Stochastic Synchronization of Reaction-Diffusion Neural Networks under General Impulsive Controller with Mixed Delays

Abstract

This paper investigates drive-response synchronization of a class of reaction-diffusion neural networks with time-varying discrete and distributed delays via general impulsive control method. Stochastic perturbations in the response system are also considered. The impulsive controller is assumed to be nonlinear and has multiple time-varying discrete and distributed delays. Compared with existing nondelayed impulsive controller, this general impulsive controller is more practical and essentially important since time delays are unavoidable in practical operation. Based on a novel impulsive differential inequality, the properties of random variables and Lyapunov functional method, sufficient conditions guaranteeing the global exponential synchronization in mean square are derived through strict mathematical proof. In our synchronization criteria, the distributed delays in both continuous equation and impulsive controller play important role. Finally, numerical simulations are given to show the effectiveness of the theoretical results.

1. Introduction

Since the pioneering work of Pecora and Carroll [1], the issue of synchronization and chaos control has been extensively studied [2] due to its potential engineering applications such as secure communication, biological systems, and information processing (see [3–10]). It is shown that neural networks exhibit chaotic behavior and provided that parameters and delays are appropriately chosen (see [11, 12]). Therefore, in recent years, synchronization and control of neural networks has been one of the hot research topics (see [13–15], etc.).

It is known that many pattern formation and wave propagation phenomena that appear in nature can be described by systems of coupled nonlinear differential equations, generally known as reaction-diffusion equations. These wave propagation phenomena are exhibited by systems belonging to very different scientific disciplines. The reaction-diffusion effects, therefore, cannot be neglected in both biological and man-made neural networks, especially when electrons are moving in noneven electromagnetic field [16]. So we must consider that the activations vary in space as well as in time, and in this case the model should be expressed by partial differential equations. There are some published papers concerning stability or synchronization of neural networks with reaction-diffusion terms and delays (see [17–25]). In [22], the authors investigated synchronization of reaction-diffusion neural networks with discrete and unbounded distributed delays. In [24], the authors investigated the boundedness and exponential stability for nonautonomous fuzzy cellular neural networks with unbounded distributed delays and reaction-diffusion terms. The authors of [25] studied exponential stability of reaction-diffusion Cohen-Grossberg neural networks with time-varying discrete delays and stochastic perturbations.

Time delays usually exist in neural networks due to finite speeds of switching of amplifiers and transmission of signals in hardware implementation. Ignoring them when studying dynamics of neural networks may lead to impractical results. Moreover, delays are commonly time varying and unknown [26]. Therefore, papers concerning synchronization or stability of neural networks with or without reaction-diffusion terms have considered various time delays. The authors in [11] studied exponential synchronization problem for coupled neural networks with constant time delay. In [27], both constant and time-varying discrete delays were considered for the synchronization of a class of delayed neural networks. In [28–31] several types of synchronization for neural networks with discrete and bounded distributed delays were studied. However, the delay kernel of the bounded distributed delays in [28–31] has to be 1 because the well-known Jensen’s inequality [32] is not applicable anymore if the delay kernel is not 1. In the case of unbounded distributed delay, it is necessary to consider the delay kernel, which satisfies the condition that its integral from zero to infinite is bounded [22, 33, 34]. But the authors in [22, 33, 34] had to use algebraic approach instead of matrix method to derive their main results which has more complex form and is more conservative than those obtained by matrix method. In [21], Wang and Zhang studied global asymptotic stability of reaction-diffusion Cohen-Grossberg neural networks with unbounded distributed delays by using a matrix decomposition method, and the obtained results were in terms of linear matrix inequality (LMI). But the Lyapunov functional and proof process used in [21] are relatively complex. Recently, authors in [35] studied global asymptotic synchronization in an array of coupled neural networks with probabilistic interval time-varying coupling delays and unbounded distributed delays; a novel integral inequality including the Jensen’s inequality as a special case was developed. By using the developed integral inequality, one can use LMI method to solve the problem of distributed delays with not-equal-to-1 delay kernel instead of the matrix decomposition method used in [21].

It should be noted that control method is of great significance to realize synchronization. Specially, in [29], the output feedback controller which has time-varying discrete and distributed delays was considered. On the other hand, impulsive control, as one of the most effective and economic control methods, has recently attracted great interests of many researchers in different fields, since it needs small control gains and acts only at discrete times, thus control cost and the amount of transmitted information can be reduced drastically (see [3, 9, 26, 36–40] and references cited therein). As for neural networks with reaction-diffusion terms, there are several results on synchronization via control. For instance, state feedback control technique is utilized in [20] to realize exponential synchronization of stochastic fuzzy cellular neural networks with time delay in the leakage term and reaction diffusion. In [22], global exponential stability and synchronization of delayed reaction-diffusion neural networks under hybrid state feedback control and impulsive control. However, to the authors knowledge, impulsive control has not been considered in the literature to realize synchronization of reaction-diffusion neural networks. Moreover, the impulsive controllers in [3, 9, 36–40] were nondelayed. Recently, in [41], global exponential stability of fuzzy reaction-diffusion cellular neural networks with time-varying discrete delays and unbounded distributed delays and impulsive perturbations were studied. Nevertheless, to the best of our knowledge, there are no results on stability or synchronization of reaction-diffusion neural networks with time-varying discrete delays and distributed delays under impulsive controller which has multiple time-varying delays, let alone impulsive controller with distributed delays. If these delays are considered in impulsive controller, the analysis methods used in [3, 9, 26, 36–40] are not applicable anymore. Considering the fact that both discrete delays and distributed delays are unavoidable in practice, it is of great importance to consider delayed impulsive control to synchronize-delayed neural neural networks.

Being motivated by the above discussions, this paper aims to study the global exponential derive-response synchronization of reaction-diffusion neural networks with multiple time-varying discrete delays and unbounded distributed delays via general impulsive control. The general impulsive controller is assumed to be nonlinear and has multiple time-varying discrete and distributed delays. Since time delays are always vary and unavoidable in practical operation, the general impulsive controller is essentially important and more practical than existing nondelayed impulsive controller. Stochastic perturbations in the response system are also considered. By using a novel integral inequality in [35], the problem of distributed delays with not-equal-to-1 delay kernel can be solved by matrix method. By utilizing the novel integral inequality, the properties of random variables and Lyapunov functional method, sufficient conditions guaranteeing the considered drive-response systems to realize synchronization in mean square are derived through strict mathematical proof. The proof process and the results are very simple. Finally, numerical simulations are given to show the effectiveness of the theoretical results.

The rest of this paper is organized as follows. In Section 2, the considered model of coupled reaction-diffusion neural networks with delays is presented. Some necessary assumptions, definitions, and lemmas are also given in this section. In Section 3, synchronization for the proposed model is studied. Then, in Section 4, simulation examples are presented to show the effectiveness of the theoretical results. Finally, Section 5 provides some conclusions.

Notations. In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. ℕ₊ denotes the set of positive integers. 𝐼_𝑛 denotes the 𝑛×𝑛 identity matrix. ℝ^𝑛 denotes the Euclidean space, and ℝ^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrix. 𝜆_max(𝐴) and 𝜆_min(𝐴) mean the largest and smallest eigenvalues of matrix 𝐴, respectively, ‖𝐴‖=𝜆_max(𝐴^𝑇𝐴), where 𝑇 denotes transposition. 𝐶=diag(𝑐₁,𝑐₂,…,𝑐_𝑛) means 𝐶 is a diagonal matrix. Moreover, let (𝑆,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) be a complete probability space with filtration {ℱ_𝑡}_t≥0 satisfying the usual conditions (i.e., the filtration contains all 𝑃-null sets and is right continuous). Denote by 𝐿^𝑃_ℱ0((−∞,0];ℝ^𝑛) the family of all ℱ₀-measurable 𝐶((−∞,0];ℝ^𝑛)-valued random variables 𝜉={𝜉(𝑠)∶𝑠≤0} such that sup_𝑠≤0𝔼(‖𝜉(𝑠)‖^𝑝)<∞, where 𝔼{⋅} stands for mathematical expectation operator with respect to the given probability measure 𝑃. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.

