1.Introduction BingLi andQiankunSong SynchronizationofChaoticDelayedFuzzyNeuralNetworksunderImpulsiveandStochasticPerturbations ResearchArticle

Volume 2013, Article ID 543549,14pages http://dx.doi.org/10.1155/2013/543549

Research Article

Synchronization of Chaotic Delayed Fuzzy Neural Networks under Impulsive and Stochastic Perturbations

Bing Li

and Qiankun Song

1College of Science, Chongqing Jiaotong University, Chongqing 400074, China

2Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China

Correspondence should be addressed to Bing Li; [email protected]

Received 4 October 2012; Revised 3 December 2012; Accepted 14 December 2012 Academic Editor: Xiaodi Li

Copyright © 2013 B. Li and Q. Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The synchronization problem of chaotic fuzzy cellular neural networks with mixed delays is investigated. By an impulsive integrodifferential inequality and the Itˆo’s formula, some sufficient criteria to synchronize the networks under both impulsive and stochastic perturbations are obtained. The example and simulations are given to demonstrate the efficiency and advantages of the proposed results.

1. Introduction

Fuzzy cellular neural network (FCNN), which integrated fuzzy logic into the structure of a traditional cellular neural networks (CNNs) and maintained local connectivity among cells, was first introduced by T. Yang and L. Yang [1] to deal with some complexity, uncertainty, or vagueness in CNNs.

Lots of studies have illustrated that FCNNs are a useful paradigm for image processing and pattern recognition [2].

So far, many important results on stability analysis and state estimation of FCNNs have been reported (see [3–12] and the references therein).

Recently, it has been revealed that if the network’s parameters and time delays are appropriately chosen, then neural networks can exhibit some complicated dynamics and even chaotic behaviors [13, 14]. The chaotic system exhibits unpredictable and irregular dynamics, and it has been found in many fields. Since the drive-response con- cept was proposed by Pecora and Carroll [15] in 1990 for constructing the synchronization of coupled chaotic systems, the control and synchronization problems of chaotic systems have been extensively investigated. In recent years, various synchronization schemes for chaotic neural networks have derived and demonstrated potential applications in many areas such as secure communication, image processing and harmonic oscillation generation; see [16–32].

Although there have been many results which can be applied to synchronization problems of a broad class of FCNNs [25–32], there are some disadvantages that need attention.

(1)Synchronization procedures and schemes are rather sensitive to the unavoidable channel disturbances which are usually presented in two forms: impulse and random noise.

However, in [25–27], authors provided some new schemes to synchronize the chaotic systems without considering both impulse and random noise. In [28,29], under the condition of no channel disturbance, Yu et al. and Xing and Peng studied the lag synchronization problems of FCNNs, respectively. In [30, 31], authors studied the synchronization of impulsive fuzzy cellular neural networks (IFCNNs) with delays. In [32], authors derived some synchronization schemes for FCNNs with random noise. In fact, in real system, it is more reasonable that the two perturbations coexist simultaneously.

(2)The criteria proposed in [25–32] are valid only for FCNNs with discrete delays. For example, in [25,28,30,31], the involved delays are constants. In [26,27,32], the involved delays are time-varying delays which are continuously differ- entiable, and the corresponding derivatives are required to be finite or not greater than 1. In [29], Xing and Peng provided some new criteria on lag synchronization problem of FCNNs but they only considered the case for bounded time-varying delays. In fact, time delays may occur in an irregular fashion,

and sometimes they may be not continuously differentiable.

Besides this, distribution delays may also exist when neural networks have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths.

(3)Some conditions imposed on the impulsive perturba- tions are too strong. For instance, Feng et al. [31] required the magnitude of jumps not to be smaller than 0 and not greater than 2. However, the disturbance in the real environment may be very intense.

Therefore, it is of great theoretical and practical signif- icance to investigate synchronization problems of IFCNNs with mixed delays and random noise. However, up to now, to the best of our knowledge, no result for synchronization of IFNNs with mixed delays and random noise has been reported.

Inspired by the above discussion, this paper addresses the exponential synchronization problem of IFCNNs with mixed delays and random noise. Based on the properties of nonsingular M-matrix and the It̂𝑜’s formula, we design some synchronization schemes with a state feedback con- troller to ensure the exponential synchronization control.

Our method does not resort to complicated Lyapunov- Krasovskii functional which is widely used. The proposed synchronization schemes are novel and improve some of the previous literature.

This paper is organized as follows. In Section 2, we introduce the drive-response models and some preliminaries.

InSection 3, some synchronization criteria for FCNNs with mixed delays are derived. InSection 4, an example and its simulations are given to illustrate the effectiveness of theo- retical results. Finally, conclusions are drawn inSection 5.

2. Model Description and Preliminaries

LetR^𝑛 be the space of𝑛-dimensional real column vectors, and letR^𝑚×𝑛represent the class of𝑚 × 𝑛matrices with real components. | ⋅ | denotes the Euclidean norm in R^𝑛. The inequality “≤” (“>”) between matrices or vectors such as𝐴 ≤ 𝐵(𝐴 > 𝐵) means that each pair of corresponding elements of𝐴and 𝐵satisfies the inequality “≤” (“>”).𝐴 ∈ R^𝑚×𝑛 is called a nonnegative matrix if𝐴 ≥ 0, and𝑥 ∈ R^𝑛is called a positive vector if 𝑥 > 0. The transpose of 𝐴 ∈ R^𝑚×𝑛 or𝑥 ∈ R^𝑛 is denoted by𝐴^𝑇 or𝑥^𝑇. Let𝐸 denote the unit matrix with appropriate dimensions.N := {1, 2, . . . , 𝑛}, and N:= {1, 2, . . .},R₊ := [0, +∞).

PC[𝐽,R^𝑛] = {𝜓 : 𝐽 → R^𝑛|𝜓(𝑠) is continuous and bounded for all but at most countable points𝑠 ∈ 𝐽and at these points,𝜓(𝑠⁺)and𝜓(𝑠⁻)exist,𝜓(𝑠) = 𝜓(𝑠⁺)}. Here,𝐽 ⊂Ris an interval;𝜓(𝑠⁺)and𝜓(𝑠⁻)denote the right-hand and left-hand limits of the function 𝜓(𝑠), respectively. Especially PC :=

PC[(−∞, 0],R^𝑛]with the norm‖𝜓‖ =sup_{−∞<𝑠≤0} |𝜓(𝑠)|for 𝜓 ∈PC.

