Adaptive Synchronization for a Class of Cellular Neural Networks with Pantograph Delays

Abstract

This paper deals with the synchronization problem for a class of cellular neural networks with pantograph delays. By using Lyapunov functional theory and inequality technique, some new and useful results are obtained for asymptotical synchronization under adaptive feedback controller.

1. Introduction

In recent several decades, since Chua and Yang [1, 2] proposed cellular neural networks in 1988, they have attracted considerable attention due to their potential applications in signal processing, image processing, and pattern recognition. So, the dynamical analysis of cellular neural networks is important and interesting from both theoretical and applied points of view; many effective research methods and important results have been presented in [3–13] and references cited therein.

Synchronization, as one of the most important collective behaviors, has attracted a wide range of research actively. Recently, synchronization of neural networks delays has been extensively studied in [11–29]. Several control schemes have been introduced to realize the network synchronization, for example, fuzzy control [25, 26], impulsive control [27, 29], intermittent control [11, 28], and adaptive control [12, 13].

On the other hand, time delays are unavoidablely encountered in the signal transmission among the neurons due to the finite switching speed of neurons and amplifiers, which will affect the stability of the neural networks and may lead to some complex dynamic behaviors, such as instability, chaos, oscillation, and other performance of the neural network. Therefore, it is a very important research on the delayed neural networks. In view of the significance of the control for delayed neural networks, in recent years, there were many good results [3, 4, 6–11, 13, 15–24, 28, 30–32]. As we know, pantograph delays are playing more and more important role in some fields. For example, pantograph delay is usually required in Web Quality of Service (QoS) routing decision. However, few authors have taken care of neural networks with pantograph delays.

In [8], Ding studied the synchronization of delayed fuzzy cellular neural networks with impulses by using a nonimpulsive system to replace the impulsive system and some synchronization criteria were obtained by the well-known Lasalle invariant principle. In [11], the authors studied the lag synchronization for delayed fuzzy cellular neural networks via periodically intermittent control. The synchronization for a class of delayed fuzzy cellular neural networks with the parameters unknown was investigated (see [13]), by means of the Lasalle invariant principle of functional equations and the adaptive control method. In [30], the synchronization control of stochastic neural networks with time-varying delays is discussed by the Lyapunov functional method and linear inequality approach.

To the best of our knowledge, the adaptive synchronization results on neural networks with delays are usually based on Lasalle invariant principle, and there are few or even no results with regard to the adaptive synchronization for cellular neural networks with pantograph delays by applying adaptive feedback control. Based on the above analysis, in this paper, we will investigate the adaptive synchronization for cellular neural networks with pantograph delays by using Lyapunov functional theory, inequality technique, and Barbalat lemma, some new and useful criteria are derived.

This paper is organized as follows. In Section 2, model description and preliminaries are given. Some synchronization criteria are obtained in Section 3 under the adaptive feedback controller we assumed. In Section 4, the effectiveness and feasibility of the developed methods are shown by a numerical example.

2. Preliminaries

Motivated by the analysis in the above section, in this paper, we consider a class of cellular neural networks with pantograph delays, which is described by the following model: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑞_𝑗𝑡+𝐼_𝑖(𝑡),𝑖∈ℐ, (1) for𝑡>0, where𝑖,𝑗∈ℐ={1,2,…,𝑛};𝑥_𝑖(𝑡)denotes the state of the𝑖th neuron at the time 𝑡;𝑎_𝑖>0represents the rate with which the𝑖th neuron will reset its potential to the resting state in isolation when disconnected from the network and external input;𝑏_𝑖𝑗denotes the strength of the𝑗th neuron on the𝑖th neuron at the time 𝑡;𝑐_𝑖𝑗denotes the strength of the𝑗th neuron on the𝑖th neuron at the time𝑞_𝑗𝑡;𝑓_𝑗(⋅)corresponds to the output of the𝑗th neuron; and𝑞_𝑗(𝑗∈ℐ)is constant and satisfies0<𝑞_𝑗<1,𝑞_𝑗𝑡=𝑡−(1−𝑞_𝑗)𝑡, in which(1−𝑞_𝑗)𝑡denotes the pantograph delays along the axon of the𝑗th node and𝐼_𝑖(𝑡)corresponds to the external bias on the𝑖th neuron.