2. Preliminaries

Consider a delayed neural network with reaction-diffusion terms which is described as follows: 𝜕𝑦_𝑖(𝑡,𝑥)=𝜕𝑡^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝑟_𝑖𝑙𝜕𝑦_𝑖(𝑡,𝑥)𝜕𝑥_𝑙−𝑐_𝑖𝑦_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑦_𝑗(+𝑡,𝑥)_𝑛𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏₁(𝑡),𝑥+^𝑛_𝑗=1𝑑_𝑖𝑗^𝑡_−∞𝐾(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠,𝑥)d𝑠+𝐼_𝑖(𝑡), (2.1) or in a compact form 𝜕𝑦(𝑡,𝑥)=𝜕𝑡^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝑅_𝑙𝜕𝑦(𝑡,𝑥)𝜕𝑥_𝑙−𝐶𝑦(𝑡,𝑥)+𝐴𝑓𝑦(𝑡,𝑥)+𝐵𝑓𝑦𝑡−𝜏₁(𝑡),𝑥+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑓𝑦(𝑠,𝑥)d𝑠+𝐼(𝑡), (2.2) where 𝑖=1,2,…,𝑛, 𝑅_𝑙=diag(𝑟_1𝑙,𝑟_2𝑙,…,𝑟_𝑛𝑙), 𝑙=1,2,…,𝑚, 𝑟_𝑖𝑙≥0 means the transmission diffusion coefficient along the 𝑖th neuron; 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚)^𝑇∈Ω⊂ℝ^𝑚, Ω={𝑥∣|𝑥_𝑘|≤𝑧_𝑙,𝑙=1,2,…,𝑚}, and 𝑧_𝑙 is a constant. 𝑦(𝑡,𝑥)=(𝑦₁(𝑡,𝑥),𝑦₂(𝑡,𝑥),…,𝑦_𝑛(𝑡,𝑥))^𝑇∈ℝ^𝑛 represents the state vector of the network at time 𝑡 and in space 𝑥; 𝑛 corresponds to the number of neurons; 𝑓(𝑦(𝑡,𝑥))=(𝑓₁(𝑦₁(𝑡,𝑥)),…,𝑓_𝑛(𝑦_𝑛(𝑡,𝑥)))^𝑇 is the neuron activation function at time 𝑡 and in space 𝑥; 𝐶=diag(𝑐₁,𝑐₂,…,𝑐_𝑛) with 𝑐_𝑖>0; 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛 and 𝐷=(𝑑_𝑖𝑗)_𝑛×𝑛 are the connection weight matrix; 𝐼(𝑡)=(𝐼₁(𝑡),𝐼₂(𝑡),…,𝐼_𝑛(𝑡))^𝑇∈ℝ^𝑛 is an external input vector. The bounded function 𝜏₁(𝑡) represents unknown time-varying discrete delay of the system with 0<𝜏₁(𝑡)≤𝜏₁, in which 𝜏₁ is a constant, 𝐾(𝑡) is a nonnegative bounded scalar function defined on [0,+∞) describing the delay kernel of the unbounded distributed delay.

We suppose that system (2.2) has an unique continuous solution for any initial condition of the following form: 𝑦(𝑠,𝑥)=𝜙(𝑠,𝑥)∈𝐶([−∞,0]×Ω,ℝ^𝑛), where 𝐶([−∞,0]×Ω,ℝ^𝑛) denotes the Banach space of all continuous functions from [−∞,0]×Ω to ℝ^𝑛 with the norm 𝜙(𝑠,𝑥)=_Ω𝜙^𝑇(𝑠,𝑥)𝜙(𝑠,𝑥)d𝑥^1/2. (2.3) It is assumed that (2.2) satisfies the following Dirichlet boundary condition: 𝑦(𝑡,𝑥)=0,(𝑡,𝑥)∈[−∞,+∞]×𝜕Ω. (2.4)

Based on the concept of drive-response synchronization, we take (2.2) as the driver system and design the following controlled response system: d𝑢(𝑡,𝑥)=^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝑅_𝑙𝜕𝑢(𝑡,𝑥)𝜕𝑥_𝑙−𝐶𝑢(𝑡,𝑥)+𝐴𝑓(𝑢(𝑡,𝑥))+𝐵𝑓𝑢𝑡−𝜏₁(𝑡),𝑥+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑓(𝑢(𝑠,𝑥))d𝑠+𝐼(𝑡)+^+∞_𝑘=1𝛿𝑡−𝑡_𝑘𝑈_𝑘(𝑡,𝑥)d𝑡+𝜎(𝑡,𝑥)d𝜔(𝑡), (2.5) where 𝑒(𝑡,𝑥)=𝑢(𝑡,𝑥)−𝑦(𝑡,𝑥), 𝛿(𝑡) is the Dirac delta function, the time sequence {𝑡_𝑘} satisfies 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘−1<𝑡_𝑘<⋯, and lim_𝑘→+∞𝑡_𝑘=+∞. 𝑈_𝑘(𝑡,𝑥) is the control input. 𝜔(𝑡)=(𝜔₁(𝑡),…,𝜔_𝑛(𝑡))^𝑇∈ℝ^𝑛 is a 𝑛-dimensional Brown motion defined on (𝑆,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃). Here, the white noise d𝜔_𝑖(𝑡) is independent of d𝜔_𝑗(𝑡) for 𝑖≠𝑗, and 𝜎(𝑡,𝑥)≜𝜎(𝑡,𝑒(𝑡,𝑥),𝑒(𝑡−𝜏₂(𝑡),𝑥),∫^𝑡_{𝑡−𝜏3(𝑡)}𝑒(𝑠,𝑥)d𝑠) is the noise intensity function matrix, in which the bounded functions 𝜏₂(𝑡) and 𝜏₃(𝑡) represent unknown discrete and distributed delays of the system in the stochastic perturbation with 0<𝜏_𝑖(𝑡)≤𝜏_𝑖, 𝑖=2,3. This type of stochastic perturbation can be regarded as a result from the occurrence of random uncertainties during the process of transmission. We assume that the output signals of (2.2) can be received by (2.5).

In the present paper, the control input 𝑈_𝑘(𝑡,𝑥) is assumed to be the following form: 𝑈_𝑘(𝑡,𝑥)=ℎ_𝑘𝑒(𝑡,𝑥),𝑒𝑡−𝜂₁(𝑡),𝑥,…,𝑒𝑡−𝜂_𝑞(𝑡),𝑥,^𝑡_{𝑡−𝜂𝑞+1(𝑡)}𝑒(𝑠,𝑥)d𝑠−𝑒(𝑡,𝑥), (2.6) where 𝜂_𝑖(𝑡)  𝑖=1,2,…,𝑞+1 are unknown time-varying delays with 0<𝜂_𝑖(𝑡)≤𝜂_𝑖.

Integrating from 𝑡_𝑘−𝜀 to 𝑡_𝑘+𝜀 (𝜀>0 is a sufficient small constant) on both sides of system (2.5) and letting 𝜀→0⁺, one gets from the property of the Dirac delta function that 𝑢𝑡⁺_𝑘,𝑥−𝑢𝑡⁻_𝑘,𝑥=ℎ_𝑘𝑒𝑡_𝑘,𝑥,𝑒𝑡_𝑘−𝜂₁𝑡_𝑘,𝑥,…,𝑒𝑡_𝑘−𝜂_𝑞𝑡_𝑘,𝑥,^𝑡𝑘_{𝑡𝑘−𝜂𝑞+1𝑡𝑘}𝑒()𝑡𝑠,𝑥d𝑠−𝑒_𝑘,𝑥, (2.7) where 𝑢(𝑡⁺_𝑘,𝑥)=lim_𝑡→𝑡⁺_𝑘𝑢(𝑡,𝑥), 𝑢(𝑡⁻_𝑘,𝑥)=lim_𝑡→𝑡⁻_𝑘𝑢(𝑡,𝑥). In the following, we use ℎ_𝑘(𝑡_𝑘,𝑥) to denote ℎ_𝑘(𝑒(𝑡_𝑘,𝑥),𝑒(𝑡_𝑘−𝜂₁(𝑡_𝑘),𝑥),…,𝑒(𝑡_𝑘−𝜂_𝑞(𝑡_𝑘),𝑥),∫^𝑡𝑘_{𝑡𝑘−𝜂𝑞+1(𝑡𝑘)}𝑒(𝑠,𝑥)d𝑠) for short.

Remark 2.1. Equation (2.7) is actually the impulsive controller of response system (2.5). To the best of our knowledge, result on synchronization of reaction-diffusion neural networks under impulsive control is seldom. In [22], global exponential synchronization of delayed reaction-diffusion neural networks was studied. However, the control scheme in [22] is hybrid nondelayed state feedback control and nondelayed impulsive control, and the continuous state feedback controller is indispensable. Moreover, the impulsive controller (2.7) is very general, since it includes information of multiple time-varying discrete delays and time-varying distributed delays. Nevertheless, most of published paper concerning impulsive control including [3, 9, 26, 36–40] did not consider time delay in the impulsive function, let alone multiple time-varying discrete delays and time-varying distributed delays. It is known that both discrete delays and distributed delays are unavoidable and often time-varying in neural networks, hence considering impulsive control with time-varying discrete delays and time-varying distributed delays is essentially important. However, when time-varying discrete delays and time-varying distributed delays are considered in impulsive control, the results in [3, 9, 26, 36–40] is not applicable anymore.

From (2.7), the controlled system (2.5) can be rewritten as d𝑢(𝑡,𝑥)=^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝑅_𝑙𝜕𝑢(𝑡,𝑥)𝜕𝑥_𝑙−𝐶𝑢(𝑡,𝑥)+𝐴𝑓(𝑢(𝑡,𝑥))+𝐵𝑓𝑢𝑡−𝜏₁(𝑡),𝑥+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑓(𝑢(𝑠,𝑥))d𝑠+𝐼(𝑡)d𝑡+𝜎(𝑡,𝑥)d𝜔(𝑡),𝑡≠𝑡_𝑘,𝑢𝑡⁺_𝑘,𝑥=𝑢𝑡⁻_𝑘,𝑥+ℎ_𝑘𝑡_𝑘,𝑥−𝑒𝑡_𝑘,𝑥,𝑡=𝑡_𝑘,𝑘∈ℕ₊. (2.8)

To maintain consistency with above definitions, the initial value and the boundary condition of (2.8) are given in the following form: 𝑢(𝑠,𝑥)=𝜑(𝑠,𝑥)∈𝐶[−∞,0]×Ω,ℝ^𝑛,𝑢(𝑡,𝑥)=0,(𝑡,𝑥)∈[−∞,+∞]×𝜕Ω. (2.9)(2.10)

Throughout this paper, we always assume that 𝑢(𝑡,𝑥) is left continuous at 𝑡_𝑘, that is, 𝑢(𝑡⁻_𝑘,𝑥)=𝑢(𝑡_𝑘,𝑥). Then subtracting (2.2) from (2.8) gets the following error dynamical system: d𝑒(𝑡,𝑥)=^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝑅_𝑙𝜕𝑒(𝑡,𝑥)𝜕𝑥_𝑙−𝐶𝑒(𝑡,𝑥)+𝐴𝑔(𝑒(𝑡,𝑥))+𝐵𝑔𝑒𝑡−𝜏₁(𝑡),𝑥+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑔(𝑒(𝑠,𝑥))d𝑠d𝑡+𝜎(𝑡,𝑥)d𝜔(𝑡),𝑡≠𝑡_𝑘,𝑒𝑡⁺_𝑘,𝑥=ℎ_𝑘𝑡_𝑘,𝑥,𝑡=𝑡_𝑘,𝑘∈ℕ₊, (2.11) where 𝑔(𝑒(𝑡,𝑥))=𝑓(𝑢(𝑡,𝑥))−𝑓(𝑦(𝑡,𝑥)).

