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

Authors: Bing Li; Qiankun Song
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The synchronization problem of chaotic fuzzy cellular neural networks with mixed delays is investigated. By an impulsive integrodifferential inequality and the Itô'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.


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## Body

## 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 concept 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 differentiable, 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 perturbations 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 significance 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 nonsingularℳ-matrix and the Ito^’s formula, we design some synchronization schemes with a state feedback controller 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 Section2, we introduce the drive-response models and some preliminaries. In Section 3, some synchronization criteria for FCNNs with mixed delays are derived. In Section 4, an example and its simulations are given to illustrate the effectiveness of theoretical results. Finally, conclusions are drawn in Section 5.


## 2. Model Description and Preliminaries

Letℝn be the space of n-dimensional real column vectors, and let ℝm×n represent the class of m×n matrices with real components. |·| denotes the Euclidean norm in ℝn. The inequality “≤” (“>”) between matrices or vectors such as A≤B (A>B) means that each pair of corresponding elements of A and B satisfies the inequality “≤” (“>”). A∈ℝm×n is called a nonnegative matrix if A≥0, and x∈ℝn is called a positive vector if x>0. The transpose of A∈ℝm×n or x∈ℝn is denoted by AT or xT. Let E denote the unit matrix with appropriate dimensions. 𝒩:={1,2,…,n}, and ℕ:={1,2,…}, ℝ+:=[0,+∞).ℙ
ℂ
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ψ
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ψ
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) is piecewise continuous and satisfies ∫0+∞|ψ(s)|eσ0sds<+∞ for some constant σ0>0}.ForA,B∈ℝn×n and ϕ:ℝ→ℝn, we denote that
(1)[A]+=(|aij|)n×n,A∘B=(aijbij)n×n,[ϕ]+=(|ϕ1|,…,|ϕn|)T,[ϕ(t)]τ=([ϕ1(t)]τ,…,[ϕn(t)]τ)T,where[ϕi(t)]τ=sup-τ≤s≤0ϕi(t+s),i∈𝒩,
and D+ϕ(t) denotes the upper-right derivative of ϕ(t) at time t.Consider IFCNNs with mixed delays as follows:(2)dxi(t)dt=-cixi+∑j=1n‍aijfj(xj)+∑j=1n‍bijνj+Ji+⋀j=1nTijμj+⋀j=1nαijfj(xj(t-τij(t)))+⋀j=1nγij∫0+∞kij(s)fj(xj(t-s))ds+⋁j=1nSijμj+⋁j=1nβijfj(xj(t-τij(t)))+⋁j=1nθij∫0+∞kij(s)fj(xj(t-s))ds,t≥t0,t≠tk,Δxi(tk)=xi(tk+)-xi(tk-)=Iik(xi(tk-)),k∈ℕ,xi(t0+s)=ϕi(s),-∞<s≤0,
where i=1,2,…,n, n denotes the number of units in the neural network. x(t)=(x1(t),…,xn(t))T represents the state variable. fj(·) is the activation function of the jth neuron. ci represents the passive decay rate to the state of ith neuron at time t. αij and γij are elements of the fuzzy feedback MIN template. βij and θij are elements of the fuzzy feedback MAX template. Tij and Sij are elements of fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively. aij and bij are elements of feedback and feed-forward template, respectively. ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operations, respectively. νi and Ji denote input and bias of the ith neuron, respectively. For any i,j∈𝒩, τij(t) corresponding to the transmission delay satisfies 0≤τij(t)≤τ, and kij∈𝕃e is the feedback kernel. For any k∈ℕ, Ik(·) represents the impulsive perturbation, and tk denotes impulsive moment satisfying tk<tk+1, limk→+∞tk=+∞.We make the following assumptions throughout this paper.(A1)
