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Pinning Impulsive Synchronization of Reaction–Diffusion Neural Networks With Time-Varying Delays Xinzhi Liu, Kexue Zhang, and Wei-Chau Xie Abstract— This paper investigates the exponential synchronization of reaction–diffusion neural networks with time-varying delays subject to Dirichlet boundary conditions. A novel type of pinning impulsive controllers is proposed to synchronize the reaction–diffusion neural networks with time-varying delays. By applying the Lyapunov functional method, sufficient verifiable conditions are constructed for the exponential synchronization of delayed reaction–diffusion neural networks with large and small delay sizes. It is shown that synchronization can be realized by pinning impulsive control of a small portion of neurons of the network; the technique used in this paper is also applicable to reaction–diffusion networks with Neumann boundary conditions. Numerical examples are presented to demonstrate the effectiveness of the theoretical results. Index Terms— Exponential synchronization, Lyapunov– Krasovskii functional, neural network, pinning impulsive control, reaction–diffusion, time delay.

I. I NTRODUCTION

A

S MATHEMATICAL or computational models inspired by the structural and functional mechanism of animals’ central nervous systems, neural networks have been extensively investigated in the past decades. Due to its wide applications in character recognition, image compression, stock market prediction, signal processing, and so on (see [1], [2], and references therein), various neural networks have been studied, such as BAM neural network [3], genetic regulatory network [4], Cohen–Grossborg neural network [5], Hopfield neural network [6], and cellular neural network [7]. Because of the wide applications of neural networks, numerous control problems about neural networks, such as stability analysis, synchronization, H∞ control, tracking control, and optimal control, have been reported recently in the literature. Among these control problems, synchronization of a group of artificial neurons in a network topology is one of the most significant and interesting collective behaviors of neural networks (see [8]–[10]) and has a tremendous application potential in many engineering fields, such as secure communication [11], information processing [12], and pin-fin heat Manuscript received March 26, 2015; revised November 11, 2015; accepted January 7, 2016. X. Liu and K. Zhang are with the Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]; [email protected]). W.-C. Xie is with the Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2016.2518479

transfer prediction [13]. In recent years, various types of conventional and novel control methods have been successfully applied to achieve network synchronization, including adaptive control [14], impulsive control [15], intermittent control [16], pinning control [17], and randomly occurring control [18]. Among these synchronization methods, the impulsive control approach has been proved to be an effective method to achieve network synchronization by using small impulses, which are samples of the state variables of the network neurons, at a sequence of discrete moments to control the state of each neuron. Results have been reported to design appropriate impulsive control instants and suitable impulsive control gains to achieve the synchronization of various kinds of neural networks (see [19]–[23]). The traditional method to synchronize a neural network is to add a controller to each of the network neurons to tame the neuron dynamics to approach a desired synchronization trajectory. However, a neural network is normally composed of a large number of high-dimensional neurons, and it is expensive and infeasible to control all the neurons. Motivated by this practical consideration, the idea of controlling a small portion of nodes, named pinning control, was introduced in [17] and [24], and many pinning algorithms have been reported for the synchronization of dynamical networks (see [25], [30]–[33]). Obviously, the pinning control method reduces the control cost to a certain extent by reducing the amount of controllers added to the neurons. It is worth noting that the cost of control can be further reduced by combining pinning control and impulsive control, i.e., adding the impulsive controllers to a small fraction of network neurons. Hence, the notion of pinning impulsive control has stimulated many interesting pinning impulsive control strategies for the synchronization of dynamical networks with and without delays (see [34]–[37]). The reason that time-delay is significant to the study of neural networks lies in the communication time between neurons and the finite switching speed of amplifiers in hardware implementations of neural networks. Moreover, it is well known that the existence of time-delay may cause oscillation, instability, and poor performance of the dynamical systems (see [38], [39]). Therefore, it is crucial to investigate neural networks subject to delays. In the past decades, many researchers have contributed to the area of synchronization of delayed neural networks (see [5], [16], [20], [21], [23], [25]–[29]). However, very few works have been done on pinning impulsive synchronization of dynamical networks

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with time-delay (see [37], [40]–[42]). Pinning impulsive synchronization of stochastic delayed coupled networks was investigated in [40], and a hybrid adaptive controller, consisting a pinning adaptive controller and a pinning impulsive controller, was designed to achieve synchronization. Actually, the pinning adaptive controller played a key role in the synchronization process, and the pinning impulsive controller cannot be used separately to synchronize the network. In [41], a pinning impulsive control algorithm was introduced to stabilize a class of nonlinear dynamical networks with a time-varying delay. However, time-delay appeared only in the mathematical model of the single node, and no delay was considered in the interconnections among different nodes of the network. Furthermore, the synchronization results in [37] and [42] have some fatal errors. In [37], pinning synchronization of dynamical networks was achieved by a single impulsive controller, but, as pointed out in [42], there are some fatal errors in the proof of the main result of [37] (see [42, Remark 3.11] for details). Yang et al. [42] then investigated the pinning impulsive synchronization of neural networks with reaction–diffusion terms and time-varying delays, using a Halanay-type inequality and comparison method. However, the proof in [42] is not sufficient, and further discussions need to be done to complete the proof of the main result in [42] (see Remark 11 for details). Thus, the results in [42] lack theoretical support. Based on the above literature review, it can be seen that pinning impulsive control for synchronization of delayed neural networks needs further investigation, which motivates the research in this paper. In practice, reaction–diffusions are inevitable in some applications of neural networks due to, for example, the noneven electromagnetic field in which electrons are moving [43] and the diffusion effects in biological systems (see [44], [45]). Therefore, it is necessary to consider the state activations that vary in both space and time, leading to neural networks in the form of partial differential equations. In recent years, impulsive partial differential equations have received a great deal of attention (see [46]–[49]), and impulsive control and stabilization has been shown to be a powerful tool in the applications of various neural networks with reaction–diffusions (see [42], [43], [50], [51]). Due to the advantages of the pinning impulsive control and the existence of time-delay, the study of pinning impulsive control and synchronization of reaction–diffusion neural networks with delays is an interesting and challenging research area that is yet to be fully developed. However, to the best of our knowledge, very few works have been done on this topic, and the existing results in [42] is inconvincible as mentioned before. Moreover, the pinning impulsive control schemes for delayed dynamical networks obtained in [40] and [41] cannot be applied and extended directly to the synchronization problem of neural networks with both reaction–diffusion terms and time-varying delays. In order to fill the research gap discussed above, this paper studies the pinning impulsive synchronization problem of reaction–diffusion neural networks with time-varying delays. There are several difficulties to conduct this research. First, to apply pinning impulsive control method, it is necessary

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and difficult to select appropriate neurons to control at each impulsive instant (see Example 15 for a detailed discussion). Second, for networks with delays, it is practically needed to establish sufficient conditions to guarantee the synchronization of networks with various delay sizes. The Lyapunov–Krasovskii functional method is one of the main approaches to study the stability and synchronization of dynamical networks with time-varying delays. However, the previous two difficulties tighten the restriction on constructing feasible functional candidates when applying the method of Lyapunov–Krasovskii functionals. In this paper, we introduce a type of Lyapunov–Krasovskii functionals and a pinning algorithm to overcome the above difficulties, and sufficient conditions are derived to guarantee the synchronization of neural networks with small and large time-delay, respectively. The pinning algorithm in our paper is more general than the one in [34] and [35] for the synchronization of stochastic dynamical networks, since we can control different amounts of nodes at distinct impulsive instants, while the number of nodes to be controlled is fixed for all impulsive times in [34] and [35]. The Lyapunov–Krasovskii functional candidate is divided into a function part and a functional part. The function part plays an important role to carry over the pinning algorithm and handle the effects that impulses act on the Lyapunov–Krasovskii functional. The idea of constructing Lyapunov–Krasovskii functional candidate and the mathematical analysis approach used in this paper can also be applied to extend our pinning impulsive control scheme to control problems of various dynamical systems with time-delay. The rest of this paper is organized as follows. In Section II, the control problem of the reaction–diffusion neural networks with time-varying delays is formulated, and the pinning algorithm on selecting neurons to add the impulsive controllers is introduced. The main synchronization results are presented in Section III with some discussions. Numerical simulations and further discussions are conducted in Section IV. The detailed proofs of the main results are given in Section V. Finally, some conclusions and possible future research are drawn in Section VI. II. P ROBLEM F ORMULATION Let R denote the set of real numbers, Rm the m-dimensional real space equipped with the Euclidean norm, and N the set of positive integers. C m (W ) represents the set of continuous m-time differentiable real-valued functions on the domain W . G denotes the cardinality of set G (that is, the number of elements of set G if G is finite). Consider the following reaction–diffusion neural network with time-varying delays:   m  ∂ ∂u i (t, x) ∂u i (t, x) = dil − ci u i (t, x) ∂t ∂ xl ∂ xl l=1 n  ai j f j (u j (t, x)) + j =1