It is obvious that system (2.11) satisfies the Dirichlet boundary condition, and its initial condition is 𝑒(𝑠,𝑥)=𝜑(𝑠,𝑥)−𝜙(𝑠,𝑥)=𝜑(𝑠,𝑥)∈𝐶[−∞,0]×Ω,ℝ^𝑛,𝑖=1,2,…,𝑁. (2.12)

It is easy to see that the error system (2.11) admits a zero solution. Clearly, if the zero solution is globally exponentially stable, then the controlled system (2.8) is globally exponentially synchronized with system (2.2).

Throughout this paper, we assume that (H₁) for any 𝑢,𝑣∈ℝ, there exist constants 𝜇_𝑖  (𝑖=1,2,…,𝑛) such that |𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)|≤𝜇_𝑖|𝑢−𝑣|; (H₂) there is a positive constant 𝑘 such that ∫₀^+∞𝐾(𝑢)d𝑢=𝑘; (H₃) there exist positive constants 𝜌₁, 𝜌₂ and 𝜌₃ such that trace𝜎^𝑇(𝑡,𝑥)𝜎(𝑡,𝑥)≤𝜌₁𝑒^𝑇(𝑡,𝑥)𝑒(𝑡,𝑥)+𝜌₂𝑒^𝑇𝑡−𝜏₂(𝑡),𝑥𝑒𝑡−𝜏₂(𝑡),𝑥+𝜌₃^𝑡_{𝑡−𝜏3(𝑡)}𝑒^𝑇(𝑠,𝑥)𝑒(𝑠,𝑥)d𝑠; (2.13)(H₄) there exist nonnegative constants 𝛼_𝑘, 𝛽^𝑗_𝑘, 𝑗=1,2,…,𝑞+1 such that ℎ^𝑇_𝑘𝑡_𝑘,𝑥ℎ_𝑘𝑡_𝑘,𝑥≤𝛼_𝑘𝑒^𝑇𝑡_𝑘,𝑥𝑒𝑡_𝑘,𝑥+𝛽¹_𝑘𝑒^𝑇𝑡_𝑘−𝜂₁𝑡_𝑘,𝑥𝑒𝑡_𝑘−𝜂₁𝑡_𝑘,𝑥+⋯+𝛽^𝑞_𝑘𝑒^𝑇𝑡_𝑘−𝜂_𝑞𝑡_𝑘𝑒𝑡_𝑘−𝜂_𝑞𝑡_𝑘+𝛽_𝑘^{𝑞+1𝑡𝑘}_{𝑡𝑘−𝜂𝑞+1(𝑡𝑘)}𝑒^𝑇(𝑠,𝑥)𝑒(𝑠,𝑥)d𝑠. (2.14)

The following basic definitions and lemmas are needed in this paper to get main results.

Definition 2.2 (see [9]). The dynamical network (2.9) is said to be globally exponentially synchronized with system (2.2) in mean square if there exist constants 𝑀>1 and 𝜃>0 such that for any initial values (2.12) 𝔼‖𝑒(𝑡,𝑥)‖²≤max_𝑠≤0𝔼𝜑(𝑠,𝑥)²𝑀𝑒^−𝜃𝑡 (2.15) hold for 𝑡≥0.

Lemma 2.3 (see [17]). Let Ω be a cube |𝑥_𝑘|<𝑙_𝑘(𝑘=1,2,…,𝑚), and let 𝑣(𝑥) be a real-valued function belonging to 𝐶¹(Ω) which vanish on the boundary 𝜕Ω of Ω, that is, 𝑣(𝑥)|_𝜕Ω=0. Then _Ω𝑣²(𝑥)d𝑥≤𝑙²_𝑘Ω𝜕𝑣(𝑥)𝜕𝑥_𝑘²d𝑥. (2.16)

Lemma 2.4 (see [42]). If 𝑋,𝑌 are real matrices with appropriate dimensions, then there exist number 𝜀>0 such that 𝑋^𝑇𝑌+𝑌^𝑇𝑋≤𝜀𝑋^𝑇1𝑌+𝜀𝑌^𝑇𝑌. (2.17)

Lemma 2.5 2.5 (see [35]). Suppose that 𝐾(𝑡) is a nonnegative bounded scalar function defined on [0,+∞), and there exists a positive constant 𝑘 such that ∫₀^+∞𝐾(𝑢)d𝑢=𝑘. For any constant matrix 𝐷∈ℝ^𝑛×𝑛, 𝐷>0, and vector function 𝑥∶(−∞,𝑡]→ℝ^𝑛 for 𝑡≥0, one has 𝑘^𝑡_−∞𝐾(𝑡−𝑠)𝑥^𝑇(𝑠)𝐷𝑥(𝑠)d𝑠≥^𝑡_−∞𝐾(𝑡−𝑠)𝑥(𝑠)d𝑠^𝑇𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑥(𝑠)d𝑠 (2.18) provided the integrals are all well defined.

Remark 2.6. When there is a positive bounded function 𝑘(𝑡) such that ∫₀^𝜃(𝑡)𝐾(𝑢)d𝑢=𝑘(𝑡), where 0<𝜃(𝑡)≤𝜃, then the inequality (2.18) becomes the following from: 𝑘(𝑡)^𝑡_{𝑡−𝜃(𝑡)}𝐾(𝑡−𝑠)𝑥^𝑇(𝑠)𝐷𝑥(𝑠)d𝑠≥^𝑡_{(𝑡)𝑡−𝜃}𝐾(𝑡−𝑠)𝑥(𝑠)d𝑠^𝑇𝐷^𝑡_{𝑡−𝜃(𝑡)}𝐾(𝑡−𝑠)𝑥(𝑠)d𝑠. (2.19) Specially, when 𝐾(𝑡)=1 for 𝑡≥0, then 𝑘(𝑡)=𝜃(𝑡) in (2.19). In this case, the inequality (2.19) turns out to the well-known Jensen’s inequality [32]. In the literature, there were many results concerning stability or synchronization of neural networks with bounded distributed delays, for instance, see [28–31]. However, the delay kernels in [28–31] were all assumed to be 1. Obviously, the unbounded distributed delays in this paper include those [28–31] as a special case. It is easy to see from inequalities (2.18) and (2.19) that results of this paper are also applicable to neural networks with bounded distributed delays, no matter whether 𝐾(𝑡) is equal to 1 or not. In this sense, models in this paper are more general than those those in [28–31].

Lemma 2.7. Consider the following impulsive differential inequalities: 𝐷⁺𝑣(𝑡)≤𝑎𝑣(𝑡)+𝑏₁[𝑣(𝑡)]_𝜏1+𝑏₂[𝑣(𝑡)]_𝜏2+⋯+𝑏_𝑚[𝑣(𝑡)]_𝜏𝑚,𝑡≠𝑡_𝑘,𝑡≥𝑡₀,𝑣𝑡⁺_𝑘≤𝑝_𝑘𝑣𝑡⁻_𝑘+𝑞¹_𝑘𝑣𝑡⁻_𝑘𝜏1+𝑞²_𝑘𝑣𝑡⁻_𝑘𝜏2+⋯+𝑞^𝑚_𝑘𝑣𝑡⁻_𝑘𝜏𝑚,𝑘∈ℕ₊,𝑣(𝑡)=𝜙(𝑡),𝑡∈𝑡₀−𝜏,𝑡₀, (2.20) where 𝑎, 𝑏_𝑖, 𝑝_𝑘, 𝑞^𝑖_𝑘, and 𝜏_𝑖 are constants, 𝑏_𝑖≥0, 𝑝_𝑘≥0, 𝑞^𝑖_𝑘≥0, 𝜏_𝑖≥0, 𝑖=1,2,…,𝑚, 𝑣(𝑡)≥0, [𝑣(𝑡)]_𝜏𝑖=sup_{𝑡−𝜏𝑖≤𝑠≤𝑡}𝑣(𝑠), [𝑣(𝑡⁻_𝑘)]_𝜏𝑖=sup_{𝑡𝑘−𝜏𝑖(𝑡𝑘)≤𝑠<𝑡𝑘}𝑣(𝑠), 𝜙(𝑡) is continuous on [𝑡₀−𝜏,𝑡₀], and 𝑣(𝑡) is continuous except 𝑡_𝑘, 𝑘∈ℕ₊, where it has jump discontinuities. The consequence {𝑡_𝑘} satisfies 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<𝑡_𝑘+1<⋯, and lim_𝑘→+∞𝑡_𝑘=+∞. Suppose that 𝑝_𝑘+^𝑚_𝑖=1𝑞^𝑖_𝑘∑<1,𝑎+^𝑚𝑖=1𝑏_𝑖𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘+ln𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘𝑡_𝑘+1−𝑡_𝑘<0. (2.21)(2.22) Then there exist constants 𝛽>1 and 𝜆>0 such that 𝑣(𝑡)≤𝜙_𝜏𝛽𝑒^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀, (2.23) where ‖𝜙‖_𝜏=sup_{𝑡0−𝜏≤𝑠≤𝑡0}‖𝜙(𝑠)‖, 𝜏=max{𝜏_𝑖,𝑖=1,2,…,𝑚}.