fi is globally Lipschitz continuous, that is, for any i∈𝒩, there exists nonnegative constant Li such that
(3)|fi(u)-fi(v)|≤Li|u-v|foru,v∈ℝ.(A2)
For anyk∈ℕ, there is a nonnegative constant ηk such that
(4)|u+Iik(u)-v-Iik(v)|≤ηk|u-v|foru,v∈ℝ,i∈𝒩.Let (2) be the drive system, and let the response system with random noise be described by
(5)dyi(t)=[-ciyi+∑j=1n‍aijfj(yj)+∑j=1n‍bijνj+Ji+⋀j=1nTijμj+⋀j=1nαijfj(yj(t-τij(t)))+⋀j=1nγij∫0+∞kij(s)fj(yj(t-s))ds+⋁j=1nSijμj+⋁j=1nβijfj(yj(t-τij(t)))+⋁j=1nθij∫0+∞kij(s)fj(yj(t-s))ds+Ui(t)]dt+∑j=1n‍σij(t,xj(t)-yj(t),xj(t-τij(t))mmmmmm-yj(t-τij(t)))dwj(t),t≥t0,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-)=Iik(yi(tk-)),k∈ℕ,yi(t0+s)=ψi(s),-∞<s≤0,
where w(t)=(w1(t),…,wn(t))T is an n-dimensional standard Brownian motion defined on a complete probability space (Ω,ℱ,𝒫) with a natural filtration {ℱt}t≥0 generated by {w(s):0≤s≤t} and satisfying the usual conditions (i.e., it is right continuous, and ℱ0 contains all 𝒫-null sets). The initial value ψ=(ψ1(s),…,ψn(s))T∈ℙℂℱ0b[(-∞,0],ℝn] which denotes the family of all bounded ℱ0-measurable and ℙℂ-valued random variables ψ with the norm ∥ψ∥ℱp=sup-∞<s≤0E|ψ(s)|p, where E denotes the expectation of stochastic process. U(t)=(U1(t),…,Un(t))T is the state feedback controller designed by
(6)Ui(t)=∑j=1n‍Mij(yj(t)-xj(t))+∑j=1n‍Nij(yj(t-τij(t))-xj(t-τij(t))),
where M=(Mij)n×n, N=(Nij)n×n are the controller gain matrices to be scheduled. The diffusion coefficient matrix (or noise intensity matrix) σ:ℝ×ℝn×ℝn→ℝn×n satisfies the local Lipschitz condition and the linear growth condition (see [33]). In addition,(A3)
fori∈𝒩, there exist nonnegative constants cij, dij such that
(7)σiσiT≤∑j=1n‍cij|xj-yj|2+∑j=1n‍dij|xj(t-τij(t))-yj(t-τij(t))|2,where σi=(σi1,…,σin).Lete(t)=(e1(t),…,en(t))T, where ei(t)=yi(t)-xi(t), be the synchronization error. Then, the error dynamical system between (2) and (5) is given by
(8)dei(t)=[-ciei+∑j=1n‍aij(fj(yj)-fj(xj))+∑j=1n‍Mijej(t)+∑j=1n‍Nijej(t-τij(t))+⋀j=1nαijfj(yj(t-τij(t)))-⋀j=1nαijfj(xj(t-τij(t)))+⋁j=1nβijfj(yj(t-τij(t)))-⋁j=1nβijfj(xj(t-τij(t)))+⋀j=1nγij∫0+∞kij(s)fj(yj(t-s))ds-⋀j=1nγij∫0+∞kij(s)fj(xj(t-s))ds+⋁j=1nθij∫0+∞kij(s)fj(yj(t-s))ds-⋁j=1nθij∫0+∞kij(s)fj(xj(t-s))ds]dt+∑j=1n‍σij(t,ej(t),ej(t-τij(t)))dwj(t),t≥t0,t≠tk,Δei(tk)=ei(tk+)-ei(tk-)=Iik(yi(tk-))-Iik(xi(tk-)),k∈ℕ,ei(t0+s)=ψi(s)-ϕi(s),-∞<s≤0.For convenience, we use the following notations:D1=diag{-c1,…,-cn}, L=diag{L1,…,Ln}, K(s)=(kij(s))n×n, A=(aij)n×n, M-=(M-ij)n×n with M-ii=Mii, M-ij=|Mij| for i≠j, α=(αij)n×n, β=(βij)n×n, Γ=(γij)n×n, Θ=(θij)n×n, C=(cij)n×n, and D=(dij)n×n.The following definition and lemmas will be employed.Definition 1.
The systems (2) and (5) are called to be globally exponentially synchronized in p-moment, if there exist positive constants λ, K such that
(9)E|e(t)|p≤K∥ψ-ϕ∥ℱpe-λ(t-t0),t≥t0.
It is said especially to be globally exponentially synchronized in mean square when p=2.