+ +

n  j =1 Ji ,

bi j f j (u j (t − τi j (t), x)) i = 1, 2, . . . , n

(1)

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where x = (x 1 , x 2 , . . . , x m )T ∈  ⊂ Rm is the space variable with  = {x = (x 1 , x 2 , . . . , x m )T : |x k | ≤ h k , k = 1, 2, . . . , m}, and h k (k = 1, 2, . . . , m) are positive constants; u i (t, x) denotes the state of the i th neuron at time t and in space x; the activation function f j (u j (t, x)) stands for the output of the j th neuron at time t and in space x. Ji , ci , ai j and bi j are constants: Ji is the external bias or input to the i th neuron; ci > 0 represents the rate with which the i th neuron will reset its potential to the resting state when disconnected from the network and under external input Ji ; ai j and bi j are the connection weights between neurons. τi j denotes the transmission time-varying delay from the j th neuron to the i th neuron; dil > 0 is the transmission diffusion coefficient along the i th neuron. Throughout this paper, we make the following assumptions on time-varying delays and activation functions. (A1 ) There exist positive constants τ and δi j such that 0 ≤ τi j (t) ≤ τ and τ˙i j (t) ≤ δi j < 1 for all i, j ∈ {1, 2, . . . , n}. (A2 ) There exists a constant L i such that | f i (u) − f i (v)| ≤ L i |u − v| for all u, v ∈ R and i = 1, 2, . . . , n. Remark 1: Assumption (A1 ) implies that the time-delay in network (1) is bounded and (d(t − τi j (t))/dt) > 0, i.e., t − τi j (t) is increasing. Intuitively, as time t increases, the delay dependence of the state u i (t, x) is increasing. Thus, it is straightforward to make this assumption. In terms of the activation function, various neural networks possess the properties concluded in assumption (A2 ) (see bidirectional memory networks [52] and BAM networks [53]). The Dirichlet boundary condition of system (1) is given by u i (t, x) = 0

(2)

for (t, x) ∈ [t0 − τ, +∞) × ∂ and i = 1, 2, . . . , n, where t0 is the initial time, and ∂ denotes the boundary of . The initial value of system (1) is given as follows: u i (t0 + s, x) = φi (s, x)

(3)

for (s, x) ∈ [−τ, 0] ×  and i = 1, 2, . . . , n, where φ = (φ1 , φ2 , . . . φn )T ∈ C([−τ, 0] × , Rn ). For φ ∈ C([−τ, 0] × , Rn ) and a given s ∈ [−τ, 0], define the following norm:  n  1/2  2 φi (s, x)d x . φ(s, ·)2 = i=1



Similarly, for u = (u 1 , u 2 , . . . , u n )T ∈ C([0, ∞)×, Rn ) and a given t ≥ 0, we define the following norm:  n  1/2  2 u i (t, x)d x . u(t, ·)2 = i=1



Let us introduce a drive system in the form of (1) with Dirichlet boundary condition (2) and initial condition (3),

3

and a response system in the form of ⎧   m ⎪ ⎪ ∂v i (t, x) =  ∂ d ∂v i (t, x) − c v (t, x) ⎪ ⎪ il i i ⎪ ∂t ∂ xl ∂ xl ⎪ ⎪ l=1 ⎪ ⎪ n ⎪  ⎪ ⎪ ⎪ + ai j f j (v j (t, x)) ⎪ ⎪ ⎪ ⎪ ⎨ j =1 n  ⎪ + bi j f j (v j (t − τi j (t), x)) ⎪ ⎪ ⎪ ⎪ j =1 ⎪ ⎪ ⎪ ⎪ ⎪ + Ji + Ui (t, x) ⎪ ⎪ ⎪ ⎪ v (t + s, x) = ϕi (s, x), (s, x) ∈ [−τ, 0] ×  ⎪ i 0 ⎪ ⎪ ⎩v (t, x) = 0, (t, x) ∈ [t − τ, ∞) × ∂ i 0

(4)

where Ui is the control input to the i th neuron yet to be designed, and ϕ = (ϕ1 , ϕ2 , . . . , ϕn )T ∈ C([−τ, 0] × , Rn ). Without loss of generality, we assume that ϕ(s, x) ≡ φ(s, x) for (s, x) ∈ [−τ, 0] ×  so that the synchronization behavior can be observed. Definition 2: The drive system (1) and the response system (4) are said to be exponentially synchronized under controller U (t, x), if there exist constants μ > 0 and M ≥ 1 such that u(t, ·) − v(t, ·)2 ≤ Me−μ(t −t0 ) sup ϕ(s, ·) − φ(s, ·)2 s∈[−τ,0]

for all t ≥ t0 , where U = (U1 , U2 , . . . , Un )T , u = (u 1 , u 2 , . . . , u n )T , and v = (v 1 , v 2 , . . . , v n )T . The objective of this paper is to exponentially synchronize system (4) with (1) by designing a suitable impulsive controller U (t, x). In order to force the trajectory of network (4) to approach the trajectory of network (1) exponentially, we design the following pinning impulsive controller: ⎧∞  ⎪ ⎨ −q e (t , x)δ(t −t ), i ∈ D and D = l k i k k k k k Ui (t, x) = k=1 ⎪ ⎩ 0, i ∈ Dk (5) where i = 1, 2, . . . , n, and qk ∈ (0, 1) is the impulsive control gain to be determined. The impulsive instant sequence {tk } satisfies {tk } ⊂ R, 0 ≤ t0 < t1 < t2 < · · · < tk < · · · ·, and limk→∞ tk = ∞. δ(·) is the Dirac delta function. The error state ei (t, x) = v i (t, x) − u i (t, x) represents the state difference of the two networks (1) and (4). lk denotes the number of neurons to be pinned at each impulsive instant. The index set Dk is defined as follows: at the impulsive instant tk , we can reorder the scalar states e1 (tk , x), e2 (tk , x), . . . , en (tk , x) such that e p1 (tk , ·)2 ≥ e p2 (tk , ·)2 ≥ · · · ≥ e plk (tk , ·)2 ≥ · · · ≥ e pn (tk , ·)2 , then define Dk = { p1, p2, . . . , plk } and Dk = lk , where 0 < lk ≤ n. The pinning impulsive synchronization mechanism is illustrated in Fig. 1. The response system (4) with the pinning impulsive controller (5) can be rewritten in the form of an

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Fig. 1. Pinning impulsive synchronization mechanism. Si and Ri represent the ith neurons of network (1) and (4), respectively. f i depicts all the activation functions corresponding to the ith neuron of the network.

