_5)$

$$D_j=(z_2-b_j)2q+(2q-1)\sum _{i=1}^n{G}_i(|p_{ji}|+|q_{ji}|)+F_j\sum _{i=1}^n(|c_{ij}|+\frac{1}{1-{\tau }^{*}}|d_{ij}|)<0.$$

Proof. 

Without loss of generalization, we assume that $e_i(t)\ne 0,r_j(t)\ne 0$ (if $e_i(t)=0,r_j(t)=0,$ then the finite-time synchronization has been proved; if $e_i(t)=0$ or $r_j(t)=0,$ then the proof is a special case of the following proof).

We construct a Lyapunov function as follows:

$$M(t)=M_1(t)+M_2(t),$$

where,

$$M_1(t)=\sum _{i=1}^n{\left[e_i(t)\right]}^{2q}+\sum _{j=1}^m{\left[r_j(t)\right]}^{2q},$$

$$M_2(t)=\frac{1}{1-{\tau }^{*}}\sum _{i=1}^n\sum _{j=1}^m|d_{ij}|F_j{\int }_{t-{\tau }_{ji}(t)}^t{\left[r_j(s)\right]}^{2q}ds+\frac{1}{1-{\sigma }^{*}}\sum _{i=1}^n\sum _{j=1}^m|q_{ji}|G_i{\int }_{t-{\sigma }_{ij}(t)}^t{\left[e_i(s)\right]}^{2q}ds.$$

From system (3), Lemma 1, assumption $(H_1)$ and the controllers (5), we have

$$\begin{array}{ccc}& & \frac{d{M}_1(t)}{dt}\hfill \\ & =& \sum _{i=1}^{n}2q{\left[e_i(t)\right]}^{2q-1}\frac{d{e}_i(t)}{dt}+\sum _{j=1}^{m}2q{\left[r_j(t)\right]}^{2q-1}\frac{d{r}_j(t)}{dt}\hfill \\ & =& \sum _{i=1}^{n}2q{\left[e_i(t)\right]}^{2q-1}\{-a_i{e}_i(t)+\sum _{j=1}^m{c}_{ij}\left[f_j(v_j(t))-f_j(y_j(t))\right]+\sum _{j=1}^m{d}_{ij}[f_j(v_j(t-{\tau }_{ji}(t)))\hfill \\ & & -f_j(y_j(t-{\tau }_{ji}(t)))]+P_i(t)\}+\sum _{j=1}^{m}2q{\left[r_j(t)\right]}^{2q-1}\{-b_j{r}_j(t)+\sum _{i=1}^n{p}_{ji}[g_i(u_i(t))\hfill \\ & & -g_i(x_i(t))]+\sum _{i=1}^n{q}_{ji}\left[g_i(u_i(t-{\sigma }_{ij}(t)))-g_i(x_i(t-{\sigma }_{ij})(t))\right]+Q_j(t)\}\hfill \\ & \le & \sum _{i=1}^n\{(w_2-a_i)2q{\left[e_i(t)\right]}^{2q}+2q{w}_1{\left[e_i(t)\right]}^{2q-2}+2q|e_i(t){|}^{2q-1}(\sum _{j=1}^m|c_{ij}||f_j(v_j(t))\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & -f_j(y_j(t))|+\sum _{j=1}^m|d_{ij}||f_j(v_j(t-{\tau }_{ji}(t)))-f_j(y_j(t-{\tau }_{ji}(t)))|)+2q{w}_3\}+\sum _{j=1}^m\{(z_2-b_j)\hfill \\ & & \times 2q{\left[r_j(t)\right]}^{2q}+2q{z}_1{\left[r_j(t)\right]}^{2q-2}+2q|r_j{(t)|}^{2q-1}(\sum _{i=1}^n|p_{ji}||g_i(u_i(t))-g_i(x_i(t))|\hfill \\ & & +\sum _{i=1}^n|q_{ji}||g_i(u_i(t-{\sigma }_{ij}(t)))-g_i(x_i(t-{\sigma }_{ij}(t)))|)+\psi (t)+2q{z}_3\}\hfill \\ & \le & \sum _{i=1}^n\{(w_2-a_i)2q{\left[e_i(t)\right]}^{2q}+2q{w}_1{\left[e_i(t)\right]}^{2q-2}+2q|e_i(t){|}^{2q-1}(\sum _{j=1}^m|c_{ij}|F_j|r_j(t)|\hfill \\ & & +\sum _{j=1}^m|d_{ij}|F_j|r_j(t-{\tau }_{ji}(t))|)+2q{w}_3\}+\sum _{j=1}^m\{(z_2-b_j)2q{\left[r_j(t)\right]}^{2q}+2q{z}_1{\left[r_j(t)\right]}^{2q-2}\hfill \\ & & +2q|r_j{(t)|}^{2q-1}\left(\sum _{i=1}^n|p_{ji}|G_i|e_i(t)|+\sum _{i=1}^n|q_{ji}|G_i|e_i(t-{\sigma }_{ij}(t))|\right)+\psi (t)+2q{z}_3\}.\hfill \end{array}$$