System (1) is supplemented with initial value given by 𝑥_𝑖(0)=𝑥_𝑖0,𝑖∈ℐ. (2)

In order to observe the synchronization behavior of system (1), the slaver system is designed as follows: 𝑦̇_𝑖(𝑡)=−𝑎_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑞_𝑗𝑡+𝐼_𝑖(𝑡)+𝐾_𝑖(𝑡),𝑖∈ℐ, (3) for𝑡>0, where𝑦_𝑖(𝑡)denotes the state of the slave system, the rest of the notations are the same as in system (1), and𝐾_𝑖(𝑡)is a control input to be designed.

The initial value of system (3) is given by 𝑦_𝑖(0)=𝑦_𝑖0,𝑖∈ℐ. (4)

Assume that𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇and𝑦(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))^𝑇are solutions of systems (1) and (3) with different initial values𝑥_𝑖(0)=𝑥_𝑖0and𝑦_𝑖(0)=𝑦_𝑖0at the time𝑡=0. Let𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝑥_𝑖(𝑡)be the synchronization error between the states of the drive system (1) and response system (3); subtracting (1) from (3) yields the following error system: 𝑒_𝑖̇(𝑡)=−𝑎_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)−𝑓_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑞_𝑗𝑡−𝑓_𝑗𝑥_𝑗𝑞_𝑗𝑡+𝐾_𝑖(𝑡),𝑖∈ℐ, (5) where the adaptive feedback controller𝐾_𝑖(𝑡)is designed by 𝐾_𝑖(𝑡)=𝜀_𝑖(𝑡)𝑒_𝑖(𝑡). (6) The feedback strength𝜀(𝑡)=(𝜀₁(𝑡),…,𝜀_𝑛(𝑡))^𝑇is adapted by the following law: 𝜀_𝑖̇(𝑡)=−𝜆_𝑖𝑒²_𝑖(𝑡), (7) where𝜆_𝑖>0(𝑖=1,2,…,𝑛).

In order to further study systems (1) and (3) and obtain the main results, the following assumptions are necessary.

Assumption 1. We assume that there exist positive constants𝑀_𝑖(𝑖∈ℐ)such that activation function𝑓_𝑖(⋅)satisfies the following condition: 𝑓_𝑖(𝑥)≤𝑀_𝑖,𝑖∈ℐ. (8)

Assumption 2. For each𝑖∈ℐ, we assume activation function𝑓_𝑖(⋅)satisfies the Lipschitz condition; that is, there exists constant𝐿_𝑖such that 𝑓_𝑖(𝑥)−𝑓_𝑖𝑦≤𝐿_𝑖𝑥−𝑦,∀𝑥,𝑦∈𝑅,𝑖∈ℐ. (9)
On the synchronization of drive-response system (1) and (3), we have the following definition.

Definition 3. The master system (1) and the response system (3) are said to be synchronized, if for any solution𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇of system (1) and any solution𝑦(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))^𝑇of system (3), we have lim_𝑡→∞𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡)=0. (10) In addition, the following lemma is essential in establishing our main results.

Lemma 4 (Barbalat Lemma). Function𝑓(𝑥) defined on interval[0,+∞); if𝑓(𝑥)is uniformly continuous and∫₀^+∞𝑓(𝑥)𝑑𝑥<+∞ is satisfied. Then lim_𝑥→+∞𝑓(𝑥)=0.

3. Adaptive Synchronization

In this section, we will use Lyapunov functional theory, inequality technique, and Barbalat lemma to study the error system (5) realize globally asymptotical stable under adaptive feedback controller (6), that is, to realize the adaptive synchronization of drive system (1) and response system (3).

The following theorem is given to guarantee the synchronization of system (1) and (3).

Theorem 5. Assume that Assumptions 1 and 2 hold, and the feedback strength𝜀_𝑖(𝑡)is adapted by (7); then the error system (5) is globally asymptotical stable; that is, the drive system (1) and response system (3) are synchronized under adaptive feedback controller (6).