The proof of Lemma 2.7 is given in the appendix, which is partly similarly to that of Lemma  1 in [43].

Remark 2.8. Lemma 2.7 actually provides stability criterion for impulsive differential equations with multiple time-varying delays, and impulsive function is related to the same multiple time-varying delays. Actually, Lemma 2.7 can be written in a more general form. Let 𝑏_𝑖=0, 𝑖=ℎ+1,…,𝑚, 𝑞^𝑗_𝑘=0, 𝑗=1,…,ℎ, 1<ℎ<𝑚−1, 𝜏_ℎ+1=𝜎₁,…,𝜏_{ℎ+𝑚−ℎ}=𝜎_𝑚−ℎ=𝜎_𝑟, 𝑞_𝑘^ℎ+1=𝑞¹_𝑘,…,𝑞_𝑘^{ℎ+𝑚−ℎ}=𝑞_𝑘^𝑚−ℎ=𝑞^𝑟_𝑘, the other parameters are the same as those in Lemma 2.7. Then one can get the following Lemma 2.9.

Lemma 2.9. Consider the following impulsive differential inequality: 𝐷⁺𝑣(𝑡)≤𝑎𝑣(𝑡)+𝑏₁[𝑣(𝑡)]_𝜏1+𝑏₂[𝑣(𝑡)]_𝜏2+⋯+𝑏_ℎ[𝑣(𝑡)]_𝜏ℎ,𝑡≠𝑡_𝑘,𝑡≥𝑡₀,𝑣𝑡⁺_𝑘≤𝑝_𝑘𝑣𝑡⁻_𝑘+𝑞¹_𝑘𝑣𝑡⁻_𝑘𝜎1+𝑞²_𝑘𝑣𝑡⁻_𝑘𝜎2+⋯+𝑞^𝑟_𝑘𝑣𝑡⁻_𝑘𝜎𝑟,𝑘∈ℕ₊,𝑣(𝑡)=𝜙(𝑡),𝑡∈𝑡₀−𝜏,𝑡₀. (2.24) Suppose that 𝑝_𝑘+^𝑟_𝑖=1𝑞^𝑖_𝑘∑<1,𝑎+^ℎ𝑖=1𝑏_𝑖𝑝_𝑘+∑^𝑟_𝑗=1𝑞^𝑗_𝑘+ln𝑝_𝑘+∑^𝑟_𝑗=1𝑞^𝑗_𝑘𝑡_𝑘+1−𝑡_𝑘<0. (2.25) Then there exist constants 𝛽>1 and 𝜆>0 such that 𝑣(𝑡)≤𝜙_𝜏𝛽𝑒^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀, (2.26) where ‖𝜙‖_𝜏=sup_{𝑡0−𝜏≤𝑠≤𝑡0}‖𝜙(𝑠)‖, 𝜏=max{𝜏_𝑖,𝜎_𝑗,𝑖=1,2,…,ℎ,𝑗=1,2,…,𝑟}.

Remark 2.10. Lemmas 2.7 and 2.9 are general. Specially, if 𝑞^𝑖_𝑘=0, 𝑖=1,2,…,𝑟, then the inequalities in (2.25) becomes 𝑝_𝑘∑<1,𝑎+^ℎ𝑖=1𝑏_𝑖𝑝_𝑘+ln𝑝_𝑘𝑡_𝑘+1−𝑡_𝑘<0. (2.27)

Take 𝑝=max{𝑝_𝑘,𝑘∈ℕ₊}, 𝜌=sup_𝑘∈ℕ+{𝑡_𝑘−𝑡_𝑘−1}. Then 𝑝<1 and ∑𝑎+^ℎ𝑖=1𝑏_𝑖𝑝_𝑘+ln𝑝_𝑘𝑡_𝑘+1−𝑡_𝑘∑≤𝑎+^ℎ𝑖=1𝑏_𝑖𝑝_𝑘+ln𝑝𝑡_𝑘+1−𝑡_𝑘∑≤𝑎+^ℎ𝑖=1𝑏_𝑖𝑝_𝑘+ln𝑝𝜌. (2.28) Therefore, ∑𝑝<1,𝑎+^ℎ𝑖=1𝑏_𝑖𝑝+ln𝑝𝜌<0 (2.29) implies (2.27); that is, the inequality (2.27) is less conservative than (2.29). In fact, the inequality (2.29) is exactly the inequalities (5) and (6) in Theorem 3.1 of [26]. (In the proof in Theorem  3.1 in [26], one can get from 𝑏_𝑖=(𝐿_𝑖/𝛼)𝜆_max(𝑃)/𝜆_min(𝑃) that (𝐿²_𝑖𝜆_max(𝑃))/(𝑏_𝑖𝜆_min(𝑃))=𝛼𝐿_𝑖𝜆_max(𝑃)/𝜆_min(𝑃). By comparing the coefficients in the first two inequalities in the proof of Theorem  3.1 in [26] with those in the inequalities (5) and (6) in [26], the conclusion can be easily achieved). Hence, Lemmas 2.7 and 2.9 improve and extend the Theorem  3.1 in [26]. In the literature, many results including those in [3, 9, 38, 40] were derived by using similar method used in [26]. Since Lemmas 2.7 and 2.9 include corresponding results in [26] as a special case and are less conservative than them, Lemmas 2.7 and 2.9 are very useful for stabilization and synchronization of impulsive control system.

3. Main Results

In this section, the global exponential synchronization criteria for system (2.8) and (2.2) are derived through strict mathematical reasoning.

Theorem 3.1. Suppose that conditions (H₁)–(H₄) hold. If there exists constants 𝜀₁>0, 𝜀₂>0 and 𝜀₃>0 such that 0<𝛼_𝑘+^𝑞_𝑖=1𝛽^𝑖_𝑘+𝛽_𝑘^𝑞+1𝜂_𝑞+1<1,𝑘∈ℕ₊,𝜀𝑎+²𝜇+𝜌₂+𝜀₃𝑘²𝜇+𝜌₃𝜏₃𝛼_𝑘+∑^𝑞_𝑖=1𝛽^𝑖_𝑘+𝛽_𝑘^𝑞+1𝜂_𝑞+1+ln𝛼_𝑘+∑^𝑞_𝑖=1𝛽^𝑖_𝑘+𝛽_𝑘^𝑞+1𝜂_𝑞+1𝑡_𝑘+1−𝑡_𝑘<0, (3.1)(3.2) where 𝑎=−2𝜆_min(𝑅+𝐶)+𝜀₁⁻¹‖𝐴‖²+𝜀₁𝜇+𝜀₂⁻¹‖𝐵‖²+𝜀₃⁻¹‖𝐷‖²+𝜌₁, 𝑅=diag(∑^𝑚_𝑙=1(𝑟_1𝑙/𝑧²_𝑙),∑^𝑚_𝑙=1(𝑟_2𝑙/𝑧²_𝑙),…,∑^𝑚_𝑙=1(𝑟_𝑛𝑙/𝑧²_𝑙)), 𝜇=max{𝜇²_𝑖,𝑖=1,2,…,𝑛}. Then, under the impulsive controller (2.7), the controlled system (2.8) is globally exponentially synchronized with system (2.2) in mean square.