(18)

Since $q\ge 1,$ applying Lemma 1 gives

$$\begin{array}{c}\hfill |e_i{(t)|}^{2q-1}|r_j(t)|\le \frac{(2q-1){\left[e_i(t)\right]}^{2q}}{2q}+\frac{{\left[r_j(t)\right]}^{2q}}{2q},\end{array}$$

(19)

$$\begin{array}{c}\hfill |e_i{(t)|}^{2q-1}|r_j(t-{\tau }_{ji}(t))|\le \frac{(2q-1){\left[e_i(t)\right]}^{2q}}{2q}+\frac{{\left[r_j(t-{\tau }_{ji}(t))\right]}^{2q}}{2q},\end{array}$$

(20)

$$\begin{array}{c}\hfill |r_j{(t)|}^{2q-1}|e_i(t)|\le \frac{(2q-1){\left[r_j(t)\right]}^{2q}}{2q}+\frac{{\left[e_i(t)\right]}^{2q}}{2q},\end{array}$$

(21)

and

$$\begin{array}{c}\hfill |r_j{(t)|}^{2q-1}|e_i(t-{\sigma }_{ij}(t))|\le \frac{(2q-1){\left[r_j(t)\right]}^{2q}}{2q}+\frac{{\left[e_i(t-{\sigma }_{ij}(t))\right]}^{2q}}{2q}.\end{array}$$

(22)

Substituting (19)–(22) into (18), it follows that

$$\begin{array}{ccc}\hfill & & M_1^{\prime }\left(t\right)\hfill \\ & \le & \sum _{i=1}^n\{(w_2-a_i)2q{\left[e_i(t)\right]}^{2q}+2q{w}_1{\left[e_i(t)\right]}^{2q-2}+\sum _{j=1}^m(|c_{ij}|[(2q-1){\left[e_i(t)\right]}^{2q}+{\left[r_j(t)\right]}^{2q}]\hfill \\ & & +|d_{ij}|[(2q-1){\left[e_i(t)\right]}^{2q}+{\left[r_j(t-{\tau }_{ji}(t))\right]}^{2q}])F_j+2q{w}_3\}+\sum _{j=1}^m\{(z_2-b_j)2q{\left[r_j(t)\right]}^{2q}\hfill \\ & & +2q{z}_1{\left[r_j(t)\right]}^{2q-2}+\sum _{i=1}^n(|p_{ji}|[(2q-1){\left[r_j(t)\right]}^{2q}+{\left[e_i(t)\right]}^{2q}]+|q_{ji}|[(2q-1){\left[r_j(t)\right]}^{2q}\hfill \\ & & +{\left[e_i(t-{\sigma }_{ij}(t))\right]}^{2q}])G_i+\psi (t)+2q{z}_3\}.\hfill \end{array}$$

(23)

