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

and Applied Analysis 3 Theorem 5. Assume that Assumptions 1 and 2 hold, and the feedback strength ε 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:


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][4][5][6][7][8][9][10][11][12][13] and references cited therein.
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.

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: for  > 0, where ,  ∈ I = {1, 2, . . ., };   () denotes the state of the th neuron at the time ;   > 0 represents the rate with which the th neuron will reset its potential to the resting state in isolation when disconnected from the network and external input;   denotes the strength of the th neuron on the th neuron at the time ;   denotes the strength of the th neuron on the th neuron at the time   ;   (⋅) corresponds to the output of the th neuron; and   ( ∈ I) is constant and satisfies 0 <   < 1,    =  − (1 −   ), in which (1−  ) denotes the pantograph delays along the axon of the th node and   () corresponds to the external bias on the th neuron.System (1) is supplemented with initial value given by In order to observe the synchronization behavior of system (1), the slaver system is designed as follows: for  > 0, where   () denotes the state of the slave system, the rest of the notations are the same as in system (1), and   () is a control input to be designed.
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 constants   ( ∈ I) such that activation function   (⋅) satisfies the following condition: 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 solution () = ( 1 (), . . .,   ())  of system (1) and any solution () = ( 1 (), . . .,   ())  of system (3), we have lim In addition, the following lemma is essential in establishing our main results.

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 strength   () 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: where ℎ is a constant and  = min ∈I {  }, which will be given in the following.From ( 5) and Assumption 2, we have In the following, we calculate the upper right derivative of () along the solution of error system (5).From (11) Then, we finally obtain For  ≥ 0, integrating both sides of inequality ( 15) over [0, ], we get Then In view of (0) > 0, From ( 11) and ( 17), we have Therefore,   () is bounded on [0, +∞).From (12), we get that ė  () is bounded on [0, +∞).On the other hand, in view of (18), applying Lemma 4, we have lim 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 masterresponse system (1) and (3) can be synchronized under the adaptive feedback controller (6) and (7), if there exists a constant ℎ satisfying the following condition.
In particular, when   =  (0 <  < 1), the masterresponse systems (1) and ( 3) have the following special case: 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 constant ℎ satisfying the following condition.Corollary 10.Under Assumptions 1 and 2, the masterresponse system (1) and (23) can be synchronized under the adaptive feedback controller (6), if there exists a constant  > 0 such that feedback strength () is adapted duly according to the following updated law: Remark 11.In [13]

Numerical Simulations
In this section, one chaotic network is given to show the effectiveness of our results obtained in this paper.
where 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 where  = 1, 2; the parameters   ,   , and   are defined as in system (25); and We choose the following initial condition associated with response system (26): Furthermore, we choose in error system (27) the initial conditions   (0) = 0 ( = 1, 2) and  1 = 0.3,  2 = 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.

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.

( 23 )
Thus, we obtain the following corollary as another special case of Theorem 5.
, 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.