L^𝑒 = {𝜓 : R₊ → R|𝜓(𝑠)is piecewise continuous and satisfies∫₀^+∞|𝜓(𝑠)|𝑒^𝜎0𝑠𝑑𝑠 < +∞for some constant𝜎₀> 0}.

For𝐴, 𝐵 ∈R^𝑛×𝑛and𝜙 :R → R^𝑛, we denote that [𝐴]⁺= (󵄨󵄨󵄨󵄨󵄨𝑎^𝑖𝑗󵄨󵄨󵄨󵄨󵄨)𝑛×𝑛,

𝐴 ∘ 𝐵 = (𝑎_𝑖𝑗𝑏_𝑖𝑗)_𝑛×𝑛, [𝜙]⁺= (󵄨󵄨󵄨󵄨𝜙¹󵄨󵄨󵄨󵄨,...,󵄨󵄨󵄨󵄨𝜙^𝑛󵄨󵄨󵄨󵄨)^𝑇,

[𝜙 (𝑡)]_𝜏= ([ 𝜙₁(𝑡)]_𝜏, . . . , [ 𝜙_𝑛(𝑡)]_𝜏)^𝑇, where[𝜙_𝑖(𝑡)]_𝜏 = sup

−𝜏≤𝑠≤0𝜙_𝑖(𝑡 + 𝑠) , 𝑖 ∈N,

(1)

and𝐷⁺𝜙(𝑡)denotes the upper-right derivative of𝜙(𝑡)at time 𝑡.

Consider IFCNNs with mixed delays as follows:

𝑑𝑥_𝑖(𝑡)

𝑑𝑡 = −𝑐_𝑖𝑥_𝑖+∑^𝑛

𝑗=1

𝑎_𝑖𝑗𝑓_𝑗(𝑥_𝑗) +∑^𝑛

𝑗=1

𝑏_𝑖𝑗𝜈_𝑗+ 𝐽_𝑖

+⋀^𝑛

𝑗=1

𝑇_𝑖𝑗𝜇_𝑗+⋀^𝑛

𝑗=1

𝛼_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

+⋀^𝑛

𝑗=1

𝛾_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑥_𝑗(𝑡 − 𝑠)) 𝑑𝑠 +⋁^𝑛

𝑗=1

𝑆_𝑖𝑗𝜇_𝑗+⋁^𝑛

𝑗=1

𝛽_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

+⋁^𝑛

𝑗=1

𝜃_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑥_𝑗(𝑡 − 𝑠)) 𝑑𝑠, 𝑡 ≥ 𝑡₀, 𝑡 ̸= 𝑡_𝑘, Δ𝑥_𝑖(𝑡_𝑘) = 𝑥_𝑖(𝑡⁺_𝑘) − 𝑥_𝑖(𝑡⁻_𝑘)

= 𝐼_𝑖𝑘(𝑥_𝑖(𝑡⁻_𝑘)) , 𝑘 ∈N, 𝑥_𝑖(𝑡₀+ 𝑠) = 𝜙_𝑖(𝑠) , −∞ < 𝑠 ≤ 0,

(2)

where 𝑖 = 1, 2, . . . , 𝑛, 𝑛 denotes the number of units in the neural network. 𝑥(𝑡) = (𝑥₁(𝑡), . . . , 𝑥_𝑛(𝑡))^𝑇 represents the state variable.𝑓_𝑗(⋅)is the activation function of the𝑗th neuron.𝑐_𝑖 represents the passive decay rate to the state of 𝑖th neuron at time 𝑡. 𝛼_𝑖𝑗 and 𝛾_𝑖𝑗 are elements of the fuzzy feedback MIN template.𝛽_𝑖𝑗and𝜃_𝑖𝑗are elements of the fuzzy feedback MAX template.𝑇_𝑖𝑗 and 𝑆_𝑖𝑗 are elements of fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively. 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 are elements of feedback and feed-forward template, respectively. ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operations, respectively. 𝜈_𝑖 and𝐽_𝑖denote input and bias of the𝑖th neuron, respectively.

For any𝑖, 𝑗 ∈ N, 𝜏_𝑖𝑗(𝑡) corresponding to the transmission delay satisfies0 ≤ 𝜏_𝑖𝑗(𝑡) ≤ 𝜏, and𝑘_𝑖𝑗 ∈ L^𝑒 is the feed- back kernel. For any𝑘 ∈ N, 𝐼_𝑘(⋅)represents the impulsive perturbation, and 𝑡_𝑘 denotes impulsive moment satisfying 𝑡_𝑘< 𝑡_𝑘+1, lim_{𝑘 → +∞}𝑡_𝑘= +∞.

We make the following assumptions throughout this paper.

(𝐴₁) 𝑓_𝑖is globally Lipschitz continuous, that is, for any𝑖 ∈ N, there exists nonnegative constant𝐿_𝑖such that

󵄨󵄨󵄨󵄨𝑓^𝑖(𝑢) − 𝑓_𝑖(𝑣)󵄨󵄨󵄨󵄨 ≤ 𝐿^𝑖|𝑢 − 𝑣| for𝑢, 𝑣 ∈R. (3) (𝐴₂)For any𝑘 ∈N, there is a nonnegative constant𝜂_𝑘such

that

󵄨󵄨󵄨󵄨𝑢 + 𝐼^𝑖𝑘(𝑢) − 𝑣 − 𝐼_𝑖𝑘(𝑣)󵄨󵄨󵄨󵄨 ≤ 𝜂^𝑘|𝑢 − 𝑣| for𝑢, 𝑣 ∈R, 𝑖 ∈N.