On the other hand, we have

$$\begin{array}{ccc}& & \frac{d{V}_2(t)}{dt}\hfill \\ & =& \frac{1}{1-{\tau }^{*}}\sum _{i=1}^n\sum _{j=1}^m|d_{ij}|F_j\left({\left[r_j(t)\right]}^{2q}-(1-{\tau }_{ji}^{\prime }(t)){\left[r_j(t-{\tau }_{ji(t)})\right]}^{2q}\right)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & +\frac{1}{1-{\sigma }^{*}}\sum _{i=1}^n\sum _{j=1}^m|q_{ji}|G_i\left({\left[e_i(t)\right]}^{2q}-(1-{\sigma }_{ij}^{\prime }(t)){\left[e_i(t-{\sigma }_{ij(t)})\right]}^{2q}\right)\hfill \\ & \le & \frac{1}{1-{\tau }^{*}}\sum _{i=1}^n\sum _{j=1}^m|d_{ij}|F_j{\left[r_j(t)\right]}^{2q}+\frac{1}{1-{\sigma }^{*}}\sum _{i=1}^n\sum _{j=1}^m|q_{ji}|G_i{\left[e_i(t)\right]}^{2q}\hfill \\ & & -\left(\sum _{i=1}^n\sum _{j=1}^m|d_{ij}|F_j{\left[r_j(t-{\tau }_{ji(t)})\right]}^{2q}+\sum _{i=1}^n\sum _{j=1}^m|q_{ji}|G_i{\left[e_i(t-{\sigma }_{ij(t)})\right]}^{2q}\right).\hfill \end{array}$$

(24)

From (23) and (24), we have

$$\begin{array}{ccc}\hfill & & M^{\prime }(t)=M_1^{\prime }(t)+M_2^{\prime }(t)\hfill \\ & \le & \sum _{i=1}^n\{(w_2-a_i)2q{\left[e_i(t)\right]}^{2q}+2q{w}_1|{\left[e_i(t)\right]}^{2q-2}+\sum _{j=1}^m(|c_{ij}|[(2q-1){\left[e_i(t)\right]}^{2q}+{\left[r_j(t)\right]}^{2q}]+\hfill \\ & & |d_{ij}|[(2q-1){\left[e_i(t)\right]}^{2q}+\frac{1}{1-{\tau }^{*}}{\left[r_j(t)\right]}^{2q}])F_j+2q{w}_3\}+\sum _{j=1}^m\{(z_2-b_j)2q{\left[r_j(t)\right]}^{2q}\hfill \\ & & +2q{z}_1{\left[r_j(t)\right]}^{2q-2}+\sum _{i=1}^n(|p_{ji}|[(2q-1){\left[r_j(t)\right]}^{2q}+{\left[e_i(t)\right]}^{2q}]+|q_{ji}|[(2q-1){\left[r_j(t)\right]}^{2q}\hfill \\ & & +\frac{1}{1-{\sigma }^{*}}{\left[e_i(t)\right]}^{2q}])G_i+\psi (t)+2q{z}_3\}\hfill \\ & =& \sum _{i=1}^n\{\left[(w_2-a_i)2q+(2q-1)\sum _{j=1}^m{F}_j(|c_{ij}|+|d_{ij}|)+G_i\sum _{j=1}^m(|p_{ji}|+\frac{1}{1-{\sigma }^{*}}|q_{ji}|)\right]{\left[e_i(t)\right]}^{2q}+\hfill \\ & & 2q{w}_1{\left[e_i(t)\right]}^{2q-2}+2q{w}_3\}+\sum _{j=1}^m\{[(z_2-b_j)2q+(2q-1)\sum _{i=1}^n{G}_i(|p_{ji}|+|q_{ji}|)+F_j\sum _{i=1}^n(|c_{ij}|\hfill \\ & & +\frac{1}{1-{\tau }^{*}}|d_{ij}|)]{\left[r_j(t)\right]}^{2q}+2q{z}_1{\left[r_j(t)\right]}^{2q-2}+\psi (t)+2q{z}_3\}\hfill \\ & =& \sum _{i=1}^n\left(C_i{\left[e_i(t)\right]}^{2q}+2q{w}_1{\left[e_i(t)\right]}^{2q-2}+2q{w}_3\right)+\sum _{j=1}^m\left(D_j{\left[r_j(t)\right]}^{2q}+2q{z}_1{\left[r_j(t)\right]}^{2q-2}+2q{z}_3\right)\hfill \\ & & +m\psi (t).\hfill \end{array}$$

(25)

Denote

$$\widehat{F}(e_i^2(t))=C_i{\left[e_i^2(t)\right]}^q+2q{w}_1{\left[e_i^2(t)\right]}^{q-1}+2q{w}_3,$$

$$\widehat{G}(r_j^2(t))=D_j{\left[r_j^2(t)\right]}^q+2q{z}_1{\left[r_j^2(t)\right]}^{q-1}+2q{z}_3.$$