Proof . In order to establish the result of Theorem 5, we introduce the following Lyapunov functional: 𝑉(𝑡)=^𝑛_𝑖=1𝑒_𝑖(𝑡)²+14^𝑛_𝑖=11𝜆_𝑖2𝜀_𝑖(𝑡)+ℎ²+1𝑞^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗^𝑡_𝑞𝑗𝑡𝑒²_𝑗(𝑠)𝑑𝑠, (11) whereℎis a constant and 𝑞=min_𝑖∈ℐ{𝑞_𝑗}, which will be given in the following.
From (5) and Assumption 2, we have 𝐷⁺𝑒_𝑖(𝑡𝑒)=sgn_𝑖(𝑡)−𝑎_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)−𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑞_𝑗𝑡−𝑓_𝑗𝑥_𝑗𝑞_𝑗𝑡+𝜀_𝑖(𝑡)𝑒_𝑖(𝑡)≤−𝑎_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)−𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑞_𝑗𝑡−𝑓_𝑗𝑥_𝑗𝑞_𝑗𝑡+𝜀_𝑖(𝑡)𝑒_𝑖(𝑡≤)−𝑎_𝑖+𝜀_𝑖(𝑡)𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝑒_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝐿_𝑗𝑒_𝑗𝑞_𝑗𝑡. (12)
In the following, we calculate the upper right derivative of𝑉(𝑡)along the solution of error system (5). From (11) and (12), we have 𝐷⁺𝑉(𝑡)=2^𝑛_𝑖=1𝑒_𝑖(𝑡)𝐷⁺𝑒_𝑖(𝑡)−^𝑛_𝑖=12𝜀_𝑖(𝑡)+ℎ𝑒²_𝑖(𝑡)+1𝑞^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗𝑒²_𝑗(𝑡)−𝑞_𝑗𝑒²_𝑗𝑞_𝑗𝑡≤2^𝑛_𝑖=1𝑒_𝑖(𝑡)−𝑎_𝑖+𝜀_𝑖(𝑡)𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝑒_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝐿_𝑗𝑒_𝑗𝑞_𝑗𝑡−2^𝑛_𝑖=1𝜀_𝑖(𝑡)𝑒²_𝑖(𝑡)−^𝑛_𝑖=1ℎ𝑒²_𝑖(𝑡)+1𝑞^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗𝑒²_𝑗(𝑡)−𝑞_𝑗𝑒²_𝑗𝑞_𝑗𝑡=2^𝑛_𝑖=1𝑒_𝑖(𝑡)−𝑎_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝑒_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝐿_𝑗𝑒_𝑗𝑞_𝑗𝑡−^𝑛_𝑖=1ℎ𝑒²_𝑖(𝑡)+1𝑞^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗𝑒²_𝑗(𝑡)−𝑞_𝑗𝑒²_𝑗𝑞_𝑗𝑡=−^𝑛_𝑖=12𝑎_𝑖𝑒²_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=12𝑏_𝑖𝑗𝐿_𝑗𝑒_𝑖(𝑡)𝑒_𝑗(𝑡+_{𝑛𝑛𝑖=1𝑗=1})2𝑐_𝑖𝑗𝐿_𝑗𝑒_𝑖(𝑡)𝑒_𝑗𝑞_𝑗𝑡−^𝑛_𝑖=1ℎ𝑒²_𝑖(𝑡)+1𝑞^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗𝑒²_𝑗(𝑡)−𝑞_𝑗𝑒²_𝑗𝑞_𝑗𝑡≤−^𝑛_𝑖=12𝑎_𝑖𝑒²_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝑒²_𝑖(𝑡)+𝑒²_𝑗(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗𝑒²_𝑖(𝑡)+𝑒²_𝑗𝑞_𝑗𝑡−^𝑛_𝑖=1ℎ𝑒²_𝑖(𝑡)+1𝑞^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗𝑒²_𝑗(𝑡)−𝑞_𝑗𝑒²_𝑗𝑞_𝑗𝑡=−^𝑛_𝑖=12𝑎_𝑖+ℎ𝑒²_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝑏_𝑖𝑗𝐿_𝑗+𝑏_𝑗𝑖𝐿_𝑖𝑒²_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗+1𝑞𝑐_𝑗𝑖𝐿_𝑖𝑒²_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗𝑞1−^𝑗𝑞≤−^𝑛_𝑖=12𝑎_𝑖+ℎ𝑒²_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝑏_𝑖𝑗𝐿_𝑗+𝑏_𝑗𝑖𝐿_𝑖𝑒²_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝐿_𝑗+1𝑞𝑐_𝑗𝑖𝐿_𝑖𝑒²_𝑖(𝑡)=−^𝑛_𝑖=12𝑎_𝑖+ℎ−^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗+𝑏_𝑗𝑖𝐿_𝑖−^𝑛_𝑗=1𝑐_𝑖𝑗𝐿_𝑗+1𝑞𝑐_𝑗𝑖𝐿_𝑖𝑒²_𝑖(𝑡). (13)
We choose ℎ=^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗+𝑏_𝑗𝑖𝐿_𝑖+^𝑛_𝑗=1𝑐_𝑖𝑗𝐿_𝑗+1𝑞𝑐_𝑗𝑖𝐿_𝑖−2𝑎_𝑖+1. (14) Then, we finally obtain 𝐷⁺𝑉(𝑡)≤−^𝑛_𝑖=1𝑒²_𝑖(𝑡), (15) For𝑡≥0, integrating both sides of inequality (15) over[0,𝑡], we get 𝑉(𝑡)−𝑉(0)≤−^𝑡₀^𝑛_𝑖=1𝑒²_𝑖(𝑠)𝑑𝑠. (16) Then 𝑉(𝑡)+^𝑡₀^𝑛_𝑖=1𝑒²_𝑖(𝑠)𝑑𝑠≤𝑉(0), (17) In view of𝑉(0)>0, lim_𝑡→∞sup^𝑡₀^𝑛_𝑖=1𝑒²_𝑖(𝑠)𝑑𝑠≤𝑉(0)<+∞. (18) From (11) and (17), we have ^𝑛_𝑖=1𝑒_𝑖(𝑡)²≤𝑉(𝑡)≤𝑉(0). (19) Therefore,𝑒_𝑖(𝑡)is bounded on[0,+∞). From (12), we get that𝑒_𝑖̇(𝑡)is bounded on[0,+∞). On the other hand, in view of (18), applying Lemma 4, we have lim_𝑡→∞𝑒_𝑖(𝑡)=0, (20) which implies that the error system (5) is globally asymptotically stable; that is, the drive-response systems (1) and (3) are synchronized under adaptive feedback controller (6). The proof of Theorem 5 is completed.