(4)

Let (2) be the drive system, and let the response system with random noise be described by

𝑑𝑦_𝑖(𝑡) = [ [

−𝑐_𝑖𝑦_𝑖+∑^𝑛

𝑗=1

𝑎_𝑖𝑗𝑓_𝑗(𝑦_𝑗) +∑^𝑛

𝑗=1

𝑏_𝑖𝑗𝜈_𝑗

+ 𝐽_𝑖+⋀^𝑛

𝑗=1

𝑇_𝑖𝑗𝜇_𝑗+⋀^𝑛

𝑗=1

𝛼_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

+⋀^𝑛

𝑗=1

𝛾_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑦_𝑗(𝑡 − 𝑠)) 𝑑𝑠 +⋁^𝑛

𝑗=1𝑆_𝑖𝑗𝜇_𝑗+⋁^𝑛

𝑗=1𝛽_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))) +⋁^𝑛

𝑗=1

𝜃_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑦_𝑗(𝑡−𝑠)) 𝑑𝑠+𝑈_𝑖(𝑡)]

] 𝑑𝑡

+∑^𝑛

𝑗=1

𝜎_𝑖𝑗(𝑡, 𝑥_𝑗(𝑡) − 𝑦_𝑗(𝑡) , 𝑥_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))

−𝑦_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))) 𝑑𝑤_𝑗(𝑡) , 𝑡 ≥ 𝑡₀, 𝑡 ̸= 𝑡_𝑘, Δ𝑦_𝑖(𝑡_𝑘) = 𝑦_𝑖(𝑡⁺_𝑘) − 𝑦_𝑖(𝑡⁻_𝑘) = 𝐼_𝑖𝑘(𝑦_𝑖(𝑡⁻_𝑘)) , 𝑘 ∈N, 𝑦_𝑖(𝑡₀+ 𝑠) = 𝜓_𝑖(𝑠) , −∞ < 𝑠 ≤ 0,

(5) where𝑤(𝑡) = (𝑤₁(𝑡), . . . , 𝑤_𝑛(𝑡))^𝑇is an𝑛-dimensional stand- ard Brownian motion defined on a complete probability space(Ω,F,P)with a natural filtration{F_𝑡}_𝑡≥0 generated by {𝑤(𝑠) : 0 ≤ 𝑠 ≤ 𝑡} and satisfying the usual conditions (i.e., it is right continuous, andF₀ contains all P-null sets). The initial value 𝜓 = (𝜓₁(𝑠), . . . , 𝜓_𝑛(𝑠))^𝑇 ∈ PC^𝑏_F0[(−∞, 0],R^𝑛]which denotes the family of all bounded F₀-measurable andPC-valued random variables𝜓with the norm ‖𝜓‖^𝑝_F = sup_{−∞<𝑠≤0}E|𝜓(𝑠)|^𝑝, where E denotes the

expectation of stochastic process.𝑈(𝑡) = (𝑈₁(𝑡), . . . , 𝑈_𝑛(𝑡))^𝑇 is the state feedback controller designed by

𝑈_𝑖(𝑡) =∑^𝑛

𝑗=1

𝑀_𝑖𝑗(𝑦_𝑗(𝑡) − 𝑥_𝑗(𝑡))

+∑^𝑛

𝑗=1

𝑁_𝑖𝑗(𝑦_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)) − 𝑥_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))) , (6)

where𝑀 = (𝑀_𝑖𝑗)_𝑛×𝑛,𝑁 = (𝑁_𝑖𝑗)_𝑛×𝑛 are the controller gain matrices to be scheduled. The diffusion coefficient matrix (or noise intensity matrix)𝜎 : R×R^𝑛×R^𝑛 → R^𝑛×𝑛satisfies the local Lipschitz condition and the linear growth condition (see [33]). In addition,

(𝐴₃)for𝑖 ∈ N, there exist nonnegative constants𝑐_𝑖𝑗,𝑑_𝑖𝑗 such that

𝜎_𝑖𝜎^𝑇_𝑖 ≤∑^𝑛

𝑗=1

𝑐_𝑖𝑗󵄨󵄨󵄨󵄨󵄨𝑥^𝑗− 𝑦_𝑗󵄨󵄨󵄨󵄨󵄨² +∑^𝑛

𝑗=1

𝑑_𝑖𝑗󵄨󵄨󵄨󵄨󵄨𝑥^𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)) − 𝑦_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))󵄨󵄨󵄨󵄨󵄨², (7)

where𝜎_𝑖= (𝜎_𝑖1, . . . , 𝜎_𝑖𝑛).

Let𝑒(𝑡) = (𝑒₁(𝑡), . . . , 𝑒_𝑛(𝑡))^𝑇, where𝑒_𝑖(𝑡) = 𝑦_𝑖(𝑡)−𝑥_𝑖(𝑡), be the synchronization error. Then, the error dynamical system between (2) and (5) is given by

𝑑𝑒_𝑖(𝑡) = [ [

−𝑐_𝑖𝑒_𝑖+∑^𝑛

𝑗=1𝑎_𝑖𝑗(𝑓_𝑗(𝑦_𝑗) − 𝑓_𝑗(𝑥_𝑗)) +∑^𝑛

𝑗=1

𝑀_𝑖𝑗𝑒_𝑗(𝑡) +∑^𝑛

𝑗=1

𝑁_𝑖𝑗𝑒_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))

+⋀^𝑛

𝑗=1

𝛼_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

−⋀^𝑛

𝑗=1

𝛼_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

+⋁^𝑛

𝑗=1

𝛽_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

−⋁^𝑛

𝑗=1

𝛽_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

+⋀^𝑛

𝑗=1

𝛾_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑦_𝑗(𝑡 − 𝑠)) 𝑑𝑠

−⋀^𝑛

𝑗=1𝛾_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑥_𝑗(𝑡 − 𝑠)) 𝑑𝑠

+⋁^𝑛

𝑗=1

𝜃_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑦_𝑗(𝑡 − 𝑠)) 𝑑𝑠

−⋁^𝑛

𝑗=1

𝜃_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑥_𝑗(𝑡 − 𝑠)) 𝑑𝑠]

] 𝑑𝑡

+∑^𝑛

𝑗=1

𝜎_𝑖𝑗(𝑡, 𝑒_𝑗(𝑡) , 𝑒_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))) 𝑑𝑤_𝑗(𝑡) , 𝑡 ≥ 𝑡₀, 𝑡 ̸= 𝑡_𝑘, Δ𝑒_𝑖(𝑡_𝑘) = 𝑒_𝑖(𝑡_𝑘⁺) − 𝑒_𝑖(𝑡⁻_𝑘)

= 𝐼_𝑖𝑘(𝑦_𝑖(𝑡⁻_𝑘)) − 𝐼_𝑖𝑘(𝑥_𝑖(𝑡⁻_𝑘)) , 𝑘 ∈N, 𝑒_𝑖(𝑡₀+ 𝑠) = 𝜓_𝑖(𝑠) − 𝜙_𝑖(𝑠) , −∞ < 𝑠 ≤ 0.