For any nonsingularℳ-matrix A (see [34]), we define that
(10)ℳA={z∈ℝn∣Az>0,z>0}.Lemma 2 (see [35]).
For a nonsingularℳ-matrix A, ℳA is a nonempty cone without conical surface.Lemma 3 (see [36]).
Forxi≥0, αi>0, and ∑i=1nαi=1,
(11)∏i=1n‍xiαi≤∑i=1n‍αixi.
The sign of equality holds if and only if xi=xj for all i,j∈𝒩.Lemma 4 (see [1]).
Letαij, βij∈ℝ and x, y∈ℝn be the two states of the system (2). Then, one has
(12)|⋀j=1nαijfj(xj)-⋀j=1nαijfj(yj)|≤∑j=1n‍|αij||fj(xj)-fj(yj)|,|⋁j=1nβijfj(xj)-⋁j=1nβijfj(yj)|≤∑j=1n‍|βij||fj(xj)-fj(yj)|.Lemma 5 (see [36]).
For the integerp≥2 and x=(x1,…,xn)T∈ℝn, there exists a positive constant ep(n) such that
(13)ep(n)(∑i=1n‍xi2)p/2≤∑i=1n‍|xi|p.Lemma 6.
Fork∈ℕ, assume that v=(v1(t),…,vn(t))T∈ℙℂ[(-∞,+∞),ℝn] satisfies
(14)D+v(t)≤A0v(t)+Pv(t)+Q[v(t)]τ+∫0+∞Υ(s)v(t-s)ds,t≥t0,t≠tk,v(tk+)≤ρkv(tk-),ρk≥0,vt0∈ℙℂ,wherevt0=v(t0+s),s∈(-∞,0],
in which(C1)
A0=diag{a1,…,an}, P=(pij)n×n with pij≥0 for i≠j, Q=(qij)n×n≥0, Υ(s)=(υij(s))n×n≥0, and υij∈𝕃e, i,j∈𝒩.(C2)
Π=-(A0+P+Q+∫0+∞Υ(s)ds) is a nonsingular ℳ-matrix.Then, there must exist z=(z1,…,zn)T∈ℳΠ and λ∈(0,σ0] such that
(15)v(t)≤(∏j=0k-1‍ϱj)ze-λ(t-t0),tk-1≤t<tk,k∈ℕ,
provided that the initial value vt0 satisfies
(16)v(t)≤ze-λ(t-t0),-∞<t≤t0,
where ϱ0=1, ϱk=max{1,ρk}, and z, λ can be determined by
(17)(λE+A0+P+Qeλτ+∫0+∞Υ(s)eλsds)z<0.Proof.
By condition(C2) and Lemma 2, we can find z~=(z~1,…,z~n)T∈ℳΠ such that Πz~>0, namely, (A0+P+Q+∫0+∞Υ(s)ds)z~<0. By the continuity, there must be some positive constant λ∈(0,σ0] satisfying
(18)(λE+A0+P+Qeλτ+∫0+∞Υ(s)eλsds)z~<0.
Noting that vt0∈ℙℂ, we can find a constant B>0 such that ∥vt0∥≤B. Denote that z:=(B/mini∈𝒩z~i)z~=(z1,…,zn)T. Obviously, z and λ satisfy (16) and (17).
Letwi(t):=zie-λ(t-t0) for t∈ℝ, i∈𝒩. For any small enough ϵ>0, (16) implies that vi(t)≤wi(t)<(1+ϵ)wi(t), t∈(-∞,t0]. Next, we claim that for any t∈[t0,t1),
(19)vi(t)<(1+ϵ)wi(t),i∈𝒩.
If inequality (19) is not true, then there must exist some m∈𝒩 and t*∈(t0,t1) such that
(20)vm(t*)=(1+ϵ)wm(t*),vi(t)<(1+ϵ)wi(t),t∈(-∞,t*),i∈𝒩,(21)D+vm(t*)≥(1+ϵ)wm′(t*).