Then $\frac{d\left[\widehat{F}(e_i^2(t)|)\right]}{d(e_i^2(t))}={\left[e_i^2(t)\right]}^{q-2}2q[C_i{e}_i^2(t)+w_1(q-1)].$ From $\frac{d\left[\widehat{F}(e_i^2(t))\right]}{d(e_i^2(t))}=0,$ we have $e_i^2(t)=\frac{2w_1(q-1)}{-C_i}=x_{*}.$ Since when $e_i^2(t)>x_{*},\frac{d\left[\widehat{F}(e_i^2(t))\right]}{d(e_i^2(t))}<0$, $e_i^2(t)>x_{*},\frac{d\left[\widehat{F}(e_i^2(t))\right]}{d(e_i^2(t))}<0$, then $x_{*}$ is a unique local maximum point of $\widehat{F}(e_i^2(t)|)$. It is clear that $\widehat{F}(x_{*})=\frac{w_1^q{2}^q{(q-1)}^{q-1}}{{(-C_i)}^{q-1}}+2q{w}_3<0.$

In addition, because $C_i<0$, then we have $\underset{e_i(t)\to -\infty }{lim}\widehat{F}(e_i^2(t))=\underset{e_i(t)\to +\infty }{lim}\widehat{F}(e_i^2(t))=-\infty <0$. So, from Lemma 4, we get $max\{\widehat{F}(e_i^2(t)\}<0.$ Then, $\widehat{F}(e_i^2(t))<0.$ Similarly, we can prove $\widehat{G}(r_j^2(t))<0.$

Substituting $\widehat{F}(e_i^2(t))<0$ and $\widehat{G}(r_j^2(t))<0$ into (25) yields

$$\begin{array}{c}\hfill M^{\prime }(t)<m\psi (t).\end{array}$$

(26)

Integrating (26) over $[0,t]$ gives

$$\begin{array}{ccc}\hfill & & M(t)\hfill \\ & <& M(0)+m\left[t-a^{*}t{e}^t+a^{*}t+1-e^t-3M(0)t\right],(t\ge \frac{1}{e})\hfill \\ & <& m\left[t-a^{*}t{e}^t+a^{*}t+1\right]+(1-3mt)M(0).\hfill \end{array}$$

(27)

According to Lemma 3, we have

$$\begin{array}{c}\hfill t+1+a^{*}t-a^{*}t{e}^t<0,t\ge \frac{1}{e}.\end{array}$$

(28)

It is clear that when $t>\frac{1}{3m},$

$$\begin{array}{c}\hfill (1-3mt)M(0)<0.\end{array}$$

(29)

Substituting (28) and (29) into (27) yields that when $t>t_2=max\{\frac{1}{3m},\frac{1}{e}\},$

$$0\le M(t)\le 0,t>t_2,$$

that is, $\underset{t\to t_2}{lim}|e_i(t)|=0,\underset{t\to t_2}{lim}|r_j(t)|=0,|e_i(t)|=0,|r_j(t)|=0,t>t_2.$

Namely,

$$\underset{t\to t_2}{lim}|u_i(t)-x_i(t)|=0,|u_i(t)-x_i(t)|=0,t>t_2;$$

$$\underset{t\to t_2}{lim}|v_j(t)-y_j(t)|=0,|v_j(t)-y_j(t)|=0,t>t_2.$$

□

Remark 1.

Without applying some finite-time stability theorems, the inequality approach used in [14,15,16,17,18,19,21,22] or the integral inequality method used in [48,49,50], in this paper, by applying the maximum-value approach used in [42], we studied the finite-time synchronization of BAM neural networks. However, the concrete inequality techniques via the maximum-value approach in our paper are different from those in [42].

Remark 2.

When ${\tau }_{ji}^{\prime }(t)>1,{\sigma }_{ij}^{\prime }(t)>1,$ then ${\tau }^{*}>1,{\sigma }^{*}>1.$ Thus, $V(t)$ and $M(t)$ are not positive so the results obtained do not hold. When delays are equal to 1, the derived conditions become delay-independent.

Remark 3.