Proof. Consider the following Lyapunov function: 𝑉(𝑡)=_Ω12𝑒^𝑇(𝑡,𝑥)𝑒(𝑡,𝑥)d𝑥. (3.3)
We use ℒ𝑉(𝑡) to denote the infinitesimal operator of 𝑉(𝑡) [44], which is defined as ℒ𝑉(𝑡)=lim_Δ→0+Δ⁻¹[𝔼{𝑉(𝑡+Δ)∣𝑡}−𝑉(𝑡)]. (3.4)
Based on the property of Wiener process [11], differentiating 𝑉(𝑡) along the solution of the error system (2.11) for 𝑡∈(𝑡_𝑘−1,𝑡_𝑘], 𝑘∈ℕ₊ obtains that d𝑉(𝑡)=ℒ𝑉(𝑡)d𝑡+𝑒(𝑡,𝑥)𝜎(𝑡,𝑥)d𝜔(𝑡), (3.5) where ℒ𝑉(𝑡)=_Ω𝑒^𝑇(𝑡,𝑥)^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝑅_𝑙𝜕𝑒(𝑡,𝑥)𝜕𝑥_𝑙−𝑒^𝑇(𝑡,𝑥)𝐶𝑒(𝑡,𝑥)+𝑒^𝑇(𝑡,𝑥)𝐴𝑔(𝑒(𝑡,𝑥))+𝑒^𝑇(𝑡,𝑥)𝐵𝑔𝑒𝑡−𝜏₁(𝑡),𝑥+𝑒^𝑇(𝑡,𝑥)𝐷^𝑡_−∞𝐾()𝑔(𝑒(+1𝑡−𝑠𝑠,𝑥))d𝑠2trace𝜎^𝑇(𝑡,𝑥)𝜎(𝑡,𝑥)d𝑥. (3.6)
From the Green’s formula and the Dirichlet boundary condition, we have (see [17–19]) _Ω𝑒^𝑇(𝑡,𝑥)^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝑅_𝑙𝜕𝑒(𝑡,𝑥)𝜕𝑥_𝑙d𝑥=_Ω^𝑛_𝑖=1𝑒_𝑖(𝑡,𝑥)^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝑟_𝑖𝑙𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑙=d𝑥_𝑛𝑖=1Ω𝑒_𝑖(𝑡,𝑥)∇𝑟_𝑖𝑙𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑙^𝑚_𝑙=1=d𝑥_𝑛𝑖=1Ω∇𝑒_𝑖(𝑡,𝑥)𝑟_𝑖𝑙𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑙^𝑚_𝑙=1−d𝑥_𝑛𝑗=1Ω𝑟_𝑖𝑙𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑙^𝑚_𝑙=1∇𝑒_𝑖()=𝑡,𝑥d𝑥_{𝑛𝑖=1𝜕Ω}𝑒_𝑖(𝑡,𝑥)𝑟_𝑖𝑙𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑙^𝑚_𝑙=1−d𝑥_{𝑛𝑖=1Ω𝑚𝑙=1}𝑟_𝑖𝑙𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑙²d𝑥=−^𝑛_𝑖=1Ω^𝑚_𝑙=1𝑟_𝑖𝑙𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑙²d𝑥, (3.7) in which ∇=(𝜕/𝜕𝑥₁,𝜕/𝜕𝑥₂,…,𝜕/𝜕𝑥_𝑚) is the gradient operator, and 𝑟_𝑖𝑙𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑙^𝑚_𝑙=1=𝑟_𝑖1𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥₁,𝑟_𝑖2𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥₂,…,𝑟_𝑖𝑚𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑚^𝑇. (3.8)
In view of Lemma 2.3, it is derived that −^𝑛_𝑖=1Ω^𝑚_𝑙=1𝑟_𝑖𝑙𝜕𝑒_𝑖(𝑡,𝑥)𝜕𝑥_𝑙²d𝑥≤−^𝑛_𝑖=1Ω^𝑚_𝑙=1𝑟_𝑖𝑙𝑧²_𝑙𝑒²_𝑖(𝑡,𝑥)d𝑥=−_Ω𝑒^𝑇(𝑡,𝑥)𝑅𝑒(𝑡,𝑥)d𝑥. (3.9)
For any positive constants 𝜀₁, 𝜀₂, and 𝜀₃, it follows from (H₁) and Lemma 2.4 that 𝑒^𝑇()(𝑒(≤1𝑡,𝑥𝐴𝑔𝑡,𝑥))2𝜀₁⁻¹𝑒^𝑇(𝑡,𝑥)𝐴𝐴^𝑇𝑒()+1𝑡,𝑥2𝜀₁𝑔^𝑇(𝑒(𝑔(𝑒(≤1𝑡,𝑥))𝑡,𝑥))2𝜀₁⁻¹‖𝐴‖²+𝜀₁𝜇𝑒^𝑇()𝑒(),𝑒𝑡,𝑥𝑡,𝑥_𝑇(𝑡,𝑥)𝐵𝑔𝑒𝑡−𝜏₁(𝑡)≤1,𝑥2𝜀₂⁻¹‖𝐵‖²𝑒^𝑇()𝑒()+1𝑡,𝑥𝑡,𝑥2𝜀₂𝜇𝑒^𝑇𝑡−𝜏₁(𝑡),𝑥𝑒𝑡−𝜏₁(𝑡),𝑒,𝑥_𝑇(𝑡,𝑥)𝐷^𝑡_−∞𝐾()𝑔(𝑒(1𝑡−𝑠𝑠,𝑥))d𝑠≤2𝜀₃^𝑡_−∞𝐾(𝑡−𝑠)𝑔(𝑒(𝑠,𝑥))d𝑠^𝑇×^𝑡_−∞𝐾()𝑔(𝑒(1𝑡−𝑠𝑠,𝑥))d𝑠+2𝜀₃⁻¹‖𝐷‖²𝑒^𝑇(𝑡,x)𝑒(𝑡,𝑥). (3.10)(3.11)(3.12)
By using condition (H₂) and Lemma 2.5, one obtains from (3.12) that𝑒^𝑇(𝑡,𝑥)𝐷^𝑡_−∞𝐾()𝑔(𝑒(1𝑡−𝑠𝑠,𝑥))d𝑠≤2𝜀₃𝑘^𝑡_−∞𝐾(𝑡−𝑠)𝑔^𝑇(𝑒(𝑔(𝑒(+1𝑠,𝑥))𝑠,𝑥))d𝑠2𝜀₃⁻¹‖𝐷‖²𝑒^𝑇()𝑒()≤1𝑡,𝑥𝑡,𝑥2𝜀₃𝑘𝜇^𝑡_−∞𝐾(𝑡−𝑠)𝑒^𝑇()𝑒()+1𝑠,𝑥𝑠,𝑥d𝑠2𝜀₃⁻¹‖𝐷‖²𝑒^𝑇(𝑡,𝑥)𝑒(𝑡,𝑥). (3.13)
Considering condition (H₃) and substituting (3.9)–(3.11) and (3.13) into (3.6) derive that ℒ𝑉(𝑡)≤_Ω𝑎2𝑒^𝑇()𝑒()+1𝑡,𝑥𝑡,𝑥2𝜀₂𝜇𝑒^𝑇𝑡−𝜏₁(𝑡),𝑥𝑒𝑡−𝜏₁(𝑡)+𝜌,𝑥₂2𝑒^𝑇𝑡−𝜏₂(𝑡),𝑥𝑒𝑡−𝜏₂(𝑡)+1,𝑥2𝜀₃𝑘𝜇^𝑡_−∞𝐾(𝑡−𝑠)𝑒^𝑇()𝑒()+𝜌𝑠,𝑥𝑠,𝑥d𝑠₃2^𝑡_{𝑡−𝜏3(𝑡)}𝑒^𝑇()𝑒()(𝑡)𝑠,𝑥𝑠,𝑥𝑑𝑠d𝑥=𝑎𝑉+𝜀₂𝜇𝑉𝑡−𝜏₁(𝑡)+𝜌₂𝑉𝑡−𝜏₂(𝑡)+𝜀₃𝑘𝜇^𝑡_−∞𝐾(𝑡−𝑠)𝑉(𝑠)d𝑠+𝜌₃^𝑡_{𝑡−𝜏3(𝑡)}𝑉(s)d𝑠. (3.14)
Taking mathematical expectations on both sides of (3.5), it can be derived from inequations (3.14) and (H₂) that d𝔼{𝑉(𝑡)}{𝑉(𝑡d𝑡≤𝑎𝔼)}+𝜀₂𝜇[𝔼{𝑉(𝑡)}]_𝜏1+𝜌₂[𝔼{𝑉(𝑠)}]_𝜏2+𝜀₃𝑘²𝜇[𝔼{𝑉(𝑠)}]_−∞+𝜌₃𝜏₃[𝔼{𝑉(𝑠)}]_𝜏3,𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈ℕ₊, (3.15) where [𝔼{𝑉(𝑠)}]_−∞=max_𝑠≤𝑡𝔼{𝑉(𝑠)}.
On the other hand, it is obtained from (H₄) and the second equation of (2.11) that 𝑉𝑡⁺_𝑘=_Ω12𝑒^𝑇𝑡⁺_𝑘,𝑥𝑒𝑡⁺_𝑘,𝑥d𝑥=_Ω12ℎ^𝑇_𝑘𝑡_𝑘,𝑥ℎ_𝑘𝑡_𝑘,𝑥d𝑥≤𝛼_𝑘𝑉𝑡_𝑘+𝛽¹_𝑘𝑉𝑡_𝑘−𝜂₁𝑡_𝑘+⋯+𝛽^𝑞_𝑘𝑉𝑡_𝑘−𝜂_𝑞𝑡_𝑘+𝛽_𝑘^𝑞+1𝑡_{𝑡−𝜂𝑞+1(𝑡)}𝑉(𝑠)d𝑠, (3.16) which means that 𝔼𝑉𝑡⁺_𝑘≤𝛼_𝑘𝔼𝑉𝑡_𝑘+𝛽¹_𝑘𝔼𝑉𝑡_𝑘𝜂1+⋯+𝛽^𝑞_𝑘𝔼𝑉𝑡_𝑘𝜂𝑞+𝛽_𝑘^𝑞+1𝜂_𝑞+1[𝑉(𝑠)]_𝜂𝑞+1. (3.17)
By virtue of Lemma 2.7, if the inequalities (3.1) and (3.2) hold, then it follows from (3.15) and (3.17) that there exist constants 𝑀>1 and 𝜃>0 such that 𝔼{𝑉(𝑡)}≤max_𝑠≤0𝔼𝜑₁(𝑠,𝑥)²𝑀𝑒^−𝜃𝑡,𝑡≥0. (3.18)
By Definition 2.2, the controlled system (2.8) is globally exponentially synchronized with system (2.2) in mean square. This completes the proof.