(8)

For convenience, we use the following notations:𝐷₁ = diag{−𝑐₁, . . . , −𝑐_𝑛},𝐿 = diag{𝐿₁, . . . , 𝐿_𝑛},𝐾(𝑠) = (𝑘_𝑖𝑗(𝑠))_𝑛×𝑛, 𝐴 = (𝑎_𝑖𝑗)_𝑛×𝑛,𝑀 = (𝑀_𝑖𝑗)_𝑛×𝑛with𝑀_𝑖𝑖 = 𝑀_𝑖𝑖,𝑀_𝑖𝑗= |𝑀_𝑖𝑗|for 𝑖 ̸= 𝑗,𝛼 = (𝛼_𝑖𝑗)_𝑛×𝑛,𝛽 = (𝛽_𝑖𝑗)_𝑛×𝑛,Γ = (𝛾_𝑖𝑗)_𝑛×𝑛,Θ = (𝜃_𝑖𝑗)_𝑛×𝑛, 𝐶 = (𝑐_𝑖𝑗)_𝑛×𝑛, and𝐷 = (𝑑_𝑖𝑗)_𝑛×𝑛.

The following definition and lemmas will be employed.

Definition 1. The systems (2) and (5) are called to be globally exponentially synchronized in𝑝-moment, if there exist posi- tive constants𝜆,𝐾such that

E|𝑒 (𝑡)|^𝑝≤ 𝐾󵄩󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩󵄩^𝑝F𝑒^{−𝜆(𝑡−𝑡0)}, 𝑡 ≥ 𝑡₀. (9) It is said especially to be globally exponentially synchronized in mean square when𝑝 = 2.

For any nonsingular M-matrix𝐴(see [34]), we define that

M_𝐴= {𝑧 ∈R^𝑛| 𝐴𝑧 > 0, 𝑧 > 0} . (10)

Lemma 2 (see [35]). For a nonsingularM-matrix𝐴,M_𝐴is a nonempty cone without conical surface.

Lemma 3 (see [36]). For𝑥_𝑖≥ 0,𝛼_𝑖> 0, and∑^𝑛_𝑖=1𝛼_𝑖= 1,

∏𝑛 𝑖=1

𝑥^𝛼_𝑖^𝑖 ≤∑^𝑛

𝑖=1

𝛼_𝑖𝑥_𝑖. (11)

The sign of equality holds if and only if𝑥_𝑖= 𝑥_𝑗for all𝑖, 𝑗 ∈N.

Lemma 4 (see [1]). Let𝛼_𝑖𝑗,𝛽_𝑖𝑗∈Rand𝑥,𝑦 ∈R^𝑛be the two states of the system(2). Then, one has

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨

⋀𝑛 𝑗=1

𝛼_𝑖𝑗𝑓_𝑗(𝑥_𝑗) −⋀^𝑛

𝑗=1

𝛼_𝑖𝑗𝑓_𝑗(𝑦_𝑗)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛼^𝑖𝑗󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑓^𝑗(𝑥_𝑗) − 𝑓_𝑗(𝑦_𝑗)󵄨󵄨󵄨󵄨󵄨,

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨

⋁𝑛 𝑗=1

𝛽_𝑖𝑗𝑓_𝑗(𝑥_𝑗) −⋁^𝑛

𝑗=1

𝛽_𝑖𝑗𝑓_𝑗(𝑦_𝑗)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛽^𝑖𝑗󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑓^𝑗(𝑥_𝑗) − 𝑓_𝑗(𝑦_𝑗)󵄨󵄨󵄨󵄨󵄨.

(12)

Lemma 5 (see [36]). For the integer 𝑝 ≥ 2 and 𝑥 = (𝑥₁, . . . , 𝑥_𝑛)^𝑇 ∈R^𝑛, there exists a positive constant𝑒_𝑝(𝑛)such that

𝑒_𝑝(𝑛) (∑^𝑛

𝑖=1

𝑥_𝑖²)

𝑝/2

≤∑^𝑛

𝑖=1󵄨󵄨󵄨󵄨𝑥^𝑖󵄨󵄨󵄨󵄨^𝑝. (13) Lemma 6. For𝑘 ∈N, assume that𝑣 = (𝑣₁(𝑡), . . . , 𝑣_𝑛(𝑡))^𝑇 ∈ PC[(−∞, +∞),R^𝑛]satisfies

𝐷⁺𝑣 (𝑡) ≤ 𝐴₀𝑣 (𝑡) + 𝑃𝑣 (𝑡) + 𝑄 [𝑣 (𝑡)]_𝜏 + ∫^+∞

0 Υ (𝑠) 𝑣 (𝑡 − 𝑠) 𝑑𝑠, 𝑡 ≥ 𝑡₀, 𝑡 ̸= 𝑡_𝑘, 𝑣 (𝑡⁺_𝑘) ≤ 𝜌_𝑘𝑣 (𝑡⁻_𝑘) , 𝜌_𝑘 ≥ 0,

𝑣_𝑡0 ∈PC, where𝑣_𝑡0= 𝑣 (𝑡₀+ 𝑠) , 𝑠 ∈ (−∞, 0] , (14)

in which

(𝐶₁) 𝐴₀ = diag{𝑎₁, . . . , 𝑎_𝑛},𝑃 = (𝑝_𝑖𝑗)_𝑛×𝑛with𝑝_𝑖𝑗 ≥ 0for 𝑖 ̸= 𝑗,𝑄 = (𝑞_𝑖𝑗)_𝑛×𝑛 ≥ 0,Υ(𝑠) = (𝜐_𝑖𝑗(𝑠))_𝑛×𝑛 ≥ 0, and 𝜐_𝑖𝑗∈L^𝑒,𝑖, 𝑗 ∈N.