The controllers in our paper are different from those in [42]. Firstly, the error items in the controllers in our paper are different from those in [42]; namely, fractional-type functions are designed in the error items of the controllers in our paper, while exponential- and logarithm-type functions are designed in the error items in the controllers from [42]. Secondly, the time t items are different from those in [42]; namely, fractional- and exponential-type functions are designed for the time items of the controllers in our paper, while logarithm-, fractional- and exponential-type functions are designed for the time items in the controllers from [42].

4. Numerical Examples

A BAM neural network composed of neurons arranged in two layers possesses good applications in the field of pattern recognition and artificial intelligence, since BAM neural networks have the capability to learn from a process. Hence, the study of synchronization in practical application for the drive–response BAM neural networks is very necessary. In this section, we give two examples to illustrate our theoretical results by intuitive images.

Example 1.

We consider system (1), response system (2) and error system (3) with controllers (4) for $i=1,2;$ $j=1,2$, where $c_{11}=3,$ $c_{12}=-2,$ $c_{21}=4,$ $c_{22}=-5;$ $d_{11}=1,$ $d_{12}=-4,$ $d_{21}=-2,$ $d_{22}=3;$ $p_{11}=-1,$ $p_{12}=3,$ $p_{21}=0.9,$ $p_{22}=-2;$ $q_{11}=4,$ $q_{12}=-3,$ $q_{21}=-5,$ $q_{22}=2.$ $a_1=3,$ $a_2=1,$ $b_1=4,$ $b_2=2,$ $p=2>1,$ $a=3>1,k_1=1>0,$ $k_2=-16<0,$ $k_3=-3<0,$ $l_1=2>0,$ $l_2=-8<0,$ $l_3=-5<0,$ $\varphi (t)=\frac{4}{4t+1}-{(t+\frac{1}{4})}^{a-1}-1,$ $I_1=1,$ $I_2=-2,$ $J_1=1,$ $J_2=2,$ $f_j(y_j(t))=0.5|y_j(t)|-1,g_i(x_i(t))=0.7|x_i(t)|+1,$ ${\tau }_{ji}(t)=0.5sint$ and ${\sigma }_{ij}(t)=0.5cost.$ The initial conditions are given as: $x_1(0)=320.5,x_2(0)=220,y_1(0)=-103.7,y_2(0)=160.4,u_1(0)=-150.8,u_2(0)=-470,v_1(0)=180.7,v_2(0)=-207.6$. Thus, we have $A_1=-19.07<0,A_2=-16.5<0,B_1=-9.8<0,B_2=-2.57<0$ and

$$p{k}_3+k_1^p{\left(\frac{p-1}{-A_1}\right)}^{p-1}=-5.9476<0,p{k}_3+k_1^p{\left(\frac{p-1}{-A_2}\right)}^{p-1}=-5.9394<0,$$

$$p{l}_3+l_1^p{\left(\frac{p-1}{-B_1}\right)}^{p-1}=-9.5918<0,p{l}_3+l_1^p{\left(\frac{p-1}{-B_2}\right)}^{p-1}=-8.4436<0.$$

The controllers from (4) for Example 1 are different from those in the literature [14,15,16,17,18,19,20,21,22,42]. It is not difficult to verify that the parameters do not satisfy the conditions of finite-time synchronization from the above papers. Therefore, the finite-time synchronization of the drive system (1) and response system (2) cannot be verified with the results in [14,15,16,17,18,19,20,21,22,42]. It is easy to verify that $(H_1)$–$(H_3)$ in Theorem 1 in our paper are satisfied. Hence, by Theorem 1 in our paper, the drive system (1) and the response system (2) are finite-time synchronized under controllers (4). The curves of variables $x_1(t),x_2(t),u_1(t)$ and $u_2(t)$ are shown in Figure 1, the curves of variables $y_1(t),y_2(t),v_1(t)$ and $v_2(t)$ are shown in Figure 2 and the error curves of the drive–response system $e_1(t),e_2(t),r_1(t)$ and $r_2(t)$ are shown in Figure 3.

Figure 1. The curves of $x_1(t),x_2(t),u_1(t),u_2(t)$ from Example 1.

Figure 2. The curves of $y_1(t),y_2(t),v_1(t),v_2(t)$ from Example 1.