On the other hand, (14) together with (17) and (20) leads to
(22)D+vm(t*)≤amvm(t*)+∑j=1n‍pmjvj(t*)+∑j=1n‍qmj[vj(t*)]τ+∑j=1n‍∫0+∞υmj(s)vj(t*-s)ds≤(1+ϵ)e-λ(t*-t0)amzm+(1+ϵ)e-λ(t*-t0)∑j=1n‍pmjzj+(1+ϵ)e-λ(t*-t0)∑j=1n‍qmjzjeλτ+(1+ϵ)e-λ(t*-t0)×∑j=1n‍∫0+∞υmj(s)eλsdszj<(1+ϵ)e-λ(t*-t0)(-λzm)=(1+ϵ)wm′(t*),
which contradicts (21). Therefore, (19) holds. Letting ϵ→0+ in (19), we get
(23)vi(t)≤wi(t)=ϱ0wi(t),t∈[t0,t1),i∈𝒩.
Suppose that for ν=1,2,…,k, the following inequalities hold
(24)vi(t)≤(∏m=0ν-1‍ϱm)wi(t),t∈[tν-1,tν),i∈𝒩.
For t=tk, from (14) and (24), we have
(25)vi(tk)≤ρkvi(tk-)≤ϱk(∏m=0k-1‍ϱm)wi(tk)vi(tk)≤(∏m=0k‍ϱm)wi(tk),i∈𝒩.
Recalling ρk≥1, it follows from (24) and (25) that
(26)vi(t)≤(∏m=0k‍ϱm)wi(t),t∈(-∞,tk],i∈𝒩.
Repeating the proof similar to (19) can yield
(27)vi(t)≤(∏m=0k‍ϱm)wi(t),t∈[tk,tk+1),i∈𝒩.
By the mathematical induction, we derive that for any k∈ℕ,
(28)vi(t)≤(∏j=0k-1‍ϱj)wi(t)≤(∏j=0k-1‍ϱj)zie-λ(t-t0),t∈[tk-1,tk),i∈𝒩.
The proof is completed.


## 3. Exponential Synchronization

In this section, by using Lemma6, 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~=-(D0+P-+Q-+∫0+∞Υ-(s)ds) is a nonsingular ℳ-matrix, where P-=[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,…,dn} with
(29)di=-pci+(p-1)∑j=1n‍[(∫0+∞(|γij|+|θij|)|kij(s)|ds|aij|+|αij|+|βij|mmmmmmmmmmm+∫0+∞(|γij|+|θij|)|kij(s)|ds)Ljmmmmmmmmmm+p-22(cij+dij)+M-ij+|Nij|],i∈𝒩,(A5)
the impulsive perturbations satisfy(30)supk∈ℕlnζktk-tk-1<λ,where ζk=max{1,ηkp}, and λ∈(0,σ0] is determined by
(31)(λE+D0+P-+Q-eλτ+∫0+∞Υ-(s)eλsds)z<0,foragivenz∈ℳD~.
Then, drive-response systems (2) and (5) are globally exponential synchronization in p-moment.Proof.
SinceD~=-(D0+P-+Q-+∫0+∞Υ-(s)ds) is a nonsingular ℳ-matrix, by Lemma 2 and the continuity, there must be a constant vector z=(z1,…,zn)T∈ℳD~ and a constant λ∈(0,σ0] such that (31) holds.
We denote bye=(e1,…,en)T the solution of error dynamical system (8) with the initial value ψ-ϕ∈ℙℂℱ0b[(-∞,0],ℝn] and let
(32)V(e)=(V1(e),…,Vn(e))T,Vi(e)=|ei|p,i∈𝒩.
Calculating the time derivative of Vi(e(t)) along the trajectory of error system (8) and by the Itô’s formula [33], we get for any k∈ℕ,
(33)dVi(e(t))=ℒVi(e(t))dt+∂Vi(e)∂eσdw(t),t≥t0,t≠tk,
where ℒVi(e(t)) is given by
(34)ℒVi(e(t))=p|ei(t)|p-2ei(t)×[-ciei+∑j=1n‍aijmm×(fj(yj)-fj(xj))+∑j=1n‍Mijej(t)mm+∑j=1n‍Nijej(t-τij(t))mm+⋀j=1nαijfj(yj(t-τij(t)))mm-⋀j=1nαijfj(xj(t-τij(t)))mm+⋁j=1nβijfj(yj(t-τij(t)))mm-⋁j=1nβijfj(xj(t-τij(t)))+⋀j=1nγijmm×∫0+∞kij(s)fj(yj(t-s))ds-⋀j=1nγijmm×∫0+∞kij(s)fj(xj(t-s))dsmm+⋁j=1nθij∫0+∞kij(s)fj(yj(t-s))dsmm-⋁j=1nθij∫0+∞kij(s)fj(xj(t-s))ds]+12p(p-1)|ei(t)|p-2σiσiT.