(𝐶₂) Π = −(𝐴₀ + 𝑃 + 𝑄 + ∫₀^+∞Υ(𝑠)𝑑𝑠)is a nonsingular M-matrix.

Then, there must exist𝑧 = (𝑧₁, . . . , 𝑧_𝑛)^𝑇∈M_Πand𝜆 ∈ (0, 𝜎₀] such that

𝑣 (𝑡) ≤ (^𝑘−1∏

𝑗=0

󰜚_𝑗) 𝑧𝑒^{−𝜆(𝑡−𝑡0)}, 𝑡_𝑘−1≤ 𝑡 < 𝑡_𝑘, 𝑘 ∈N, (15)

provided that the initial value𝑣_𝑡0satisfies

𝑣 (𝑡) ≤ 𝑧𝑒^{−𝜆(𝑡−𝑡0)}, −∞ < 𝑡 ≤ 𝑡₀, (16) where󰜚₀= 1,󰜚_𝑘=max{1, 𝜌_𝑘}, and𝑧,𝜆can be determined by

(𝜆𝐸 + 𝐴₀+ 𝑃 + 𝑄𝑒^𝜆𝜏+ ∫^+∞

0 Υ (𝑠) 𝑒^𝜆𝑠𝑑𝑠) 𝑧 < 0. (17)

Proof. By condition(𝐶₂) and Lemma 2, we can find ̃𝑧 = (̃𝑧₁, . . . , ̃𝑧_𝑛)^𝑇 ∈ M_Πsuch thatΠ̃𝑧 > 0, namely,(𝐴₀ + 𝑃 + 𝑄+∫₀^+∞Υ(𝑠)𝑑𝑠) ̃𝑧 < 0. By the continuity, there must be some positive constant𝜆 ∈ (0, 𝜎₀]satisfying

(𝜆𝐸 + 𝐴₀+ 𝑃 + 𝑄𝑒^𝜆𝜏+ ∫^+∞

0 Υ (𝑠) 𝑒^𝜆𝑠𝑑𝑠) ̃𝑧 < 0. (18) Noting that𝑣_𝑡0∈PC, we can find a constant𝐵 > 0such that

‖𝑣_𝑡0‖ ≤ 𝐵. Denote that𝑧 := (𝐵/min_𝑖∈Ñ𝑧_𝑖)̃𝑧 = (𝑧₁, . . . , 𝑧_𝑛)^𝑇. Obviously,𝑧and𝜆satisfy (16) and (17).

Let𝑤_𝑖(𝑡) := 𝑧_𝑖𝑒^{−𝜆(𝑡−𝑡0)}for𝑡 ∈ R, 𝑖 ∈ N. For any small enough𝜖 > 0, (16) implies that𝑣_𝑖(𝑡) ≤ 𝑤_𝑖(𝑡) < (1+𝜖)𝑤_𝑖(𝑡), 𝑡 ∈ (−∞, 𝑡₀]. Next, we claim that for any𝑡 ∈ [𝑡₀, 𝑡₁),

𝑣_𝑖(𝑡) < (1 + 𝜖) 𝑤_𝑖(𝑡) , 𝑖 ∈N. (19) If inequality (19) is not true, then there must exist some𝑚 ∈ Nand𝑡^∗∈ (𝑡₀, 𝑡₁)such that

𝑣_𝑚(𝑡^∗) = (1 + 𝜖) 𝑤_𝑚(𝑡^∗) , 𝑣_𝑖(𝑡) < (1 + 𝜖) 𝑤_𝑖(𝑡) , 𝑡 ∈ (−∞, 𝑡^∗) , 𝑖 ∈N, (20) 𝐷⁺𝑣_𝑚(𝑡^∗) ≥ (1 + 𝜖) 𝑤_𝑚^󸀠 (𝑡^∗) . (21) On the other hand, (14) together with (17) and (20) leads to

𝐷⁺𝑣_𝑚(𝑡^∗) ≤ 𝑎_𝑚𝑣_𝑚(𝑡^∗) +∑^𝑛

𝑗=1

𝑝_𝑚𝑗𝑣_𝑗(𝑡^∗)

+∑^𝑛

𝑗=1

𝑞_𝑚𝑗[𝑣_𝑗(𝑡^∗)]_𝜏

+∑^𝑛

𝑗=1∫^+∞

0 𝜐_𝑚𝑗(𝑠) 𝑣_𝑗(𝑡^∗− 𝑠) 𝑑𝑠

≤ (1 + 𝜖) 𝑒^{−𝜆(𝑡∗−𝑡0)}𝑎_𝑚𝑧_𝑚 + (1 + 𝜖) 𝑒^{−𝜆(𝑡∗−𝑡0)}∑^𝑛

𝑗=1

𝑝_𝑚𝑗𝑧_𝑗

+ (1 + 𝜖) 𝑒^{−𝜆(𝑡∗−𝑡0)}∑^𝑛

𝑗=1

𝑞_𝑚𝑗𝑧_𝑗𝑒^𝜆𝜏

+ (1 + 𝜖) 𝑒^{−𝜆(𝑡∗−𝑡0)}

×∑^𝑛

𝑗=1

∫^+∞

0 𝜐_𝑚𝑗(𝑠) 𝑒^𝜆𝑠𝑑𝑠𝑧_𝑗

< (1 + 𝜖) 𝑒^{−𝜆(𝑡∗−𝑡0)}(−𝜆𝑧_𝑚)

= (1 + 𝜖) 𝑤_𝑚^󸀠 (𝑡^∗) ,

(22)

which contradicts (21). Therefore, (19) holds. Letting𝜖 → 0⁺ in (19), we get

𝑣_𝑖(𝑡) ≤ 𝑤_𝑖(𝑡) = 󰜚₀𝑤_𝑖(𝑡) , 𝑡 ∈ [𝑡₀, 𝑡₁) , 𝑖 ∈N. (23)

Suppose that for𝜈 = 1, 2, . . . , 𝑘, the following inequalities hold

𝑣_𝑖(𝑡) ≤ (^𝜈−1∏

𝑚=0󰜚_𝑚) 𝑤_𝑖(𝑡) , 𝑡 ∈ [𝑡_𝜈−1, 𝑡_𝜈) , 𝑖 ∈N. (24) For𝑡 = 𝑡_𝑘, from (14) and (24), we have

𝑣_𝑖(𝑡_𝑘) ≤ 𝜌_𝑘𝑣_𝑖(𝑡_𝑘⁻) ≤ 󰜚_𝑘(^𝑘−1∏

𝑚=0

󰜚_𝑚) 𝑤_𝑖(𝑡_𝑘)

≤ (∏^𝑘

𝑚=0

󰜚_𝑚) 𝑤_𝑖(𝑡_𝑘) , 𝑖 ∈N.