Figure 3. The curves of $e_1(t),e_2(t),r_1(t),r_2(t)$ from Example 1.

Example 2.

We consider drive system (1), response system (2) and error system (3) with controllers (5) for $i=1,2;j=1,2$, where $c_{11}=0.3,$ $c_{12}=-0.2,$ $c_{21}=0.4,$ $c_{22}=-0.5;$ $d_{11}=0.6,$ $d_{12}=-0.4,$ $d_{21}=-0.2,$ $d_{22}=0.3;$ $p_{11}=-0.1,$ $p_{12}=0.6,$ $p_{21}=0.8,$ $p_{22}=-0.2;$ $q_{11}=0.8,$ $q_{12}=0.3,$ $q_{21}=0.5,$ $q_{22}=0.9.$ $a_1=1,$ $a_2=2,$ $b_1=2,$ $b_2=1,$ $q=2>1,$ $b=-5<-3,$ $w_1=1>0,$ $w_2=-10<0,$ $w_3=-3<0,$ $z_1=2>0,$ $z_2=-5<0,$ $z_3=-3<0,$ $\psi (t)=1+a^{*}-a^{*}(e^t+t{e}^t)-e^t-3M(0),t\ge \frac{1}{e},$ $a^{*}=10,$ $I_1=-1,$ $I_2=2,$ $J_1=3,$ $J_2=-2,$ $f_j(y_j)=0.5sin(y_j)+0.5,g_i(x_i)=0.25cos(x_i)-1,{\tau }_{ji}(t)=0.4cost$ and ${\sigma }_{ij}(t)=0.6sint.$ Furthermore, the initial conditions are defined as: $x_1(0)=-350.5,x_2(0)=-371.63,y_1(0)=136.48,y_2(0)=130.4,u_1(0)=-1450.8,u_2(0)=1060.63,v_1(0)=320.73$ and $v_2(0)=-238.82$. Hence, we have $C_1=-40.7125<0,C_2=-44.9500<0,D_1=-25.6333<0,D_2=-21.2667<0$ and

$$2q{w}_3+2^q{w}_1^q{\left(\frac{q-1}{-C_1}\right)}^{q-1}=-11.9996<0,2q{w}_3+2^q{w}_1^q{\left(\frac{q-1}{-C_2}\right)}^{q-1}=-11.9997<0,$$

$$2q{z}_3+2^q{z}_1^q{\left(\frac{q-1}{-D_1}\right)}^{q-1}=-11.9744<0,2q{z}_3+2^q{z}_1^q{\left(\frac{q-1}{-D_2}\right)}^{q-1}=-11.9551<0.$$

The controllers from (5), are different from those in [14,15,16,17,18,19,20,21,22,42]. We can easily verify that the parameters do not satisfy the conditions on finite-time synchronization from the above literature. Hence, the finite-time synchronization of drive system (1) and response system (2) cannot be verified with the results from [14,15,16,17,18,19,20,21,22,42]. We can verify that $(H_1),(H_4)$ and $(H_5)$ are satisfied. As a result, by Theorem 2, the drive system (1) and the response system (2) are finite-time synchronized under controllers (5).

The curves of variables $x_1(t),x_2(t),u_1(t)$ and $u_2(t)$ are shown in Figure 4, the curves of variables $y_1(t),y_2(t),v_1(t)$ and $v_2(t)$ are shown in Figure 5 and the error curves of the drive–response system $e_1(t),e_2(t),r_1(t)$ and $r_2(t)$ are shown in Figure 6.

Figure 4. The curves of $x_1(t),x_2(t),u_1(t),u_2(t)$ from Example 2.

Figure 5. The curves of $y_1(t),y_2(t),v_1(t),v_2(t)$ from Example 2.

Figure 6. The curves of $e_1(t),e_2(t),r_1(t),r_2(t)$ from Example 2.

5. Conclusions

In this paper, we focused on the finite-time synchronization for a class of drive–response BAM neural networks with time-varying delays. Furthermore, two novel finite-time synchronization conditions of the above BAM neural networks were derived to assure the finite-time synchronization between the drive system and the response system. In that process, we applied the maximum-value approach and introduced two new inequalities, which were different from those in the existing papers. In the future, we will study the fixed-time synchronization of neural networks.