By (A1) and Lemma 4, we have
(35)ℒVi(e(t))≤-pci|ei(t)|p+p|ei(t)|p-1×∑j=1n‍|aij|Lj|ej(t)|+p|ei(t)|p-1×∑j=1n‍M-ij|ej(t)|+p|ei(t)|p-1×∑j=1n‍|Nij||ej(t-τij(t))|+p|ei(t)|p-1∑j=1n‍|αij|×Lj|ej(t-τij(t))|+p|ei(t)|p-1×∑j=1n‍|βij|Lj|ej(t-τij(t))|+p|ei(t)|p-1×∑j=1n‍|γij|Lj∫0+∞|kij(s)||(t-s)|ds+p|ei(t)|p-1∑j=1n‍|θij|Lj∫0+∞|kij(s)|×|ej(t-s)|ds+12p(p-1)|ej(t)|p-2σiσiT:=-pci|ei(t)|p+I1+I2+I3+I4+I5+I6+I7+I8.
Using Lemma 3 and (A3), it is easy to get
(36)I1≤(∑j=1n‍|aij|Lj)(p-1)|ei(t)|p+∑j=1n‍|aij|Lj|ej(t)|p,I2≤(∑j=1n‍M-ij)(p-1)|ei(t)|p+∑j=1n‍M-ij|ej(t)|p,I3≤(∑j=1n‍|Nij|)(p-1)|ei(t)|p+∑j=1n‍|Nij|[|ej(t)|p]τ,I4≤(∑j=1n‍|αij|Lj)(p-1)|ei(t)|p+∑j=1n‍|αij|Lj[|ej(t)|p]τ,I5≤(∑j=1n‍|βij|Lj)(p-1)|ei(t)|p+∑j=1n‍|βij|Lj[|ej(t)|p]τ,I6≤(∑j=1n‍|γij|Lj∫0+∞|kij(s)|ds)(p-1)|ei(t)|p+∑j=1n‍|γij|Lj∫0+∞|kij(s)||ej(t-s)|pds,I7≤(∑j=1n‍|θij|Lj∫0+∞|kij(s)|ds)(p-1)|ei(t)|p+∑j=1n‍|θij|Lj∫0+∞|kij(s)||ej(t-s)|pds,I8≤12p(p-1)|ei(t)|p-2∑j=1n‍cij|ej(t)|2+12p(p-1)|ei(t)|p-2∑j=1n‍dij|ej(t-τij(t))|2≤(p-1)(p-2)2(∑j=1n‍cij)|ei(t)|p+(p-1)∑j=1n‍cij|ej(t)|p+(p-1)(p-2)2(∑j=1n‍dij)|ei(t)|p+(p-1)∑j=1n‍dij[|ej(t)|p]τ.
Thus, we have
(37)ℒVi(e(t))≤-pci|ei(t)|p+(∑j=1n‍|aij|Lj)(p-1)|ei(t)|p+∑j=1n‍|aij|Lj|ej(t)|p+(∑j=1n‍M-ij)(p-1)|ei(t)|p+∑j=1n‍M-ij|ej(t)|p+(∑j=1n‍|Nij|)(p-1)|ei(t)|p+∑j=1n‍|Nij|[|ej(t)|p]τ+(∑j=1n‍|αij|Lj)(p-1)|ei(t)|p+∑j=1n‍|αij|Lj[|ej(t)|p]τ+(∑j=1n‍|βij|Lj)(p-1)|ei(t)|p+∑j=1n‍|βij|Lj[|ej(t)|p]τ+(∑j=1n‍|γij|Lj∫0+∞|kij(s)|ds)(p-1)|ei(t)|p+∑j=1n‍|γij|Lj∫0+∞|kij(s)||ej(t-s)|pds+(∑j=1n‍|θij|Lj∫0+∞|kij(s)|ds)(p-1)|ei(t)|p+∑j=1n‍|θij|Lj∫0+∞|kij(s)||ej(t-s)|pds+(p-1)(p-2)2(∑j=1n‍cij)|ei(t)|p+(p-1)×∑j=1n‍cij|ej(t)|p+(p-1)(p-2)2×(∑j=1n‍dij)|ei(t)|p+(p-1)∑j=1n‍dij[|ej(t)|p]τ.