(25)

Recalling𝜌_𝑘≥ 1, it follows from (24) and (25) that 𝑣_𝑖(𝑡) ≤ (∏^𝑘

𝑚=0󰜚_𝑚) 𝑤_𝑖(𝑡) , 𝑡 ∈ (−∞, 𝑡_𝑘] , 𝑖 ∈N. (26) Repeating the proof similar to (19) can yield

𝑣_𝑖(𝑡) ≤ (∏^𝑘

𝑚=0

󰜚_𝑚) 𝑤_𝑖(𝑡) , 𝑡 ∈ [ 𝑡_𝑘, 𝑡_𝑘+1) , 𝑖 ∈N. (27) By the mathematical induction, we derive that for any𝑘 ∈N,

𝑣_𝑖(𝑡) ≤ (∏^𝑘−1

𝑗=0

󰜚_𝑗) 𝑤_𝑖(𝑡)

≤ (∏^𝑘−1

𝑗=0

󰜚_𝑗) 𝑧_𝑖𝑒^{−𝜆(𝑡−𝑡0)}, 𝑡 ∈ [ 𝑡_𝑘−1, 𝑡_𝑘) , 𝑖 ∈N.

(28)

The proof is completed.

3. Exponential Synchronization

In this section, by using Lemma 6, we will obtain some sufficient criteria to synchronize the drive-response systems (2) and (5).

Theorem 7. Assume that(𝐴₁)–(𝐴₃)hold and

(𝐴₄)for𝑝 ≥ 2,𝐷 = −(𝐷̃ ₀ + 𝑃 + 𝑄 + ∫₀^+∞Υ(𝑠)𝑑𝑠)is a nonsingularM-matrix, where𝑃 = [𝐴]⁺𝐿 + 𝑀 + (𝑝 − 1)𝐶 := (𝑝_𝑖𝑗)_𝑛×𝑛,𝑄 = ([𝛼]⁺+[𝛽]⁺)𝐿+[𝑁]⁺+(𝑝−1)𝐷 :=

(𝑞_𝑖𝑗)_𝑛×𝑛,Υ(𝑠) = ([Γ]⁺𝐿+[Θ]⁺𝐿)∘[𝐾(𝑠)]⁺:= (𝜐_𝑖𝑗(𝑠))_𝑛×𝑛, 𝐷₀=diag{𝑑₁, . . . , 𝑑_𝑛}with

𝑑_𝑖= − 𝑝𝑐_𝑖+ (𝑝 − 1)∑^𝑛

𝑗=1[( 󵄨󵄨󵄨󵄨󵄨𝑎^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨𝛼^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨𝛽^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 +∫^+∞

0 (󵄨󵄨󵄨󵄨󵄨𝛾^𝑖𝑗󵄨󵄨󵄨󵄨󵄨+󵄨󵄨󵄨󵄨󵄨𝜃^𝑖𝑗󵄨󵄨󵄨󵄨󵄨)󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨 𝑑𝑠)𝐿^𝑗 +𝑝 − 2

2 (𝑐_𝑖𝑗+ 𝑑_𝑖𝑗)+𝑀_𝑖𝑗+󵄨󵄨󵄨󵄨󵄨𝑁^𝑖𝑗󵄨󵄨󵄨󵄨󵄨] , 𝑖 ∈N,

(29)

(𝐴₅)the impulsive perturbations satisfy sup𝑘∈N

ln𝜁_𝑘

𝑡_𝑘− 𝑡_𝑘−1 < 𝜆, (30) where𝜁_𝑘=max{1, 𝜂_𝑘^𝑝}, and𝜆 ∈ (0, 𝜎₀]is determined by

(𝜆𝐸 + 𝐷₀+ 𝑃 + 𝑄𝑒^𝜆𝜏+ ∫^+∞

0 Υ (𝑠) 𝑒^𝜆𝑠𝑑𝑠) 𝑧 < 0, for a given𝑧 ∈M𝐷̃.

(31)

Then, drive-response systems(2)and(5)are globally exponen- tial synchronization in𝑝-moment.

Proof. Since𝐷 = −(𝐷̃ ₀+𝑃+𝑄+∫₀^+∞Υ(𝑠)𝑑𝑠)is a nonsingular M-matrix, byLemma 2and the continuity, there must be a constant vector𝑧 = (𝑧₁, . . . , 𝑧_𝑛)^𝑇 ∈ M𝐷̃and a constant𝜆 ∈ (0, 𝜎₀]such that (31) holds.

We denote by 𝑒 = (𝑒₁, . . . , 𝑒_𝑛)^𝑇 the solution of error dynamical system (8) with the initial value 𝜓 − 𝜙 ∈ PC^𝑏_F0[(−∞, 0],R^𝑛]and let

𝑉 (𝑒) = (𝑉₁(𝑒) , . . . , 𝑉_𝑛(𝑒))^𝑇,

𝑉_𝑖(𝑒) = 󵄨󵄨󵄨󵄨𝑒^𝑖󵄨󵄨󵄨󵄨^𝑝, 𝑖 ∈N. (32) Calculating the time derivative of𝑉_𝑖(𝑒(𝑡))along the trajectory of error system (8) and by the Itˆo’s formula [33], we get for any 𝑘 ∈N,

𝑑𝑉_𝑖(𝑒 (𝑡)) =L𝑉_𝑖(𝑒 (𝑡)) 𝑑𝑡 + 𝜕𝑉_𝑖(𝑒)