impulsive system ⎧   m ⎪ ∂v i (t, x) =  ∂ d ∂v i (t, x) − c v (t, x) ⎪ ⎪ il i i ⎪ ∂t ⎪ ∂ xl ∂ xl ⎪ ⎪ l=1 ⎪ ⎪ n ⎪  ⎪ ⎪ ⎪ ai j f j (v j (t, x)) + ⎪ ⎪ ⎪ ⎪ j =1 ⎪ ⎪ ⎨ n  bi j f j (v j (t − τi j (t), x)) + ⎪ ⎪ ⎪ j =1 ⎪ ⎪ ⎪ ⎪ +Ji , t = tk ⎪ ⎪ ⎪ ⎪ ⎪ v i (tk , x) = −qk ei (tk , x), i ∈ Dk , Dk = lk , k ∈ N ⎪ ⎪ ⎪ ⎪ ⎪ v i (t0 + s, x) = ϕi (s, x), (s, x) ∈ [−τ, 0] ×  ⎪ ⎪ ⎩ v i (t, x) = 0, (t, x) ∈ [t0 − τ, ∞) × ∂ (6) where v i (tk , x) = v i (tk+ , x) − v i (tk− , x), v i (tk+ , x) and v i (tk− , x) denote the right and left limit of v i at tk , respectively. Remark 3: It is shown in [54] that, under assumptions ( A1 ) and ( A2 ), system (1) with the Dirichlet boundary condition (2) and initial condition (3) admits a unique global solution. The existence of solution to system (6) can be guaranteed by the existence results of reaction–diffusion equations in [55] and the method of steps, since discrete delays are considered in the system. Throughout this paper, we always assume that v i (t, x) is left continuous at tk (k ∈ N), i.e., limt →t − v i (t, x) = v i (tk , x) k for all x ∈ . Then, by introducing the error state ei , we have the following error system: ⎧   m ⎪ ⎪ ∂ei (t, x) =  ∂ d ∂ei (t, x) − c e (t, x) ⎪ ⎪ il i i ⎪ ∂t ⎪ ∂ xl ∂ xl ⎪ l=1 ⎪ ⎪ n ⎪  ⎪ ⎪ ⎪ ai j + f j (e j (t, x)) ⎪ ⎪ ⎪ ⎪ ⎨ j =1 n  ⎪ f j (e j (t − τi j (t), x)), t = tk bi j + ⎪ ⎪ ⎪ ⎪ j =1 ⎪ ⎪ ⎪ ⎪ ⎪ ei (tk , x) = −qk ei (tk , x), i ∈ Dk , Dk = lk , k ∈ N ⎪ ⎪ ⎪ ⎪ ei (t0 + s, x) = ϕi (s, x)−φi (s, x), (s, x) ∈ [−τ, 0] ×  ⎪ ⎪ ⎪ ⎩e (t, x) = 0, (t, x) ∈ [t − τ, ∞) × ∂ i 0 (7)

f j (v j (·, x)) − f j (u j (·, x)) for where f j (e j (·, x)) = j = 1, 2, . . . , n. It can be seen from (7) and Definition 2 that if e(t, ·)2 converges to zero exponentially as t → ∞, then the drive system (1) and the response system (4) can be exponentially synchronized. Remark 4: Designing an appropriate pinning impulsive controller contains four aspects. 1) How many neurons need to be pinned? 2) Which neurons need to be selected? 3) When does the impulsive control input need to be added to the neurons? 4) How strong does the impulsive control gain need to be? The pinning impulsive control scheme (5) is inspired by the idea in [34] and [35] for nonlinear networks without delays and reaction–diffusions. The difficulties of applying the pinning impulsive control scheme (5) to networks with timedelay and reaction–diffusion terms will be discussed in detail in Remark 9. Up to now, we have designed only the control strategy about which neurons need to be controlled at each impulsive instant by control scheme (5), i.e., controlling lk neurons that have larger state difference than the other n − lk neurons. Other aspects of designing a suitable pinning impulsive controller will be investigated in Section III. Sufficient conditions on suitable relations among the impulsive instant tk , impulsive control gain qk , and the number of neurons to be pinned lk will be established. Furthermore, different neurons may be selected to pin at different impulsive instant according to our pinning control mechanism (5). Therefore, we do not need the assumption about the connectivity of the network that is necessary in [42]. However, there is a fatal error in [42], which will be discussed in Remark 11. III. M AIN R ESULTS In this section, exponential synchronization criteria for reaction–diffusion neural networks with time-varying delays are established, and these results will also be discussed. For convenience, we introduce the following notations. Let lk qk (2 − qk ), k ∈ N n  n ξ −1 |b |L  ij j ij λ = max 1≤ j ≤n 1 − δi j ⎧ i=1  m n ⎨ π 2  dil  εi j |ai j |L j c = max −2ci − + 1≤i≤n ⎩ 2 h2 l=1 l j =1

ρk = 1 −

+ ξi j |bi j |L j + ε−1 j i |a j i |L i

+

ξ −1 j i |b j i |L i



1 − δ ji

where εi j and ξi j (i, j = 1, 2, . . . , n) are positive real numbers. Now we are in the position to state our main results, proofs of which will be presented in Section V. Theorem 5: Suppose that assumptions (A1 ) and (A2 ) hold, c > 0, and τ ≤ tk −tk−1 for all k ∈ N. Moreover, if there exist positive constants ξi j , εi j (i, j = 1, 2, . . . , n), and α such that ln(ρk + λτ ) ≤ −(α + c)(tk+1 − tk ), k ∈ N

(8)

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then the drive system (1) and the response system (4) are exponentially synchronized by the pinning impulsive controller (5). Remark 6: It can be seen from the proof of Theorem 5 that the synchronization rate is α/2, which is closely related to the system parameters λ, c, controller parameters ρk and the length of impulsive intervals tk+1 − tk (k ∈ N). Since ρk depends on the impulsive control gain qk and the number of neurons to be pinned at each impulsive instant tk , inequality (8) gives us a guideline of balancing the values of qk , lk , and tk+1 − tk to obtain a suitable pinning impulsive controller. However, the condition τ ≤ tk − tk−1 implies that the size of time-delay τ is a lower bound of the impulsive intervals. Therefore, large timedelay in network (1) may render Theorem 5 invalid. In order to resolve this issue, we have the following result. Theorem 7: Suppose that assumptions (A1 ) and (A2 ) hold, and c > 0. Furthermore, if there exist positive constants ξi j , εi j (i, j = 1, 2, . . . , n), and α such that ln(ρk + λτ eατ ) ≤ −(α + c)(tk+1 − tk ), k ∈ N

(9)

Neumann boundary condition   ∂u i (t, x) ∂u i (t, x) ∂u i (t, x) T ∂u i (t, x) = , ,..., · n=0 ∂ n ∂ x1 ∂ x2 ∂ xm for (t, x) ∈ [t0 − τ, ∞) × ∂, where n is the outward unit normal vector of ∂, and the dot is the inner product. Clearly, Lemma 17 in Section V is not valid for the Neumann boundary condition. However, our method is still applicable to the synchronization problem of reaction–diffusion neural networks with Neumann boundary conditions. For instance, (17) can be replaced by the following estimation:    m  ∂ ∂ei (t, x) ei (t, x) dil d x ≤ 0. ∂ xl ∂ xl  l=1

Then replacing c by ⎧  n ⎨  εi j |ai j |L j + ξi j |bi j |L j c = max −2ci + 1≤i≤n ⎩ j =1

+ ε−1 j i |a j i |L i then the drive system (1) and the response system (4) can be exponentially synchronized by the pinning impulsive controller (5). Remark 8: In this result, we do not need the condition τ ≤ tk − tk−1 , i.e., the length of impulsive interval tk − tk−1 can be less that the size of time-delay. Therefore, Theorem 7 is applicable to networks with relatively large delays. However, it can be seen that the exponential term eατ on the left-hand side of (9) makes (9) supply a more conservative condition on the choice of ρk and tk+1 − tk than condition (8) does for networks with small-enough delays. Therefore, Theorems 5 and 7 give us sufficient conditions to design appropriate pinning impulsive controllers to synchronize networks (1) and (4) with small or large time-delay. Illustrative examples are presented in Section IV. Remark 9: The Lyapunov–Krasovskii functional candidates in proofs of Section V are divided into two parts: 1) a function part and 2) a functional part. This kind of structure has been widely used in the literature when stability of dynamical systems with delays is investigated (see [42], [50], [56]). However, no pinning impulsive synchronization result has been reported for delayed networks by the method of Lyapunov–Krasovskii functionals. In this paper, there are two reasons to consider this type of Lyapunov–Krasovskii functional. First, it is straightforward for an impulse to alter the value of a function instantaneously. Thus, the value of the function part can be effectively reduced by the impulse, whereas the functional part is not affected by the impulse (see [57] for a similar discussion of impulsive delay systems). However, this fact brings dramatic difficulties to the theoretic reasoning of our main results. Second, the quadratic form of the function part makes it possible for us to generalize the pinning impulsive control strategy in [34] for networks without time-delay to synchronization problems of reaction–diffusion networks with time-varying delays. Remark 10: Another commonly considered boundary condition for reaction–diffusion neural networks is the