It follows from (A4) and (32) that
(38)ℒVi(e(t))≤diVi(e(t))+∑j=1n(∫0+∞υ-ij(s)p-ijVj(e(t))+q-ij[Vj(e(t))]τ‍mmmm+∫0+∞υ-ij(s)Vj(e(t-s))ds).
Substituting (38) into (33) gives
(39)dVi(e(t))≤[∫0+∞υ-ij(s)diVi(e(t))m+∑j=1n‍(∫0+∞υ-ij(s)p-ijVj(e(t))+q-ij[Vj(e(t))]τm+∫0+∞υ-ij(s)Vj(e(t-s))ds)]dt+∂Vi(e)∂eσdw(t),t≥t0,t≠tk,k∈ℕ.
Integrating and taking the expectations on both sides of (39) lead to
(40)EVi(e(t+δ))-EVi(e(t))≤∫tt+δ[∑j=1n‍diEVi(e(u))mmmmmm+∑j=1n‍(∫0+∞p-ijEVj(e(u))mmmggggggggggm+q-ijE[Vj(e(u))]τmmmggggggggggm+∫0+∞υ-ij(s)mmmmmmmmmm×EVj(e(u-s))ds∫0+∞υ-ij(s))∑j=1n‍]du,
where δ>0 is small enough such that t,t+δ∈[tk-1,tk) for k∈ℕ.
By the continuity ofEVi(e(t)), we conclude that
(41)D+EVi(e(t))≤diEVi(e(t))+∑j=1n‍(∫0+∞p-ijEVj(e(t))+q-ijE[Vj(e(t))]τmmmm+∫0+∞υ-ij(s)EVj(e(t-s))ds),
which implies that for t≠tk, k∈ℕ,
(42)D+EV(e(t))≤D0EV(e(t))+P-EV(e(t))+Q-E[V(e(t))]τ+∫0+∞Υ-(s)EV(e(t-s))ds.
Meanwhile, it follows from (A2) and (32) that
(43)EVi(e(tk))=E|Iik(yi(tk-))-Iik(xi(tk-))+ei(tk-)|p≤ηkpEVi(e(tk-))
for i∈𝒩 and k∈ℕ, which means that
(44)EV(e(tk))≤ηkpEV(e(tk-)).
Obviously, (42) and (44) indicate that EV(e(t)) satisfies inequality (14) in Lemma 6.
On the other hand, noting thate(t0+s)=ψ(s)-ϕ(s) in (8) and by a simple calculation, we get
(45)E|ei(t0+s)|p=E|ψi(s)-ϕi(s)|p≤∥ψ-ϕ∥ℱp
for any s∈(-∞,0], which means E|ei(t)|p≤∥ψ-ϕ∥ℱp for t∈(-∞,t0] and i∈𝒩. Recalling the definition of V(e(t)) in (32) and z>0, we conclude that for t∈(-∞,t0](46)EV(e(t))≤∥ψ-ϕ∥ℱp(1,…,1)T≤∥ψ-ϕ∥ℱp(z1mini∈𝒩{zi},…,znmini∈𝒩{zi})T≤∥ψ-ϕ∥ℱpmini∈𝒩{zi}z,
which further indicates that
(47)EV(e(t))≤z-e-λ(t-t0),t∈(-∞,t0],
where z-=(∥ψ-ϕ∥ℱp/mini∈𝒩{zi})z. This implies that condition (16) in Lemma 6 holds.
Therefore, by Lemma6, we derive that
(48)EV(e(t))≤(∏j=0k-1‍ζj)z-e-λ(t-t0),tk-1≤t<tk,k∈ℕ,
with ζ0=1. Meanwhile, (30) implies that there is a small enough constant ϵ(0<ϵ<λ) such that
(49)ζk≤e(λ-ϵ)(tk-tk-1),k∈ℕ.
Thus, inequality (48) together with (49) shows that for k=1(50)EV(e(t))≤ζ0z-e-λ(t-t0)≤z-e-ϵ(t-t0),t0≤t<t1,
and for any k≥2,
(51)EV(e(t))≤z-ζ0ζ1⋯ζk-1e-λ(t-t0)≤z-e(λ-ϵ)(t1-t0)⋯e(λ-ϵ)(tk-1-tk-2)e-λ(t-t0)=z-e(λ-ϵ)(tk-1-t0)e-λ(t-t0)≤z-e(λ-ϵ)(t-t0)e-λ(t-t0)≤z-e-ϵ(t-t0),tk-1≤t<tk.