Note that there are three uncertain positive constants 𝜀₁, 𝜀₂, and 𝜀₃. Not making a good choice of the three constants may lead to the conservativeness of Theorem 3.1 in practical application. In order to hit off this fault, our next aim is to determine the constants 𝜀₁, 𝜀₂, and 𝜀₃ such that the conservativeness of Theorem 3.1 can be reduced as much as possible. We present the following Theorem 3.2.

Theorem 3.2. Suppose that conditions (H₁)–(H₄). Then, under the impulsive controller (2.7), the controlled system (2.8) is globally exponentially synchronized with system (2.2) in mean square if the following inequalities hold 0<𝑏_𝑘<1,𝑘∈ℕ₊,𝜉_𝑘=−𝜆_min𝑅+𝐶+‖𝐴‖√𝜇+‖𝐵‖𝜇𝑏_𝑘+𝑘‖𝐷‖𝜇𝑏_𝑘+12𝜌₁+𝜌₂𝑏_𝑘+𝜌₃𝜏₃𝑏_𝑘+ln𝑏_𝑘𝑡_𝑘+1−𝑡_𝑘<0,𝑘∈ℕ₊, (3.19)(3.20) where 𝑏_𝑘=𝛼_𝑘+∑^𝑞_𝑖=1𝛽^𝑖_𝑘+𝛽_𝑘^𝑞+1𝜂_𝑞+1, the other parameters are defined as those in Theorem 3.1.

Proof. Define the function 𝐻(𝜀₁,𝜀₂,𝜀₃) with positive variables 𝜀₁, 𝜀₂, and 𝜀₃ as follows: 𝐻𝜀₁,𝜀₂,𝜀₃𝜀=𝑎+²𝜇+𝜌₂+𝜀₃𝑘²𝜇+𝜌₃𝜏₃𝛼_𝑘+∑^𝑞_𝑖=1𝛽^𝑖_𝑘+𝛽_𝑘^𝑞+1𝜂_𝑞+1+ln𝛼_𝑘+∑^𝑞_𝑖=1𝛽^𝑖_𝑘+𝛽_𝑘^𝑞+1𝜂_𝑞+1𝑡_𝑘+1−𝑡_𝑘. (3.21)
In order that the result of Theorem 3.1 is less conservative, we only need to find out three constants 𝜀⁰₁, 𝜀⁰₂, and 𝜀⁰₃ such that the inequality (3.2) is less conservative. To achieve this goal, we will find a point (𝜀⁰₁,𝜀⁰₂,𝜀⁰₃) such that 𝐻(𝜀⁰₁,𝜀⁰₂,𝜀⁰₃) takes the minimum value and 𝐻(𝜀⁰₁,𝜀⁰₂,𝜀⁰₃)<0. By simple computation, one derives that 𝜕𝐻/𝜕𝜀₁=𝜇−(‖𝐴‖²/𝜀²₁), 𝜕𝐻/𝜕𝜀₂=(𝜇/𝑏_𝑘)−(‖𝐵‖²/𝜀²₂), 𝜕𝐻/𝜕𝜀₃=(𝑘²𝜇/𝑏_𝑘)−(‖𝐷‖²/𝜀²₃). Let 𝜕𝐻/𝜕𝜀₁=𝜕𝐻/𝜕𝜀₂=𝜕𝐻/𝜕𝜀₃=0, one gets (𝜀⁰₁,𝜀⁰₂,𝜀⁰₃)=(‖𝐴‖/√𝜇,‖𝐵‖𝑏_𝑘/𝜇,‖𝐷‖/𝑘𝑏_𝑘/𝜇). It is obvious that the Hesse matrix of 𝐻(𝜀₁,𝜀₂,𝜀₃) at (𝜀⁰₁,𝜀⁰₂,𝜀⁰₃) is positive definite. Hence, 𝐻(𝜀₁,𝜀₂,𝜀₃) takes the minimum value at (𝜀⁰₁,𝜀⁰₂,𝜀⁰₃) according to the extreme value theory of multivariate function. Taking 𝐻(𝜀⁰₁,𝜀⁰₂,𝜀⁰₃)<0 arrives at the condition (3.20). This completes the proof.

Remark 3.3. Theorems 3.1 and 3.2 are not dependent on discrete delays of both continuous equation and impulsive controller, which is consistent with results of [3, 9, 26, 38], though they did not consider delays in impulses. It should be noted that the inequalities in Theorems 3.1 and 3.2 are related to 𝑘², 𝜏₃, and 𝜂_𝑞+1, which mean that distributed delays in both continuous equation and impulsive controller have important effects on synchronization criteria in our results. This new discovery is completely different from existing results including those in [3, 9, 26, 37–40]. As was pointed out in Remark 2.8, results in [3, 9, 38, 40] were derived by using similar method used in [26], hence results of this paper improve those in [3, 9, 26, 38, 40] even when 𝐷=0, 𝜎(𝑡,𝑥)=𝜎(𝑡,𝑒(𝑡,𝑥),𝑒(𝑡−𝜏₂(𝑡),𝑥)) and ℎ_𝑘(𝑡_𝑘,𝑥)=ℎ(𝑒(𝑡_𝑘,𝑥)) in (2.11). To sum up, results of this paper are new and improve and extend most of known corresponding ones.

Remark 3.4. Lemma 2.5 is utilized in (3.13), which makes the proof process more simple than those in [21, 22, 34]. In [21], matrix decomposition method was used to deal with not-equal-to-1 delay kernel, hence the Lyapunov functional and proof process are relatively complex. Authors in [22, 34] had to utilize algebraic approach instead of matrix method to derive their main results. It is well known that results derived from algebraic approach have more complex form and is more conservative than those obtained by matrix method. Therefore, results of this paper improve those in [21, 22, 34] to some extent.

Remark 3.5. Stochastic perturbations are unavoidable in real applications of neural networks. In this paper, we synchronize a class of reaction-diffusion neural networks with stochastic perturbations via impulsive control. Although there were several results on stability of reaction-diffusion neural with stochastic perturbations [45, 46], seldom published papers considered synchronization of this kind of neural networks under impulsive control. Moreover, the stochastic perturbations of this paper are more general than those in [45, 46], since they include information of distributed delays.

4. Examples and Simulations

As applications of the the theoretical results derived above, in this section, we give numerical simulations to demonstrate that our synchronization criteria are effective.

Consider the following reaction-diffusion neural network with both discrete and unbounded distributed delays 𝜕𝑦(𝑡,𝑥)=𝜕𝜕𝑡𝑅()𝜕𝑥𝜕𝑦𝑡,𝑥()𝑦()𝑦𝜕𝑥−𝐶𝑦𝑡,𝑥+𝐴𝑓𝑡,𝑥+𝐵𝑓𝑡−𝜏₁(𝑡),𝑥+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑓𝑦(𝑠,𝑥)d𝑠+𝐼(𝑡), (4.1) where 𝑦(𝑡,𝑥)=(𝑦₁(𝑡,𝑥),𝑦₂(𝑡,𝑥))^𝑇, 𝑥∈Ω=[−2,2],𝑓(𝑦(𝑡,𝑥))=(tanh(𝑥₁(𝑡,𝑥)),tanh(𝑥₂(𝑡,𝑥)))^𝑇, 𝜏₁(𝑡)=1, 𝐾(𝑡)=𝑒^−0.5𝑡, 𝑅=diag(0.1,0.1), 𝐼(𝑡)=(1,1.2)^𝑇, 𝐶=1.2001,𝐴=3−0.345,𝐵=−1.40.10.3−8,𝐷=−1.20.1−2.8−1. (4.2)

Take the boundary condition of (4.1) as 𝑦(𝑡,𝑥)=0, (𝑡,𝑥)∈(−∞,+∞)×𝜕Ω. In the case that initial condition is chosen as 𝑦(𝑠,𝑥)=(0.4,0.6)^𝑇, (𝑠,𝑥)∈[−3,0]×Ω and 𝑦(𝑠,𝑥)=0, (𝑠,𝑥)∈(−∞,−3)×Ω, the chaotic-like trajectory of (4.1) is shown in Figures 1, 2, and 3. Taking 𝑅=0, then we get the chaotic-like trajectory of (4.1) without reaction-diffusion terms shown in Figure 4.

Figure 1 

Chaotic behavior of the state 𝑦1(𝑡,𝑥) in system (4.1).

Figure 2 

Chaotic behavior of the state 𝑦2(𝑡,𝑥) in system (4.1).

Figure 3 

Chaotic behavior of system (4.1).

Figure 4 

Chaotic behavior of system (4.1) with 𝑅=0.