5

+

ξ −1 j i |b j i |L i



1 − d ji

which is independent of the reaction–diffusion coefficients dil , Theorems 5 and 7 can be applied to design suitable pinning impulsive controllers to achieve synchronization of delayed reaction–diffusion networks (1) and (4) with Neumann boundary conditions. Furthermore, the technique that is used in this paper for designing the Lyapunov–Krasovskii functionals combined with the pinning impulsive control strategy is also applicable for neural networks with distributed delays and various types of neural networks, such as BAM neural networks, stochastic neural networks, fuzzy neural networks, and discrete-time neural networks. Remark 11: The pinning impulsive synchronization of reaction–diffusion neural networks has recently been studied in [42]. By using Lyapunov function, Halanay-type inequality, and comparison method, sufficient conditions have been obtained to design appropriate impulsive controllers to pin the same neurons at each impulsive instant, which is different from our pinning impulsive mechanism. Moreover, there is a fatal error in the proof of the main result in [42]. In [42, Proof of Th. 3.1], the estimation of (3.32) is based on the assumption that (3.14) and (3.15) are true. However, (3.13) does not imply that (3.14) and (3.15) hold, and there is no condition in [42, Th. 3.1] to guarantee (3.14) and (3.15) are true. Hence, [42, Proof of Th. 3.1] is not sufficient, and the corresponding results lack theoretical support. Remark 12: In Theorems 5 and 7, two types of assistant parameters ξi j and εi j are used to reduce the conservatism of estimations in (8) and (9). These parameters make Theorems 5 and 7 flexible in dealing with networks with various system coefficients. For instance, to use Theorem 5 or Theorem 7, the parameter λ in (8) or (9) is required to make λτ or λτ eατ less than 1. For networks with large delay size, parameters ξi j can be chosen to make λ small enough, then Theorem 5 or 7 can be applied to design suitable pinning impulsive controllers to realize network synchronization. For more details, see the examples in Section IV.

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Fig. 2. State trajectories of network (1) with the parameters given in Example 15 and initial data φ1 (s, x) = 0.5 cos (πx/8), φ2 (s, x) = 0.4 cos (πx/8), for s ∈ [−τ, 0] and x ∈ . The spatiotemporal chaotic behavior can be clearly observed in the above figures.

If the uniform impulsive controller is considered in (5), i.e., lk = l, qk = q, and tk − tk−1 = T for all k ∈ N, then we have the following synchronization result. Theorem 13: Suppose that assumptions (A1 ) and (A2 ) hold, and c > 0. Furthermore, if there exist positive constants ξi j , εi j (i, j = 1, 2, . . . , n) such that ln(ρ + λτ ) < −cT

(10)

where ρ = 1 − (l/n)q(2 − q), then the drive system (1) and the response system (4) can be exponentially synchronized by the pinning impulsive controller (5). Remark 14: It can be observed from (10) that there exists a positive constant α such that ln(ρ + λτ ) = −(α + c)T

(11)

ln(ρ + λτ eατ ) = −(α + c)T

(12)

or

which implies (8) or (9) is satisfied. Therefore, Theorem 13 can be proved. Moreover, if τ ≤ T , then the convergence rate α can be estimated by (11). Otherwise, α can be obtained by solving (12). IV. N UMERICAL S IMULATIONS AND D ISCUSSIONS In this section, we present two examples to demonstrate our main results. In order to clearly observe the pinning impulsive control process in the simulation results, we will consider neural networks with only two neurons, i.e., n = 2. In the first example, we consider the synchronization problem of reaction–diffusion neural networks with time-invariant delays. Example 15: Consider a delay reaction–diffusion neural network described by (1) with Dirichlet boundary condition (2) and initial condition (3), where t0 = 0, m = 1, n = 2,  = [−4, 4], d1 = d2 = 0.1, c1 = c2 = 1, τ11 = τ22 = 1, τ12 = τ21 = 0.5, f 1 (·) = f 2 (·) = tanh(·), J1 = J2 = 0, and     2 −0.1 −1.5 −0.1 [ai j ]2×2 = , [bi j ]2×2 = . −5 3 −0.2 −2.5 It can be easily verified that assumption (A1 ) is satisfied with τ = 1 and δi j = 0 for i, j = 1, 2, and assumption (A2 ) is satisfied with L 1 = L 2 = 1. The chaotic behavior of neural network (1) with the given initial data is shown in Fig. 2.

For [ξi j ]2×2 =



10 1

  1 1 , and [εi j ]2×2 = 10 0.36

1 1



one can get the following estimations: λ = 0.36, c1 = 31.4264, and c2 = 31.4375. Then, c = 31.4375. In this example, we first consider the impulsive controller (5) with lk = n = 2, tk − tk−1 = 0.02, and qk = 0.59 for all k ∈ N; then, (9) is satisfied with α = 0.01. Therefore, we can conclude from Theorem 7 that the drive system (1) can be exponentially synchronized with the response system (4) under impulsive controller (5). Fig. 3 shows that trajectories of the synchronization error states ei (t, x), i = 1, 2. It can be seen that there are visible serrations in the trajectories of ei when t is less than 1, which can be clearly observed in Fig. 4 for ei 2 (i = 1, 2). This phenomenon can be explained by the existence of time-delay with delay size 1 in the network, which verifies our discussion of the impact of the delay size on the synchronization rate in Remark 8. Next, consider a pinning impulsive controller with lk = 1, tk − tk−1 = 0.004, and qk = 0.9 for k ∈ N, i.e., control one neuron at each impulsive instant. Then, all the conditions of Theorem 7 are satisfied, and numerical simulations are shown in Fig. 5(a). Compared with the pinning impulsive control method in [42], we need to select a small fraction of neurons to pin at each impulsive instant according to our pinning algorithm, while no conditions are required on how to select neurons to control in [42]. Actually, selection of suitable neurons to pin is necessary when applying pinning impulsive control approach. From the simulation results in Fig. 5(b) and (c), we can conclude that the pinning neurons need to be carefully selected to achieve the network synchronization, which verifies our theoretical analysis of the deficiency of the results in [42]. It is worth noting that the pinning strategy introduced in this paper is one of the feasible selection methods of the pinning neurons to realize the network synchronization. In this sense, we have overcome the deficiency of the main result in [42]. Compared with the traditional impulsive controller, fewer neurons are required to be controlled at each impulsive instant. Moreover, by comparison of the numerical results shown in Fig. 5(b) and (c), we can see that our pinning algorithm is more efficient in synchronizing the networks. Although the pinning method considered in Fig. 5(c) (pinning a specific

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7

Fig. 3. Synchronization errors e1 and e2 . The initial data of the drive system (1) are chosen the same as that in Fig. 2, and the initial conditions of the response system (4) are given by ϕ1 (s, x) = 0.4 cos (πx/8), ϕ2 (s, x) = 0.6 cos (πx/8), for s ∈ [−τ, 0] and x ∈ . It can be seen that the synchronization between the drive and response systems can be realized. 0.45 0.2 0.4 0.18 0.35

0.14

0.3

0.12

0.25

||e ||

||e ||

1 2

2 2

0.16

0.1

0.2 0.08 0.15 0.06 0.1 0.04 0.05

0.02 0

0

0.5

1

0

1.5

0

0.5

1

t

Fig. 4.

1.5

t

Synchronization errors e1 and e2 in norm. The effects of time-delay in the synchronization process can be clearly observed. 2.5 0.03 5

||e ||

2 2

2 2

0.45 0.25

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||e ||

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||e ||

0.025 ||e1||2

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||e1||2

−3

x 10

2 2

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4

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0.2

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3

0 21

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2.5

21.02 t

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0.25

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0.1

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1.5 0.1

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0.05

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1.02

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0.1 0

1.1

t

0

0

0.2

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0

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20 t

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35

40

0

0.06

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t

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5

10

15

20 t

25

30

35

40

Fig. 5. Synchronization processes via different pinning impulsive controllers. For these three sub-figures, the impulsive time sequence and impulsive control gains are chosen to be the same: tk − tk−1 = 0.004 and qk = 0.9 for all k ∈ N. Different pinning algorithms are introduced as follows. (a) Pinning control the response system according to the pinning strategy in (5) with lk = 1. (b) Impulsive control the second neuron of the response system at each impulsive instant. (c) Impulsive control the first neuron of the response system. It can be seen that synchronization can be achieved in (a) and (c), and the synchronization time in (c) is greater than 10 units of time, which is dramatically larger than that in (a). However, the synchronization cannot be achieved in (b).

number of neurons at all the impulsive time) requires less information of the neurons’ states, our simulations have shown that we need to look more deeply into the dynamics of each isolated neuron, the network topology, and their relations to figure out how to select appropriate neurons to stimulate. Furthermore, this pinning approach only applies to a specific class of networks that need to be classified, since even a simple linear system may not be stabilizable via this pinning impulsive control method. For example, consider the

linear system y˙ = Ay, where y = [y1 , y2 ]T ∈ R2 and 2 1 A = . No matter how frequently the impulsive con12 troller is added to state y1 , state y2 will blow up (that is, as y1 approaches zero, the linear part y˙2 = 2y2 will dominate the evolution of state y2 ). Our future research will focus on figuring out conditions on the network dynamics and topologies to guarantee the validity of this pinning impulsive control method.