The following results are easily obtained from Theorem 5.

Corollary 6. Under Assumptions 1 and 2, the master-response system (1) and (3) can be synchronized under the adaptive feedback controller (6) and (7), if there exists a constantℎsatisfying the following condition.

Assumption 7. ∑^𝑛_𝑗=1(|𝑏_𝑖𝑗|𝐿_𝑗+|𝑏_𝑗𝑖|𝐿_𝑖)+∑^𝑛_𝑗=1(|𝑐_𝑖𝑗|𝐿_𝑗+(1/𝑞)|𝑐_𝑗𝑖|𝐿_𝑖)−2𝑎_𝑖−ℎ<0, where𝑞=min_𝑖∈ℐ{𝑞_𝑖}.
In particular, when𝑞_𝑗=𝑞(0<𝑞<1), the master-response systems (1) and (3) have the following special case: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑞𝑡+𝐼_𝑖(𝑡)𝑦̇,𝑖∈ℐ,_𝑖(𝑡)=−𝑎_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑞𝑡+𝐼_𝑖(𝑡)+𝐾_𝑖(𝑡),𝑖∈ℐ. (21)(22) From Theorem 5, we have the following corollary as a special case of Theorem 5.

Corollary 8. Assume Assumptions 1 and 2 hold, the master system (21) and response system (22) can be adaptively synchronized under the adaptive feedback controller (6) and (7), if there exists a constantℎsatisfying the following condition.

Assumption 9. ∑^𝑛_𝑗=1(|𝑏_𝑖𝑗|𝐿_𝑗+|𝑏_𝑗𝑖|𝐿_𝑖)+∑^𝑛_𝑗=1(|𝑐_𝑖𝑗|𝐿_𝑗+(1/𝑞)|𝑐_𝑗𝑖|𝐿_𝑖)−2𝑎_𝑖−ℎ<0.
If the feedback strength𝜀(𝑡)=(𝛼(𝑡),…,𝛼(𝑡))^𝑇, then the response system (3) has the special case as follows: 𝑦̇_𝑖(𝑡)=−𝑎_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑞_𝑗𝑡+𝐼_𝑖(𝑡)+𝛼(𝑡)𝑒_𝑖(𝑡),𝑖∈ℐ. (23) Thus, we obtain the following corollary as another special case of Theorem 5.