Let system (4.1) be the driver network, we design a response system as ()=𝜕d𝑢𝑡,𝑥𝑅()𝜕𝑥𝜕𝑢𝑡,𝑥()(𝑢(𝑢𝜕𝑥−𝐶𝑢𝑡,𝑥+𝐴𝑓𝑡,𝑥))+𝐵𝑓𝑡−𝜏₁(𝑡),𝑥+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑓(𝑢(𝑠,𝑥))d𝑠+𝐼(𝑡)d𝑡+𝜎(𝑡,𝑥)d𝜔(𝑡),𝑡≠𝑡_𝑘,𝑢𝑡⁺_𝑘,𝑥=𝑢𝑡⁻_𝑘,𝑥+ℎ_𝑘𝑡_𝑘,𝑥−𝑒𝑡_𝑘,𝑥,𝑡=𝑡_𝑘,𝑘∈ℕ₊, (4.3) where 𝑒(𝑡,𝑥)=𝑢(𝑡,𝑥)−𝑦(𝑡,𝑥), ℎ_𝑘(𝑡_𝑘,𝑥)=𝑎𝑒(𝑡_𝑘,𝑥)+𝑏𝑒(𝑡_𝑘−0.5|sin𝑡_𝑘|,𝑥)+𝑐∫^𝑡_𝑡−0.5𝑒(𝑠,𝑥)d𝑠 with positive constants 𝑎, 𝑏, and 𝑐, the noise intensity function matrix is 𝜎()𝑒𝑡,𝑥=0.1¹(𝑡,𝑥)𝑒₂()∫𝑡−1,𝑥_{𝑡𝑡−0.3}𝑒₁(𝑠,𝑥)d𝑠𝑒₂(𝑡,𝑥). (4.4) By Jensen’s inequality (which is a special case of inequality (2.19)), one has ^𝑡_𝑡−0.3𝑒₁(𝑠,𝑥)d𝑠²≤0.3^𝑡_𝑡−0.3𝑒₁(𝑠,𝑥)²d𝑠≤0.3^𝑡_𝑡−0.3𝑒^𝑇(𝑠,𝑥)𝑒(𝑠,𝑥)d𝑠. (4.5) From (4.5) one gets trace𝜎^𝑇(𝑡,𝑥)𝜎(𝑡,𝑥)≤0.01𝑒^𝑇(𝑡,𝑥)𝑒(𝑡,𝑥)+0.01𝑒^𝑇(𝑡−1,𝑥)𝑒(𝑡−1,𝑥)+0.003^𝑡_𝑡−0.3𝑒^𝑇(𝑠,𝑥)𝑒(𝑠,𝑥)d𝑠. (4.6) Similarly, by using Jensen’s inequality one derives that ℎ^𝑇_𝑘𝑡_𝑘,𝑥ℎ_𝑘𝑡_𝑘,𝑥≤(𝑎+𝑏+𝑐)𝑎𝑒^𝑇𝑡_𝑘,𝑥𝑒𝑡_𝑘,𝑥+𝑏𝑒^𝑇𝑡_𝑘−0.5sin𝑡_𝑘,𝑥𝑒𝑡_𝑘−0.5sin𝑡_𝑘,𝑥+𝑐^𝑡_𝑡−0.5𝑒^𝑇(𝑠,𝑥)𝑒(𝑠,𝑥)d𝑠. (4.7)

Obviously, 𝜇₁=𝜇₂=1, 𝑘=2, 𝜌₁=𝜌₂=0.01, 𝜌₃=0.003, 𝛼_𝑘=𝑎(𝑎+𝑏+𝑐), 𝛽¹_𝑘=𝑏(𝑎+𝑏+𝑐), 𝛽²_𝑘=𝑐(𝑎+𝑏+𝑐), 𝑞=1, 𝜏₁=𝜏₂=1, 𝜏₃=0.3, and 𝜂₁=𝜂₂=0.5. Therefore, (H₁)–(H₄) are satisfied. Choose 𝑎=0.2,𝑏=0.15,𝑐=0.2, and 𝑡_𝑘−𝑡_𝑘−1=0.02. Then the inequalities (3.19) and (3.20) are satisfied with 𝑏_𝑘=0.2475, 𝜉_𝑘=−0.4086<0, respectively. According to Theorem 3.2, the controlled system (4.3) is globally exponentially synchronized with system (4.1) in mean square. Figure 5 presents the dynamical behavior of synchronization errors 𝑒₁(𝑡,𝑥) and 𝑒₂(𝑡,𝑥), which close to zero quickly as time increases.

(a)

(b)

Figure 5 

Dynamical behavior of synchronization errors 𝑒1(𝑡,𝑥) (a) and 𝑒2(𝑡,𝑥) (b).

5. Conclusion

Delays are unavoidable in practical systems, and they are always unknown and time-varying. This paper studies stochastic synchronization of reaction-diffusion neural networks with both time-varying discrete and distributed delays via delayed impulsive control. The impulsive controller has multiple time-varying discrete and distributed delays which is very general. Based on a novel integral inequality, the problem of distributed delays with not-equal-to-1 delay kernel is well handled with matrix method. Sufficient synchronization criteria are given to guarantee the global exponential synchronization in mean square of the considered system. The function extreme value theorem is utilized to get a less conservative result. It is discovered that, in our synchronization criteria, the distributed delays in both continuous equation and impulsive controller have important effects. At last, numerical simulations show the validity of the obtained criteria.