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6

6

6

u

u

u2

u2

1

u2 5

4

4

4

3

3

3

u

5

5

u

u

u

1

1

2

2

2

1

1

1

0

0

0.2

0.4

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0.8

1

1.2

1.4

0

0

0.1

0.2

0.3

0.4

t

0.5 t

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

0.8

0.9

1

Fig. 6. Simulation results of Example 16. Parameters of the three simulations are given as follows, respectively. (a) tk − tk−1 = 0.15, qk = 0.64 and lk = 2 (k ∈ N), i.e., impulsive control of both neurons of network (13). (b) tk − tk−1 = 0.05, qk = 0.68, lk = 1 (k ∈ N), i.e., impulsive control of one neuron at each impulsive instant. (c) t2k−1 − t2k−2 = 0.05 and t2k − t2k−1 = 0.15, q2k−1 = 0.64 and q2k = 0.68, l2k−1 = 1 and l2k = 2 (k ∈ N), i.e., impulsive control of one neuron at each odd impulsive instant and two neurons at each even impulsive instant. All sufficient conditions of Theorem 5 are satisfied with α = 0.01, and simulation results imply that the equilibrium of system (13) can be exponentially stabilized.

If the initial conditions of the drive system (1) are the same as its equilibrium, then system (1) reduces to a system with constant states. Thereafter, the synchronization problem of (1) and (4) reduces to the stabilization problem of system (1). When dil = 0 (i = 1, 2, . . . , n and l = 1, 2, . . . , m), our results can be applied to delayed neural networks without reaction–diffusion terms. In order to show the effectiveness of Theorem 5, we consider the following example with a small delay size. Example 16: Consider the following delayed neural network: n  ai j f (u j (t)) u˙ i (t) = −ci u i (t) + +

n 

then  

w2 (x)d x ≤

A. Proof of Theorem 5 Consider the following Lyapunov–Krasovskii functional: V (t) = V1 (t) + V2 (t) where V1 (t) =

j =1

bi j f (u j (t − τ (t))) + Ji , i = 1, 2

  2    ∂w(x) 2 2   h 2i  ∂ x  d x. π i 

n   i=1

(13)

V. P ROOFS In this section, we present the proofs for the main results in Section III, which are based on the Lyapunov–Krasovskii functional method and a Poincare-type inequality presented in the following lemma (see [56]). Lemma 17: Let w(x) = w(x 1 , x 2 , . . . , x m ) be a real-valued function defined on . If w(x) ∈ C 1 () and w(x) |∂ = 0,

ei2 (t, x)d x

and

j =1

where c1 = 1, c2 = 0.5, J1 = J2 = 2, f (u) = (1/2)(|u + 1| − |u − 1|) for u ∈ R, τ (t) = (0.01et /1 + et ) for t ≥ 0, and     0.5 0.5 −1.5 −1.5 [ai j ]2×2 = , [bi j ]2×2 = . 0.5 1 −1 −0.5 ∗ By direct computation, we know that u = (0.5, 4.5)T is the unique equilibrium of system (13), and f satisfies assumption (A2 ) with L = 1. For the time-delay τ (t), we have τ (t) ≤ 0.01 and τ˙ (t) ≤ 0.0025. Then τ = 0.01 and δ = δi j = 0.0025. For ξi j = εi j = 1 (i, j = 1, 2), we have λ = 2.5253 and c = max{c1 , c2 } = 5.5253. Design three different impulsive controllers in the form of (5) with e1 (t) = u 1 (t)−0.5 and e2 (t) = u 2 (t)−4.5, and then numerical results are shown in Fig. 6.



V2 (t) =

n n  

 γi j

i=1 j =1

t t −τi j (t )

 

 e2j (s, x)d x ds

where γi j = (ξi−1 j |bi j |L j /1 − δi j )(i, j = 1, 2, . . . , n). The proof is divided into the following three steps. Step 1: Estimation of V˙ (t) on each impulsive interval and V (t) at each impulsive instant. First, differentiate V (t) along the trajectory of the error system (7) for t ∈ (tk−1 , tk ]. For V1 (t), we have n  

∂ei (t, x) dx 2ei (t, x) ∂t i=1    n  m   ∂ei (t, x) ∂ = 2 ei (t, x) dil ∂ xl ∂ xl i=1  l=1 n  − 2ci ei2 (t, x) + 2 ai j ei (t, x) fˆj (e j (t, x))

V˙1 (t) =

j =1

+2

n  j =1

⎫ ⎬

bi j ei (t, x) fˆj (e j (t − τi j (t), x)) d x. ⎭ (14)

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Then, we have the following estimations: n 

2

≤

n 

2|ai j |L j |ei (t, x)||e j (t, x)|   2 |ai j |L j εi j ei2 (t, x) + εi−1 j e j (t, x)

(15)

j =1



+ +

≤

+

n 

  2 |bi j |L j ξi j ei2 (t, x) + ξi−1 j e j (t − τi j (t), x) .

(16)

j =1

By the Dirichlet boundary condition (2), Divergence theorem, and Lemma 17, we get     m ∂ei (t, x) ∂ dil dx ei (t, x) ∂ xl ∂ xl  l=1     m ∂ ∂ei (t, x) = ei (t, x)dil dx ∂ xl  l=1 ∂ x l     m ∂ei (t, x) 2 − dil dx ∂ xl  l=1    ∂ei (t, x) m ei (t, x)dil = · n ds ∂ xl ∂ l=1     m ∂ei (t, x) 2 dil dx − ∂ xl  l=1     m ∂ei (t, x) 2 =− dil dx ∂ xl  l=1      m  dil 2 ∂ei (t, x) 2 =− h dx ∂ xl h l2 l  l=1  m   π 2  dil ei2 (t, x)d x. (17) ≤− 2 2 h  l=1 l For V2 (t), we have V˙2 (t) =

n n  



e2j (t, x)d x



e2j (t − τi j (t), x)(1 − τ˙i j (t))d x

n 





e2j (t, x)d x

ei2 (t, x)d x

 γi j



e2j (t, x)d x

⎫ ⎬ ⎭ (19)





j ∈D k





(20)

Hence, by (20), we obtain  (1 − ρk ) ei2 (tk , x)d x i ∈Dk



≤ (1 − ρk )(n − lk )βk = [ρk − (1 − qk )2 ]lk βk  ≤ [ρk − (1 − qk )2 ] ei2 (tk , x)d x i∈Dk



which yields V1 (tk+ )

=

n    

  ei2 tk+ , x d x

i=1

=



i∈Dk





−

 ξi j |bi j |L j



j =1



which implies that, for i ∈ Dk    ei2 (tk , x)d x ≤ min e2j (tk , x)d x = βk .

=



γi j

e2j (t − τi j (t), x)d x. (18)

Next, we investigate the impact that the impulses play on the Lyapunov–Krasovskii functional. For k ∈ N, define  βk = min j ∈Dk {  e2j (tk , x)d x}. From the pinning algorithm introduced with impulsive controller (5), one can have that, for any i ∈ Dk , and j ∈ Dk   2 ei (tk , x)d x ≤ e2j (tk , x)d x

i∈Dk

i=1 j =1

n 

j =1

= cV1 (t) ≤ cV (t).