Corollary 10. Under Assumptions 1 and 2, the master-response system (1) and (23) can be synchronized under the adaptive feedback controller (6), if there exists a constant𝜆>0such that feedback strength𝛼(𝑡)is adapted duly according to the following updated law: 𝛼̇(𝑡)=−𝜆𝑒²_𝑖(𝑡). (24)

Remark 11. In [13], based on Lasalle invariant principle of functional differential equations and the adaptive feedback control technique, the adaptive synchronization behavior for the delayed fuzzy cellular neural networks was obtained. In this paper, by applying Lyapunov functional theory, inequality technique, and Barbalat lemma, some useful results are derived for asymptotical synchronization under adaptive feedback controller.

Remark 12. Recently, research on synchronization problem for chaotic systems, authors have considered constant delay, time-varying delays, distributed delays, mixed time-varying delays, and so on. However, to the best of our knowledge, there are few results concerning pantograph delays for chaotic systems. So, in this paper, the adaptive synchronization of a class of cellular neural networks with pantograph delays was derived.

Remark 13. As far as we know, the synchronization of chaotic systems has been intensively investigated and many good results have been obtained by using control schemes such as adaptive control, pinning control, fuzzy control, impulsive control, intermittent control, and so on. However, to the best of our knowledge, there are few results concerning the synchronization for a class of unautonomous cellular neural networks with pantograph delays based on intermittent feedback control. Therefore, this is another interesting open problem that we should study in the future.

4. Numerical Simulations

In this section, one chaotic network is given to show the effectiveness of our results obtained in this paper.

Example 1. Consider the following cellular neural networks with pantograph delays described by 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑞_𝑗𝑡+𝐼_𝑖(𝑡),𝑖=1,2, (25) where𝑖=1,2, 𝑓₁(𝑥)=𝑓₂(𝑥)=(1/2)(|𝑥+1|−|𝑥−1|), 𝑎₁=𝑎₂=1, 𝑏₁₁=2.3, 𝑏₁₂=3.0, 𝑏₂₁=−5.0, 𝑏₂₂=−0.1, 𝑐₁₁=−1.5, 𝑐₁₂=0.1, 𝑐₂₁=1.0, 𝑐₂₂=2.4, and 𝑞₁=𝑞₂=0.1.
The numerical simulation of system (25) is represented in Figure 1, which shows that system (25) has a chaotic attractor.
In the following, we consider the adaptive synchronization of drive system (25) and response system described by 𝑦̇_𝑖(𝑡)=−𝑎_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑞_𝑗𝑡+𝐼_𝑖(𝑡)+𝜀_𝑖(𝑡)𝑒_𝑖(𝑡),𝑖=1,2, (26) where𝑖=1,2; the parameters𝑎_𝑖,𝑏_𝑖𝑗, and𝑐_𝑖𝑗are defined as in system (25); and 𝜀_𝑖̇(𝑡)=−𝜆_𝑖𝑒²_𝑖(𝑡),𝑖=1,2. (27)
We choose the following initial condition associated with response system (26): 𝑦₁(𝑠)=0.5−0.1𝑘,𝑦₂(𝑠)[].=−0.7+0.1𝑘,𝑘=−1,0,1,∀𝑠∈−1,0 (28) Furthermore, we choose in error system (27) the initial conditions𝜀_𝑖(0)=0(𝑖=1,2)and𝜆₁=0.3, 𝜆₂=0.2. By numerical simulation, we can see that the simulation results of master system (25) synchronize with response system (26) as shown in Figures 2, 3, 4, and 5.

Figure 1 

The chaotic behavior of system (25) with the initial values𝑥1(𝑠)=0.3,𝑥2(𝑠)=−0.5, for𝑠∈[−1,0].

Figure 2 

Synchronization errors of system (25) and system (26).

Figure 3 

Synchronization errors of system (25) and system (27).

Figure 4 

The curve of𝜀1(𝑡).

Figure 5 

The curve of𝜀2(𝑡).

5. Conclusions

In this paper, an adaptive controller has been proposed to investigate the adaptive synchronization for a class of cellular neural networks with pantograph delays by utilizing Lyapunov functional theory, inequality technique, and Barbalat lemma; some sufficient and useful conditions have been derived. Our synchronization criteria are easily verified and do not apply linear matrix inequality and the traditional Lasalle invariant principle. Finally, an example is given to verify the effectiveness and feasibility of the developed methods.

Acknowledgments

This work was supported by Natural Science Foundation of Xinjiang (Grant No. 2013211B06), China Postdoctoral Science Foundation funded project (2013M540782), Natural Science Foundation of Xinjiang University (Grant No. BS120101), and National Natural Science Foundation of People’s Republic of China (Grant No. 61164004).