𝜕𝑒 𝜎𝑑𝑤 (𝑡) , 𝑡 ≥ 𝑡₀, 𝑡 ̸= 𝑡_𝑘, (33) whereL𝑉_𝑖(𝑒(𝑡))is given by

L𝑉_𝑖(𝑒 (𝑡)) = 𝑝󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−2𝑒_𝑖(𝑡)

× [ [

−𝑐_𝑖𝑒_𝑖+∑^𝑛

𝑗=1

𝑎_𝑖𝑗

× (𝑓_𝑗(𝑦_𝑗) − 𝑓_𝑗(𝑥_𝑗)) +∑^𝑛

𝑗=1

𝑀_𝑖𝑗𝑒_𝑗(𝑡)

+∑^𝑛

𝑗=1

𝑁_𝑖𝑗𝑒_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))

+⋀^𝑛

𝑗=1

𝛼_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

−⋀^𝑛

𝑗=1

𝛼_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

+⋁^𝑛

𝑗=1𝛽_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡)))

−⋁^𝑛

𝑗=1

𝛽_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))) +⋀^𝑛

𝑗=1

𝛾_𝑖𝑗

× ∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑦_𝑗(𝑡 − 𝑠)) 𝑑𝑠 −⋀^𝑛

𝑗=1

𝛾_𝑖𝑗

× ∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑥_𝑗(𝑡 − 𝑠)) 𝑑𝑠 +⋁^𝑛

𝑗=1

𝜃_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑦_𝑗(𝑡 − 𝑠)) 𝑑𝑠

−⋁^𝑛

𝑗=1

𝜃_𝑖𝑗∫^+∞

0 𝑘_𝑖𝑗(𝑠) 𝑓_𝑗(𝑥_𝑗(𝑡 − 𝑠)) 𝑑𝑠]

] +1

2𝑝 (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−2𝜎_𝑖𝜎_𝑖^𝑇.

(34)

By(𝐴₁)andLemma 4, we have

L𝑉_𝑖(𝑒 (𝑡)) ≤ − 𝑝𝑐_𝑖󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝+ 𝑝 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−1

×∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑎^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨 + 𝑝󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−1

×∑^𝑛

𝑗=1

𝑀_𝑖𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨 +𝑝󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−1

×∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑁^𝑖𝑗󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))󵄨󵄨󵄨󵄨󵄨 + 𝑝 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−1∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛼^𝑖𝑗󵄨󵄨󵄨󵄨󵄨

× 𝐿_𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))󵄨󵄨󵄨󵄨󵄨 + 𝑝󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−1

×∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛽^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))󵄨󵄨󵄨󵄨󵄨 + 𝑝󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−1

×∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛾^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨 |(𝑡 − 𝑠)| 𝑑𝑠 + 𝑝 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−1∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝜃^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨

× 󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡 − 𝑠)󵄨󵄨󵄨󵄨󵄨 𝑑𝑠 +1

2𝑝 (𝑝 − 1) 󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝−2𝜎_𝑖𝜎_𝑖^𝑇 := −𝑝𝑐_𝑖󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝+ 𝐼₁+ 𝐼₂+ 𝐼₃+ 𝐼₄+ 𝐼₅+ 𝐼₆+ 𝐼₇+ 𝐼₈.

(35)

UsingLemma 3and (𝐴₃), it is easy to get

𝐼₁≤ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑎^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑎^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝, 𝐼₂≤ (∑^𝑛

𝑗=1𝑀_𝑖𝑗) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1

𝑀_𝑖𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝, 𝐼₃≤ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑁^𝑖𝑗󵄨󵄨󵄨󵄨󵄨) (𝑝 − 1)󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑁^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 [󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝]

𝜏, 𝐼₄≤ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛼^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛼^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗[󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝]

𝜏, 𝐼₅≤ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛽^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛽^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗[󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝]

𝜏, 𝐼₆≤ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛾^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨 𝑑𝑠) (𝑝 − 1)󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛾^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡 − 𝑠)󵄨󵄨󵄨󵄨󵄨^𝑝𝑑𝑠, 𝐼₇≤ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝜃^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨 𝑑𝑠) (𝑝 − 1)󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝜃^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡 − 𝑠)󵄨󵄨󵄨󵄨󵄨^𝑝𝑑𝑠, 𝐼₈≤ 1

2𝑝 (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−2∑^𝑛

𝑗=1

𝑐_𝑖𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨² +1

2𝑝 (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝−2∑^𝑛

𝑗=1

𝑑_𝑖𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡 − 𝜏_𝑖𝑗(𝑡))󵄨󵄨󵄨󵄨󵄨²

≤ (𝑝 − 1) (𝑝 − 2)

2 (∑^𝑛

𝑗=1𝑐_𝑖𝑗) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝

+ (𝑝 − 1)∑^𝑛

𝑗=1

𝑐_𝑖𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝 +(𝑝 − 1) (𝑝 − 2)

2 (∑^𝑛

𝑗=1

𝑑_𝑖𝑗) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝

+ (𝑝 − 1)∑^𝑛

𝑗=1𝑑_𝑖𝑗[󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝]

𝜏.

(36) Thus, we have

L𝑉_𝑖(𝑒 (𝑡)) ≤ − 𝑝𝑐_𝑖󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝+ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑎^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑎^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝 + (∑^𝑛

𝑗=1

𝑀_𝑖𝑗) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝

+∑^𝑛

𝑗=1𝑀_𝑖𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝 + (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑁^𝑖𝑗󵄨󵄨󵄨󵄨󵄨) (𝑝 − 1)󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑁^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 [󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝]

𝜏

+ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛼^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛼^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗[󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝]

𝜏

+ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛽^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛽^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗[󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝]

𝜏

+ (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛾^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨 𝑑𝑠) (𝑝 − 1)󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝛾^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡 − 𝑠)󵄨󵄨󵄨󵄨󵄨^𝑝𝑑𝑠 + (∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝜃^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨 𝑑𝑠) (𝑝 − 1)󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝 +∑^𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝜃^𝑖𝑗󵄨󵄨󵄨󵄨󵄨 𝐿^𝑗∫^+∞

0 󵄨󵄨󵄨󵄨󵄨𝑘^𝑖𝑗(𝑠)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡 − 𝑠)󵄨󵄨󵄨󵄨󵄨^𝑝𝑑𝑠