2|bi j |L j |ei (t, x)||e j (t − τi j (t), x)|

j =1

≤

εi−1 j |ai j |L j

j =1

|bi j ||ei (t, x)|| fˆj (e j (t − τi j (t), x))|

j =1 n 

n  j =1

j =1

≤2



V˙ (t) = V˙1 (t) + V˙2 (t)

 n m  π 2  dil ≤ − ei2 (t, x)d x 2 2 h  l i=1 l=1  n  2 εi j |ai j |L j ei2 (t, x)d x − 2ci ei (t, x)d x +

and n  2 bi j ei (t, x) fˆj (e j (t − τi j (t), x)) n 



Then, from (14) to (18), we get, for t ∈ (tk−1 , tk ]

j =1 n 



i=1 j =1

|ai j ||ei (t, x)|| fˆj (e j (t, x))|

j =1 n 

e2j (t, x)d x

γi j (1 − δi j )

−

≤2

≤

 γi j

i=1 j =1 n  n 

ai j ei (t, x) fˆj (e j (t, x))

j =1

≤

n  n 

9

≤ ρk





     ei2 tk+ , x d x + ei2 tk+ , x d x (1 − qk )2 ei2 (tk , x)d x



i∈Dk



i ∈Dk



ei2 (tk , x)d x

= ρk V1 (tk ), k ∈ N.

+ ρk

+



i ∈Dk  i ∈Dk





ei2 (tk , x)d x

ei2 (tk , x)d x (21)

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For V2 (t), we get

Next, we suppose (24) holds for k = j ( j > 2), that is

  V2 tk+ = V2 (tk ), k ∈ N.

Now, we conclude from (19), (21), and (22) with the following inequalities: V˙ (t) ≤ cV (t), t ∈ (tk−1 , tk ]

V1 (tk+ ) V2 (tk+ )

(23a)

≤ ρk V1 (tk )

(23b)

= V2 (tk ), k ∈ N.

(23c)

Step 2: Mathematical induction. We claim that

V (t) ≤ V (t0 )ec(t −t0 ) = V (t0 )e(α+c)(t1−t0 ) e−(α+c)(t1−t0 ) ec(t −t0 ) = Me−(α+c)(t1 −t0 ) ec(t −t0) .

≤

γi j

t1 −τ

i=1 j =1

n 

≤ max

1≤ j ≤n



=λ ≤ λτ

i=1

t1 t1 −τ





γi j





n  

j =1 t1 −τ

e(s, ·)22 ds

sup

s∈[t1 −τ,t1 ]

t1

e(s, ·)22

that is e(t, ·)2 =

Thus

  V2 t1+ ≤ λτ Me−(α+c)(t1 −t0 ) ec(t1 −t0 ) .

i.e., claim (24) holds for k = 2.

!

"

" V1 (t) ≤ V (t) ≤ M¯ sup

! α e(s, ·)2 e− 2 (t −t0)

for t ≥ t0 , where M¯ = [(1 + λτ )e(α+c)(t1−t0 ) ]1/2 > 1. This completes the proof.

(28)

Consider the same Lyapunov–Krasovskii functional given in the previous proof. Based on the discussion in the proof of Theorem 5 (Subsection V-A), we have that (23a), (23b), and (23c) are true. Since lim k→∞ tk = ∞, there exists an integer i ≥ 1 such that ti − τ ≥ t0 , and for t ∈ [t0 , ti ], we have

(29)

V (t) = V (t)eα(t −t0) e−α(t −t0) ≤ Me−α(t −t0 )

(31)

where M = supt ∈[t0−τ,ti ] {V (t)}eα(ti −t0 ) . Next, we shall show that

It follows from (8), (26), and (29) that:       V t1+ = V1 t1+ + V2 t2+ ≤ (ρ1 + λτ )Me−(α+c)(t1 −t0 ) ec(t1 −t0 ) ≤ Me−(α+c)(t2 −t0 ) ec(t1 −t0 ) . Then, for t ∈ (t1 , t2 ], we have   V (t) ≤ V t1+ ec(t −t1 ) ≤ Me−(α+c)(t2 −t0 ) ec(t −t0)

s∈[t0 −τ,t0 ]

e(s, ·)22

B. Proof of Theorem 7 (27)

s∈[t1 −τ,t1 ]

≤ Me−(α+c)(t1 −t0 ) ec(t1 −t0 ) .

sup

s∈[t0 −τ,t0 ]

and from the condition τ ≤ t1 − t0 and (25), we get ! ! sup e(s, ·)22 = sup V1 (s) s∈[t1 −τ,t1 ]

= V (t0 )e(α+c)(t1−t0 ) e−α(t −t0) · e(α+c)(t1−t0 ) e−α(t −t0 )

e2j (s, x)d x ds

!

V (t) ≤ Me−(α+c)(tk −t0 ) ec(t −t0)

≤ (1 + λτ )

 

which implies that (24) holds for all k ∈ N. Step 3: Convergence estimation. From (24), we have, for t ∈ (tk−1 , tk ]

(26)

ds



≤ Me−(α+c)(t j +1−t0 ) ec(t −t0 )

≤ Me−α(t −t0 )



e2j (s, x)d x

  c(t −t ) j V (t) ≤ V t + j e

≤ Me−α(tk −t0 )

and we can obtain from (23c) that   V2 t1+ = V2 (t1 )    t1 n n   = γi j e2j (s, x)d x ds t1

≤ Me−(α+c)(t j +1−t0 ) ec(t j −t0 )

(25)

Then, (23b) and (25) imply that   V1 t1+ ≤ ρ1 V1 (t1 ) ≤ ρ1 V (t1 ) ≤ ρ1 Me−(α+c)(t1−t0 ) ec(t1 −t0 )



≤ (ρ j + λτ )Me−(α+c)(t j −t0 ) ec(t j −t0 )

(24)

for all k ∈ N, where M = V (t0 )e(α+c)(t1−t0 ) . We use mathematical induction to show that claim (24) is true. For t ∈ [t0 , t1 ], we can get from (23a) that

t1 −τi j (t1 )

We shall show that (24) holds for k = j + 1. As discussed in (26)–(28), we can get  +  + V (t + j ) = V1 t j + V2 t j

then, for t ∈ (t j , t j +1 ]

V (t) ≤ Me−(α+c)(tk −t0 ) ec(t −t0 ) , t ∈ (tk−1 , tk ]

i=1 j =1 n n  

V (t) ≤ Me−(α+c)(t j −t0 ) ec(t −t0 ) , t ∈ (t j −1 , t j ].

(22)

V (t) ≤ Me−(α+c)(tk+1 −t0 ) ec(t −t0)

(32)

for t ∈ (tk , tk+1 ], and k ≥ i . When k = i , we obtain from (31) that t0 ≤ ti + s ≤ ti for s ∈ [−τ, 0], and ! sup e(s, ·)22 = sup {V1 (s)} (30)

s∈[ti −τ,ti ]

s∈[ti −τ,ti ]

≤ Me−α(ti −τ −t0 ) = Me−(α+c)(ti −t0 ) ec(ti −t0 ) eατ .

(33)

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Similar to the estimations of V1 (ti+ ) and V2 (ti+ ) in (26) and (27), we have       V ti+ = V1 ti+ + V2 ti+ ≤ ρi V1 (ti ) + V2 (ti ) ≤ ρi Me−α(ti −t0 ) + λτ eατ Me−(α+c)(ti −t0 ) ec(ti −t0 ) = (ρi + λτ eατ )Me−(α+c)(ti −t0 ) ec(ti −t0 ) ≤ Me−(α+c)(ti+1 −t0 ) ec(ti −t0 ) which, along with (23a), implies V (t) ≤ Me−(α+c)(ti+1 −t0 ) ec(t −t0 ) for t ∈ (ti , ti+1 ].

(34)

Similar to the proof of Theorem 5, we can conclude by mathematical induction that (32) holds for all k ≥ i . Then, for t ∈ (tk , tk+1 ] (k ≥ i ), we have V (t) ≤ Me−(α+c)(tk −t0 ) ec(t −t0) ≤ Me−α(tk −t0 ) ≤ Me−α(t −t0 ) .