Appendix

Proof of Lemma 2.7. Without loss of generality, we assume that 𝜏=𝜏₁≥𝜏₂≥⋯≥𝜏_𝑚. Consider the following scalar function: 𝑔_𝑘(𝜆)∑=2𝜆+𝑎+^𝑚𝑖=1𝑏_𝑖𝑒^𝜆𝜏𝑖𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘𝑒^𝜆𝜏𝑗+ln𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘𝑒^𝜆𝜏𝑗𝑡_𝑘+1−𝑡_𝑘. (A.1)
It follows from inequality (2.22) that 𝑔_𝑘(0)=𝑎+(∑^𝑚_𝑖=1𝑏_𝑖)/(𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘)+(ln(𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘))/(𝑡_𝑘+1−𝑡_𝑘)<0. Since 𝑔^′_𝑘(𝜆)=2+∑^𝑚_𝑖=1(𝑝_𝑘𝑏_𝑖𝜆𝑒^𝜆𝜏𝑖/(𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘𝑒^𝜆𝜏𝑗)²)+(𝜆∑^𝑚_𝑗=1𝑞^𝑗_𝑘𝑒^𝜆𝜏𝑗)/((𝑡_𝑘+1−𝑡_𝑘)(𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘𝑒^𝜆𝜏𝑗))>0 for 𝜆>0 and 𝑔_𝑘(𝜆) is continuous on (0,+∞), there exists a positive constant 𝜆 such that 𝑔_𝑘(𝜆)<0 and 𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘𝑒^𝜆𝜏𝑗≤1 for all 𝑘∈ℕ₊.
Let 𝛾=sup_𝑘∈ℕ+{1/(𝑝_𝑘+∑^𝑚_𝑗=1𝑞^𝑗_𝑘𝑒^𝜆𝜏𝑗)}≥1. Then we can select a constant 𝜎>0 such that for all 𝑘∈ℕ₊, 𝑎+^𝑚_𝑖=1𝛾𝑏_𝑖𝑒^𝜆𝜏𝑖()𝑡≤𝜎−𝜆,𝜎+𝜆_𝑘+1−𝑡_𝑘<−ln𝑝_𝑘+^𝑚_𝑗=1𝑞^𝑗_𝑘𝑒^𝜆𝜏𝑗≤ln𝛾. (A.2)(A.3)
From (A.3), we can choose 𝛽=𝛽₁≥𝛽₂≥⋯≥𝛽_𝑚>1 such that 1<𝑒^{(𝜎+𝜆)(𝑡1−𝑡0)}≤𝛽_𝑖≤𝛾𝑒^𝜆𝜏𝑖. (A.4)
It follows from the above inequality that 𝜙_𝜏<𝜙_𝜏𝑒^{𝜎(𝑡1−𝑡0)}≤𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡1−𝑡0)}. (A.5)
Next we will prove that 𝑣(𝑡)≤𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡−𝑡0)},𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈ℕ₊. (A.6)
We use mathematical induction to prove that (A.6) holds. Firstly, we prove that (A.6) holds for 𝑘=1. To do this, we only need to prove that 𝑣(𝑡)≤𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡1−𝑡0)},𝑡∈𝑡₀,𝑡₁. (A.7)
If the inequality (A.7) is not true, then there exists some 𝑡∈(𝑡₀,𝑡₁) such that 𝑣𝑡>𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡1−𝑡0)}≥𝜙_𝜏𝛽₂𝑒^{−𝜆(𝑡1−𝑡0)}≥⋯≥𝜙_𝜏𝛽_𝑚𝑒^{−𝜆(𝑡1−𝑡0)}≥𝜙_𝜏𝑒^{𝜎(𝑡1−𝑡0)}>𝜙_𝜏≥𝑣𝑡₀+𝑠,𝑠∈[−𝜏,0], (A.8) which implies that there exists 𝑡_𝑖̃∈(𝑡₀,𝑡) such that 𝑡_𝑚̃≤𝑡̃_𝑚−1≤⋯≤𝑡₁̃ and 𝑣𝑡_𝑖̃=𝜙_𝜏𝛽_𝑖𝑒^{−𝜆(𝑡1−𝑡0)},𝑣(𝑡)≤𝑣𝑡_𝑖̃,𝑡∈𝑡₀−𝜏,𝑡_𝑖̃, (A.9) and there exists 𝑡̂∈[𝑡₀,𝑡_𝑚̃) such that 𝑣𝑡̂=𝜙_𝜏,𝑣𝑡̂≤𝑣(𝑡)≤𝑣𝑡_𝑖̃,𝑡∈𝑡,̂𝑡_𝑖̃. (A.10) Therefore, one gets from (A.4), (A.9), and (A.10) that, for any 𝑠∈[−𝜏_𝑖,0], 𝑣(𝑡+𝑠)≤𝜙_𝜏𝛽_𝑖𝑒^{−𝜆(𝑡1−𝑡0)}≤𝜙_𝜏𝛾𝑒^𝜆𝜏𝑖𝑒^{−𝜆(𝑡1−𝑡0)}≤𝛾𝑒^𝜆𝜏𝑖𝑣𝑡̂≤𝛾𝑒^𝜆𝜏𝑖𝑣(𝑡),𝑡∈𝑡,̂𝑡_𝑖̃. (A.11) Thus, one has from (A.2) and (A.11) that 𝐷⁺𝑣(𝑡)≤𝑎𝑣(𝑡)+𝑏₁[𝑣(𝑡)]_𝜏1+𝑏₂[𝑣(𝑡)]_𝜏2+⋯+𝑏_𝑚[𝑣(𝑡)]_𝜏𝑚≤𝑎+^𝑚_𝑖=1𝛾𝑏_𝑖𝑒^𝜆𝜏𝑖𝑣(𝑡)≤(𝜎−𝜆)𝑣(𝑡),𝑡∈𝑡,̂𝑡₁̃. (A.12) It follows from (A.5), (A.9), (A.10), and (A.12) that 𝑣𝑡₁̃≤𝑣𝑡̂𝑒^{(𝜎−𝜆)(}𝑡̃₁₋𝑡₎̂=𝜙_𝜏𝑒^{(𝜎−𝜆)(}𝑡̃₁₋𝑡₎̂<𝜙_𝜏𝑒^{𝜎(𝑡1−𝑡0)}≤𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡1−𝑡0)}=𝑣𝑡₁̃, (A.13) which is a contradiction. Hence (A.6) holds for 𝑘=1.
Now we assume that (A.6) holds for 𝑘=1,2,…,𝑛,  𝑛∈ℕ₊,  𝑛≥1, that is, 𝑣(𝑡)≤𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡−𝑡0)},𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘=1,2,…,𝑛. (A.14) Next, we will show that (A.6) holds for 𝑘=𝑛+1, that is, 𝑣(𝑡)≤𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡−𝑡0)},𝑡∈𝑡_𝑛,𝑡_𝑛+1. (A.15)
For the sake of contradiction, suppose that (A.15) does not hold. Define 𝑡̆=inf{𝑡∈[𝑡_𝑛,𝑡_𝑛+1]∣𝑣(𝑡)>‖𝜙‖_𝜏𝛽₁𝑒^{−𝜆(𝑡−𝑡0)}}. Then one obtains from (A.3) and (A.14) that 𝑣𝑡⁺_𝑛≤𝑝_𝑛𝑣𝑡⁻_𝑛+𝑞¹_𝑛𝑣𝑡⁻_𝑛𝜏1+𝑞²_𝑛𝑣𝑡⁻_𝑛𝜏2+⋯+𝑞^𝑚_𝑛𝑣𝑡⁻_𝑛𝜏𝑚≤𝑝_𝑛𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡𝑛−𝑡0)}+𝑞¹_𝑛𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡𝑛−𝜏1−𝑡0)}+𝑞²_𝑛𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡𝑛−𝜏2−𝑡0)}+⋯+𝑞^𝑚_𝑛𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡𝑛−𝜏𝑚−𝑡0)}=𝑝_𝑛+^𝑚_𝑗=1𝑞^𝑗_𝑛𝑒^𝜆𝜏𝑗𝜙_𝜏𝛽₁𝑒^{̆𝜆(𝑡−𝑡𝑛)}𝑒^{̆−𝜆(𝑡−𝑡0)}<𝑝_𝑛+^𝑚_𝑗=1𝑞^𝑗_𝑛𝑒^𝜆𝜏𝑗𝑒^{𝜆(𝑡𝑛+1−𝑡𝑛)}𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}<𝑒^{−(𝜎+𝜆)(𝑡𝑛+1−𝑡𝑛)}𝑒^{𝜆(𝑡𝑛+1−𝑡𝑛)}𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}=𝑒^{−𝜎(𝑡𝑛+1−𝑡𝑛)}𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}<𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}, (A.16) which implies that 𝑡̆≠𝑡_𝑛. From the continuity of 𝑣(𝑡) in the interval [𝑡_𝑛,𝑡_𝑛+1), one has 𝑣𝑡̆=𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)},𝑣(𝑡)≤𝑣𝑡̆,𝑡∈𝑡_𝑛,𝑡̆. (A.17)
On the other hand, one can deduce from (A.16) that there exists 𝑡^∗∈(𝑡_𝑛,𝑡)̆ such that 𝑣𝑡^∗=𝑝_𝑛+^𝑚_𝑗=1𝑞^𝑗_𝑛𝑒^𝜆𝜏𝑗𝑒^{𝜆(𝑡𝑛+1−𝑡𝑛)}𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)},𝑣𝑡^∗≤𝑣(𝑡)≤𝑣𝑡̆,𝑡∈𝑡^∗,𝑡̆. (A.18)
For any 𝑡∈[𝑡^∗,𝑡]̆, 𝑠∈[−𝜏_𝑖,0], either 𝑡+𝑠∈[𝑡₀−𝜏_𝑖,𝑡_𝑛) or 𝑡+𝑠∈[𝑡_𝑛,𝑡]̆. Two cases will be discussed as follows.
Case 1. If 𝑡+𝑠∈[𝑡₀−𝜏_𝑖,𝑡_𝑛), then one obtains from (A.14) that 𝑣(𝑡+𝑠)≤𝜙_𝜏𝛽₁𝑒^{−𝜆(𝑡−𝑡0)}𝑒^−𝜆𝑠≤𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}𝑒^{̆𝜆(𝑡−𝑡)}𝑒^𝜆𝜏𝑖≤𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}𝑒^{𝜆(𝑡𝑛+1−𝑡𝑛)}𝑒^𝜆𝜏𝑖. (A.19)
Case 2. If 𝑡+𝑠∈[𝑡_𝑛,𝑡]̆, then it follows from (A.17) that 𝑣(𝑡+𝑠)≤𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}≤𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}𝑒^{𝜆(𝑡𝑛+1−𝑡𝑛)}𝑒^𝜆𝜏𝑖. (A.20)
In any case, one has from (A.18), (A.19), and (A.20) that, for any 𝑠∈[−𝜏_𝑖,0], 𝑣(𝑡+𝑠)≤𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}𝑒^{𝜆(𝑡𝑛+1−𝑡𝑛)}𝑒^𝜆𝜏𝑖=𝑒^𝜆𝜏𝑖𝑝_𝑛+∑^𝑚_𝑗=1𝑞^𝑗_𝑛𝑒^𝜆𝜏𝑗𝑣𝑡^∗≤𝑒^𝜆𝜏𝑖𝑝_𝑛+∑^𝑚_𝑗=1𝑞^𝑗_𝑛𝑒^𝜆𝜏𝑗𝑣(𝑡)≤𝛾𝑒^𝜆𝜏𝑖𝑣(𝑡),𝑡∈𝑡^∗,𝑡̆. (A.21) Hence, one obtains from (A.2) and (A.21) that 𝐷⁺𝑣(𝑡)≤𝑎𝑣(𝑡)+𝑏₁[𝑣(𝑡)]_𝜏1+𝑏₂[𝑣(𝑡)]_𝜏2+⋯+𝑏_𝑚[𝑣(𝑡)]_𝜏𝑚≤𝑎+^𝑚_𝑖=1𝛾𝑏_𝑖𝑒^𝜆𝜏𝑖𝑣(𝑡)≤(𝜎−𝜆)𝑣(𝑡),𝑡∈𝑡^∗,𝑡̆. (A.22) It follows from inequalities (A.3), (A.17), (A.18), and (A.19) that 𝑣𝑡̆≤𝑣𝑡^∗𝑒^{̆(𝜎−𝜆)(𝑡−𝑡∗)}=𝑝_𝑛+^𝑚_𝑗=1𝑞^𝑗_𝑛𝑒^𝜆𝜏𝑗𝑒^{𝜆(𝑡𝑛+1−𝑡𝑛)}𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}𝑒^{̆(𝜎−𝜆)(𝑡−𝑡∗)}<𝑒^{−(𝜎+𝜆)(𝑡𝑛+1−𝑡𝑛)}𝑒^{𝜆(𝑡𝑛+1−𝑡𝑛)}𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}𝑒^{̆(𝜎−𝜆)(𝑡−𝑡∗)}=𝑒^{−𝜎(𝑡𝑛+1−𝑡𝑛)}𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}𝑒^{̆(𝜎−𝜆)(𝑡−𝑡∗)}≤𝜙_𝜏𝛽₁𝑒^{̆−𝜆(𝑡−𝑡0)}=𝑣𝑡̆, (A.23) which is a contradiction. Therefore the assumption that the inequality (A.15) does not hold is not true, and hence the inequality (A.6) holds for 𝑘=𝑛+1. According to the theory of mathematical induction method, the inequality (A.6) holds for all 𝑘∈ℕ₊. This completes the proof.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (NSFC) under Grants no. 61263020, no. 11101053, and no. 10971240, the Scientific Research Fund of Yunnan Province under Grant no. 2010ZC150, the Scientific Research Fund of Chongqing Normal University under Grants no. 940115 and 12XLB031, the Key Project of Chinese Ministry of Education (211118), the Excellent Youth Foundation of Educational Committee of Hunan Provincial (10B002), Key Project of Chinese Education Ministry under Grant no. 212138, Natural Science Foundation of Chongqing under Grant CQ CSTC 2011BB0117, and Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ120630.