+(𝑝 − 1) (𝑝 − 2)

2 (∑^𝑛

𝑗=1𝑐_𝑖𝑗) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝+ (𝑝 − 1)

×∑^𝑛

𝑗=1

𝑐_𝑖𝑗󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝+(𝑝 − 1) (𝑝 − 2) 2

× (∑^𝑛

𝑗=1

𝑑_𝑖𝑗) 󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨^𝑝+ (𝑝 − 1)∑^𝑛

𝑗=1

𝑑_𝑖𝑗[󵄨󵄨󵄨󵄨󵄨𝑒^𝑗(𝑡)󵄨󵄨󵄨󵄨󵄨^𝑝]

𝜏. (37) It follows from (𝐴₄) and (32) that

L𝑉_𝑖(𝑒 (𝑡)) ≤ 𝑑_𝑖𝑉_𝑖(𝑒 (𝑡)) +∑^𝑛

𝑗=1

(𝑝_𝑖𝑗𝑉_𝑗(𝑒 (𝑡)) + 𝑞_𝑖𝑗[𝑉_𝑗(𝑒 (𝑡))]_𝜏

+ ∫^+∞

0 𝜐_𝑖𝑗(𝑠) 𝑉_𝑗(𝑒 (𝑡 − 𝑠)) 𝑑𝑠) . (38)

Substituting (38) into (33) gives 𝑑𝑉_𝑖(𝑒 (𝑡)) ≤ [𝑑_𝑖𝑉_𝑖(𝑒 (𝑡))

+∑^𝑛

𝑗=1

(𝑝_𝑖𝑗𝑉_𝑗(𝑒 (𝑡)) + 𝑞_𝑖𝑗[𝑉_𝑗(𝑒 (𝑡))]_𝜏

+ ∫^+∞

0 𝜐_𝑖𝑗(𝑠) 𝑉_𝑗(𝑒 (𝑡 − 𝑠)) 𝑑𝑠)] 𝑑𝑡 +𝜕𝑉_𝑖(𝑒)

𝜕𝑒 𝜎𝑑𝑤 (𝑡) , 𝑡 ≥ 𝑡₀, 𝑡 ̸= 𝑡_𝑘, 𝑘 ∈N.

(39) Integrating and taking the expectations on both sides of (39) lead to

E𝑉_𝑖(𝑒 (𝑡 + 𝛿)) −E𝑉_𝑖(𝑒 (𝑡))

≤ ∫^𝑡+𝛿

𝑡

[ [

𝑑_𝑖E𝑉_𝑖(𝑒 (𝑢))

+∑^𝑛

𝑗=1

(𝑝_𝑖𝑗E𝑉_𝑗(𝑒 (𝑢))

+ 𝑞_𝑖𝑗E[𝑉_𝑗(𝑒 (𝑢))]_𝜏 + ∫^+∞

0 𝜐_𝑖𝑗(𝑠)

×E𝑉_𝑗(𝑒 (𝑢 − 𝑠)) 𝑑𝑠) ] ]

𝑑𝑢,

(40)

where𝛿 > 0is small enough such that𝑡, 𝑡 + 𝛿 ∈ [𝑡_𝑘−1, 𝑡_𝑘)for 𝑘 ∈N.

0 1 2 3 4

0 2 4 6 8

10 Phase plot of drive system

−4 −3 −2 −1

−10

−6

−4

−8

−2

𝑥₁ 𝑥2

Figure 1: Chaos behavior of drive system.

By the continuity ofE𝑉_𝑖(𝑒(𝑡)), we conclude that 𝐷⁺E𝑉_𝑖(𝑒 (𝑡)) ≤ 𝑑_𝑖E𝑉_𝑖(𝑒 (𝑡))

+∑^𝑛

𝑗=1

(𝑝_𝑖𝑗E𝑉_𝑗(𝑒 (𝑡)) + 𝑞_𝑖𝑗E[𝑉_𝑗(𝑒 (𝑡))]_𝜏

+ ∫^+∞

0 𝜐_𝑖𝑗(𝑠)E𝑉_𝑗(𝑒 (𝑡 − 𝑠)) 𝑑𝑠) , (41) which implies that for𝑡 ̸= 𝑡_𝑘, 𝑘 ∈N,

𝐷⁺E𝑉 (𝑒 (𝑡)) ≤ 𝐷₀E𝑉 (𝑒 (𝑡))

+ 𝑃E𝑉 (𝑒 (𝑡)) + 𝑄E[𝑉 (𝑒 (𝑡))]_𝜏 + ∫^+∞

0 Υ (𝑠)E𝑉 (𝑒 (𝑡 − 𝑠)) 𝑑𝑠.

(42)

Meanwhile, it follows from (𝐴₂) and (32) that E𝑉_𝑖(𝑒 (𝑡_𝑘))

=E󵄨󵄨󵄨󵄨𝐼^𝑖𝑘(𝑦_𝑖(𝑡⁻_𝑘)) − 𝐼_𝑖𝑘(𝑥_𝑖(𝑡⁻_𝑘)) + 𝑒_𝑖(𝑡⁻_𝑘)󵄨󵄨󵄨󵄨^𝑝

≤ 𝜂_𝑘^𝑝E𝑉_𝑖(𝑒 (𝑡⁻_𝑘))

(43)

for𝑖 ∈Nand𝑘 ∈N, which means that

E𝑉 (𝑒 (𝑡_𝑘)) ≤ 𝜂^𝑝_𝑘E𝑉 (𝑒 (𝑡⁻_𝑘)) . (44) Obviously, (42) and (44) indicate that E𝑉(𝑒(𝑡)) satisfies inequality (14) inLemma 6.

On the other hand, noting that𝑒(𝑡₀+ 𝑠) = 𝜓(𝑠) − 𝜙(𝑠)in (8) and by a simple calculation, we get

E󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡₀+ 𝑠)󵄨󵄨󵄨󵄨^𝑝=E󵄨󵄨󵄨󵄨𝜓^𝑖(𝑠) − 𝜙_𝑖(𝑠)󵄨󵄨󵄨󵄨^𝑝≤ 󵄩󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩󵄩^𝑝F (45)