(35)

Hence, from (31) and (35), we can see that V (t) ≤ Me−α(t −t0 ) , t ≥ t0 . It follows that: e(t, ·)22 ≤ V (t) ≤ Me−α(t −t0 ) , t ≥ t0 that is e(t, ·)2 ≤ M¯ √

sup

α

{e(s, ·)2 }e− 2 (t −t0 ) , t ≥ t0

s∈[t0 −τ,t0 ]

where M¯ = ( M/sups∈[t0 −τ,t0 ] {e(s, ·)2 }) > 1. Therefore, the proof is complete. Remark 18: The main difference between this proof and the proof of Theorem 5 lies in the estimation of (33). Without the condition on the relation between the delay size and the length of impulsive interval, the impulsive interval that time ti − τ belongs to cannot be exactly located. Therefore, different estimation techniques are used in (28) and (33). VI. C ONCLUSION The exponential synchronization of reaction–diffusion neural networks with time-varying delays has been studied. A pinning impulsive control algorithm proposed for dynamical networks without time-delay has been successfully generalized to control neural networks with both reaction–diffusion terms and time-varying delays. In order to overcome the difficulty of utilizing this pinning algorithm to control networks with time-delay, a Lyapunov–Krasovskii functional with two parts (a function part and a functional part) has been constructed. The function part is chosen as a quadratic form to carry over the pinning algorithm in [34] to neural networks with time-delay and handle the impulsive effects. Two sets of sufficient conditions have been derived to design suitable pinning impulsive controllers to synchronize the delayed reaction–diffusion neural networks with small and large delay size, respectively. Examples of impulsive synchronization of delay reaction–diffusion neural networks, pinning impulsive stabilization of delay reaction–diffusion neural networks, and

11

pinning impulsive stabilization of neural networks with timevarying delays have been discussed with numerical simulations to demonstrate our results. We conclude this paper with some possible research directions. First, according to our pinning algorithm, the neurons to be pinned may not be the same at different impulsive instants. For future research, it would be interesting and challenging to study sufficient conditions on network topology to guarantee the synchronization of neural networks by pinning the same neurons at the impulsive instants. Furthermore, when individual networks are connected by means of additional links among them, networks of networks arise (see [58], [59]). Then, it may be possible to extend our approach to synchronize this type of generalized networks. Finally, when processing the impulsive information in the controller, it is natural and practical to consider the time-delay effects in the pinning impulsive controller. Future work could be done on synchronization of networks via pinning impulsive controller with delay effects. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers and the Associate Editor for their constructive comments and valuable suggestions, which have improved the quality of this paper. R EFERENCES [1] M. T. Hagan, H. B. Demuth, and M. H. Beale, Neural Network Design. Beijing, China: China Machine Press, 2002. [2] Z. Zhou and C. Gao, Neural Network With Applications. Beijing, China: Tsinghua Univ. Press, 2004. [3] A. Arunkumar, R. Sakthivel, K. Mathiyalagan, and S. M. Anthoni, “Robust state estimation for discrete-time BAM neural networks with time-varying delay,” Neurocomputing, vol. 131, pp. 171–178, May 2014. [4] P. Balasubramaniam and L. J. Banu, “Robust state estimation for discrete-time genetic regulatory network with random delays,” Neurocomputing, vol. 122, pp. 349–369, Dec. 2013. [5] T. Li, A. Song, and S. Fei, “Synchronization control for arrays of coupled discrete-time delayed Cohen–Grossberg neural networks,” Neurocomputing, vol. 74, nos. 1–3, pp. 197–204, Dec. 2010. [6] S. Guo, L. Huang, and L. Wang, “Exponential stability of discretetime Hopfield neural networks,” Comput. Math. Appl., vol. 47, nos. 8–9, pp. 1249–1256, Apr./May 2004. [7] X. Yang, F. Li, Y. Long, and X. Cui, “Existence of periodic solution for discrete-time cellular neural networks with complex deviating arguments and impulses,” J. Franklin Inst., vol. 347, no. 2, pp. 559–566, Mar. 2010. [8] D. Hansel and H. Sompolinsky, “Synchronization and computation in a chaotic neural network,” Phys. Rev. Lett., vol. 68, no. 5, pp. 718–721, Feb. 1992. [9] H. D. Abarbanel et al., “Synchronisation in neural networks,” Phys.-Uspekhi, vol. 39, no. 4, pp. 337–362, Apr. 1996. [10] Q. Wang, G. Chen, and M. Perc, “Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling,” PLoS ONE, vol. 6, no. 1, p. e15851, Jan. 2011. [11] V. Milanovi´c and M. E. Zaghloul, “Synchronization of chaotic neural networks and applications to communications,” Int. J. Bifurcation Chaos, vol. 6, no. 12b, pp. 2571–2585, Dec. 1996. [12] V. E. Bondarenko, “Information processing, memories, and synchronization in chaotic neural network with the time delay,” Complexity, vol. 11, no. 2, pp. 39–52, Nov./Dec. 2005. [13] J. K. Ostanek, “Improving pin-fin heat transfer predictions using artificial neural networks,” J. Turbomach., vol. 136, no. 5, pp. 051010-1–051010-9, Sep. 2013. [14] F.-C. Chen and H.-K. Khalil, “Adaptive control of a class of nonlinear discrete-time systems using neural networks,” IEEE Trans. Autom. Control, vol. 40, no. 5, pp. 791–801, May 1995. [15] X. Liu and K. Zhang, “Impulsive control for stabilisation of discrete delay systems and synchronisation of discrete delay dynamical networks,” IET Control Theory Appl., vol. 8, no. 13, pp. 1185–1195, Sep. 2014.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 12

[16] C. Liu, C. Li, and S. Duan, “Stabilization of oscillating neural networks with time-delay by intermittent control,” Int. J. Control Autom. Syst., vol. 9, no. 6, pp. 1074–1079, Dec. 2011. [17] X. F. Wang and G. Chen, “Pinning control of scale-free dynamical networks,” Phys. A, Statist. Mech. Appl., vol. 310, nos. 3–4, pp. 521–531, Jul. 2002. [18] Y. Tang and W. K. Wong, “Distributed synchronization of coupled neural networks via randomly occurring control,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 3, pp. 435–447, Mar. 2013. [19] W.-H. Chen, X. Lu, and W. X. Zheng, “Impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 4, pp. 734–748, Apr. 2015. [20] P. Li, J. Cao, and Z. Wang, “Robust impulsive synchronization of coupled delayed neural networks with uncertainties,” Phys. A, Statist. Mech. Appl., vol. 373, pp. 261–272, Jan. 2007. [21] H. Zhang, T. Ma, G.-B. Huang, and Z. Wang, “Robust global exponential synchronization of uncertain chaotic delayed neural networks via dualstage impulsive control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 831–844, Jun. 2010. [22] C. Li, L. Chen, and K. Aihara, “Impulsive control of stochastic systems with applications in chaos control, chaos synchronization, and neural networks,” Chaos, vol. 18, no. 2, pp. 023132-1–023132-11, 2008. [23] X. Yang, C. Huang, and Q. Zhu, “Synchronization of switched neural networks with mixed delays via impulsive control,” Chaos, Solitons Fractals, vol. 44, no. 10, pp. 817–826, Oct. 2011. [24] X. Li, X. Wang, and G. Chen, “Pinning a complex dynamical network to its equilibrium,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 10, pp. 2074–2087, Oct. 2004. [25] C. Zheng and J. Cao, “Robust synchronization of coupled neural networks with mixed delays and uncertain parameters by intermittent pinning control,” Neurocomputing, vol. 141, pp. 153–159, Oct. 2014. [26] Q. Wang, Z. Duan, M. Perc, and G. Chen, “Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability,” Europhys. Lett. Assoc., vol. 83, no. 5, p. 50008, Sep. 2008. [27] Q. Wang, M. Perc, Z. Duan, and G. Chen, “Synchronization transitions on scale-free neuronal networks due to finite information transmission delays,” Phys. Rev. E, vol. 80, no. 2, p. 026206, Aug. 2009. [28] Q. Wang, M. Perc, Z. Duan, and G. Chen, “Impact of delays and rewiring on the dynamics of small-world neuronal networks with two types of coupling,” Phys. A, Statist. Mech. Appl., vol. 389, no. 16, pp. 3299–3306, Aug. 2010. [29] D. Guo, Q. Wang, and M. Perc, “Complex synchronous behavior in interneuronal networks with delayed inhibitory and fast electrical synapses,” Phys. Rev. E, vol. 85, no. 6, p. 061905, Jun. 2012. [30] Z. Wu and X. Fu, “Cluster synchronization in community networks with nonidentical nodes via edge-based adaptive pinning control,” J. Franklin Inst., vol. 351, no. 3, pp. 1372–1385, Mar. 2014. [31] Y. Liang, X. Wang, and J. Eustace, “Adaptive synchronization in complex networks with non-delay and variable delay couplings via pinning control,” Neurocomputing, vol. 123, pp. 292–298, Jan. 2014. [32] L. F. R. Turci and E. E. N. Macau, “Adaptive node-to-node pinning synchronization control of complex networks,” Chaos, vol. 22, no. 3, pp. 033151-1–033151-6, Jul. 2012. [33] X. Liu and T. Chen, “Synchronization of linearly coupled networks with delays via aperiodically intermittent pinning control,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 10, pp. 2396–2407, Oct. 2015. [34] J. Lu, J. Kurths, J. Cao, N. Mahdavi, and C. Huang, “Synchronization control for nonlinear stochastic dynamical networks: Pinning impulsive strategy,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 2, pp. 285–292, Feb. 2012. [35] N. Mahdavi, M. B. Menhaj, and J. Kurths, “Pinning impulsive synchronization of complex dynamical networks,” Int. J. Bifurcation Chaos, vol. 22, no. 10, pp. 1250239-1–1250239-14, Oct. 2012. [36] A. Hu and Z. Xu, “Pinning a complex dynamical network via impulsive control,” Phys. Lett. A, vol. 374, no. 2, pp. 186–190, Dec. 2009. [37] J. Zhou, Q. Wu, and L. Xiang, “Pinning complex delayed dynamical networks by a single impulsive controller,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 12, pp. 2882–2893, Dec. 2011. [38] H. Zhang, Z. Wang, and D. Liu, “A comprehensive review of stability analysis of continuous-time recurrent neural networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 7, pp. 1229–1262, Jul. 2014. [39] H. Zhang, Z. Wang, and D. Liu, “Global asymptotic stability of recurrent neural networks with multiple time-varying delays,” IEEE Trans. Neural Netw., vol. 19, no. 5, pp. 855–873, May 2008.

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

[40] Y. Tang, W. K. Wong, J.-A. Fang, and Q.-Y. Miao, “Pinning impulsive synchronization of stochastic delayed coupled networks,” Chin. Phys. B, vol. 20, no. 4, pp. 040513-1–040513-10, Apr. 2011. [41] J. Lu, Z. Wang, and J. Cao, “Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay,” Int. J. Bifurcation Chaos, vol. 22, no. 7, pp. 1250176-1–1250176-12, Jul. 2012. [42] X. Yang, J. Cao, and Z. Yang, “Synchronization of coupled reactiondiffusion neural networks with time-varying delays via pinningimpulsive controller,” SIAM J. Control Optim., vol. 51, no. 5, pp. 3486–3510, Sep. 2013. [43] J. Liang and J. Cao, “Global exponential stability of reaction–diffusion recurrent neural networks with time-varying delays,” Phys. Lett. A, vol. 314, nos. 5–6, pp. 434–442, Aug. 2003. [44] J. D. Murray, Mathematical Biology. New York, NY, USA: Springer-Verlag, 1989. [45] V. Volman, M. Perc, and M. Bazhenov, “Gap junctions and epileptic seizures—Two sides of the same coin?” PLoS ONE, vol. 6, no. 5, p. e20572, May 2011. [46] E. Hernández M and H. R. Henríquez, “Impulsive partial neutral differential equations,” Appl. Math. Lett., vol. 19, no. 3, pp. 215–222, Mar. 2006. [47] R. Sakthivel and J. Luo, “Asymptotic stability of impulsive stochastic partial differential equations with infinite delays,” J. Math. Anal. Appl., vol. 356, no. 1, pp. 1–6, Aug. 2009. [48] D. N. Chalishajar, “Controllability of impulsive partial neutral functional differential equation with infinite delay,” Int. J. Math. Anal., vol. 5, nos. 5–8, pp. 369–380, 2011. [49] D. Bainov, Z. Kamont, and E. Minchev, “Stability of solutions of firstorder impulsive partial differential equations,” Int. J. Theoretical Phys., vol. 33, no. 6, pp. 1359–1370, Jun. 1994. [50] C. Hu, H. Jiang, and Z. Teng, “Impulsive control and synchronization for delayed neural networks with reaction–diffusion terms,” IEEE Trans. Neural Netw., vol. 21, no. 1, pp. 67–81, Jan. 2010. [51] H. Akca, J. Benbourenane, and V. Covachev, “Impulsive delay reaction-diffusion Cohen–Grossberg neural networks with zero Dirichlet boundary conditions,” Middle East J. Sci. Res., vol. 13, pp. 15–24, 2013. [52] K. Gopalsamy and X.-Z. He, “Delay-independent stability in bidirectional associative memory networks,” IEEE Trans. Neural Netw., vol. 5, no. 6, pp. 998–1002, Nov. 1994. [53] J. Cao and L. Wang, “Exponential stability and periodic oscillatory solution in BAM networks with delays,” IEEE Trans. Neural Netw., vol. 13, no. 2, pp. 457–463, Mar. 2002. [54] J. Wu, Theory and Applications of Partial Functional Differential Equations. New York, NY, USA: Springer-Verlag, 1996. [55] C. Kuttler, Reaction-Diffusion Equations and Applications. Munich, Germany: Technische Univ. München, 2011. [56] J. Zhou, S. Xu, B. Zhang, Y. Zou, and H. Shen, “Robust exponential stability of uncertain stochastic neural networks with distributed delays and reaction-diffusions,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 9, pp. 1407–1416, Sep. 2012. [57] J. Liu, X. Liu, and W.-C. Xie, “Input-to-state stability of impulsive and switching hybrid systems with time-delay,” Automatica, vol. 47, no. 5, pp. 899–908, May 2011. [58] X. Sun, J. Lei, M. Perc, J. Kurths, and G. Chen, “Burst synchronization transitions in a neuronal network of subnetworks,” Chaos, vol. 21, no. 1, p. 016110, Mar. 2011. [59] D. Y. Kenett, M. Perc, and S. Boccaletti, “Networks of networks— An introduction,” Chaos, Solitons Fractals, vol. 80, pp. 1–6, Nov. 2015. Xinzhi Liu received the B.Sc. degree in mathematics from Shandong Normal University, Jinan, China, in 1982, and the M.Sc. and Ph.D. degrees in applied mathematics from the University of Texas at Arlington, Arlington, TX, USA, in 1987 and 1988, respectively. He was a Post-Doctoral Fellow with the University of Alberta, Edmonton, AB, Canada, from 1988 to 1990. He joined the Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada, as an Assistant Professor in 1990, where he became an Associate Professor and a Full Professor in 1994 and 1997, respectively. He has authored or co-authored over 300 research articles, two research monographs, and 20 edited books. His current research interests include systems analysis, stability theory, hybrid dynamical systems, impulsive control, complex dynamical networks, and communication security.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LIU et al.: PINNING IMPULSIVE SYNCHRONIZATION OF REACTION–DIFFUSION NEURAL NETWORKS WITH TIME-VARYING DELAYS

Kexue Zhang is currently pursuing the Ph.D. degree with the Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada. His current research interests include hybrid systems and control, differential equations on time scales, and their various applications on complex dynamical networks.

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Wei-Chau Xie received the B.Eng. degree in precision engineering from Shanghai Jiao Tong University, Shanghai, China, in 1984, and the M.A.Sc. and Ph.D. degrees in civil engineering from the University of Waterloo, Waterloo, ON, Canada, in 1987 and 1990, respectively. He was a Stress Analyst and Design Engineer with the Atomic Energy of Canada Ltd., Mississauga, ON, Canada, from 1990 to 1991. He joined the Department of Civil Engineering, University of Waterloo, as an Assistant Professor in 1992, where he became an Associate Professor and a Full Professor in 1997 and 2002, respectively. He has authored the books entitled Dynamic Stability of Structures (Cambridge University Press, 2006) and Differential Equations for Engineers (Cambridge University Press, 2010). His current research interests include dynamic stability of structures, structural dynamics and random vibration, nonlinear dynamics, stochastic mechanics, seismic analysis and design of engineering structures, and reliability and safety analysis of engineering systems. Prof. Xie won the Distinguished Teacher Award from the University of Waterloo in 2007.