By Lemma 5, we get
(52)E|e(t)|p≤K∥ψ-ϕ∥ℱpe-ϵ(t-t0),t≥t0,
where K=(∑i=1nzi)/(ep(n)mini∈𝒩{zi}). The proof is complete.Remark 8.
In [25–32], the authors established some useful criteria for ensuring synchronization of FCNNs with delays, respectively. However, once the unbounded distributed delays are involved, all results in [25–32] will be invalid. Hence, in this sense, the proposed Theorem 7 has a wider range of applications than those in previous papers.Remark 9.
In synchronization scheme, we take both impulsive perturbations and random noise into account. Comparing with the results in [25–32], Theorem 7 can reflect a more realistic dynamical behavior and synchronization procedure.
If the random noise has not been considered, which meansσ≡0 in (5), then the response system reduces to
(53)dyi(t)dt=-ciyi+∑j=1n‍aijfj(yj)+∑j=1n‍bijνj+Ji+⋀j=1nTijμj+⋀j=1nαijfj(yj(t-τij(t)))+⋀j=1nγij∫0+∞kij(s)fj(yj(t-s))ds+⋁j=1nSijμj+⋁j=1nβijfj(yj(t-τij(t)))+⋁j=1nθij∫0+∞kij(s)fj(yj(t-s))ds+Ui(t),t≥t0,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-)=Iik(yi(tk-)),k∈ℕ,yi(t0+s)=ψi(s),-∞<s≤0.
In this case, the following globally exponential synchronization scheme for drive-response IFCNNs (2) and (53) can be derived.Theorem 10.
Assume that(A1) and (A2) hold and(A6)
D^=-(D1+P-1+Q-1+∫0+∞Υ-(s)ds) is a nonsingular ℳ-matrix, where P-1=[A]+L+M-, Q-1=([α]++[β]+)L+[N]+,(A7)
the impulsive perturbations satisfy(54)supk∈Nlnξktk-tk-1<λ,where ξk=max{1,ηk}, and λ∈(0,σ0] is determined by
(55)(λE+D1+P-1+Q-1eλτ+∫0+∞Υ-(s)eλsds)z<0,foragivenz∈ℳD^.
Then, the drive-response systems (2) and (53) are globally exponential synchronization.Proof.
LetV(e(t))=[e(t)]+=(|e1(t)|,…,|en(t)|)T. Calculating the time derivative of V(e(t)) along with the trajectory of error system can give
(56)D+V(e(t))≤D1V(e(t))+P-1V(e(t))+Q-1[V(e(t))]τ+∫0+∞Υ-(s)V(e(t-s))ds.
The rest proof is similar to Theorem 7 and omitted here. We complete the proof.Remark 11.
In [31], Feng et al. derived some criteria on the globally exponential synchronization for a special case of (2) and (53) with γij=θij=0, τij(t)=τij and Iik(xi(tk-))=-ηikxi(tk-) for i,j∈𝒩, k∈ℕ. In order to achieve synchronous control, the conditions as 0≤ηik≤2 for i∈𝒩, k∈ℕ have been imposed on the impulsive perturbations. However, Theorem 10 drops these restrictions.


## 4. Illustrative Example

In this section, a numerical example and its simulations are given to illustrate the effectiveness of our results.Example 1.
Consider the following 2-dimensional IFCNNs with mixed delays as the drive system(57)dxi(t)dt=-cixi+∑j=12‍aijfj(xj)+⋀j=12αijfj(xj(t-1))+⋁j=12βijfj(xj(t-1))+⋀j=12γij∫0+∞kij(s)fj(xj(t-s))ds+⋁j=12θij∫0+∞kij(s)fj(xj(t-s))ds,t≠tk,Δxi(tk)=xi(tk+)-xi(tk-)=-Jkxi(tk-),
where i,j=1,2, fj(u)=tanh(u). Jk=e0.15+1,tk=tk-1+4 for k∈ℕ. For the simplicity of computer simulations, we choose kij(s)=e-s for s∈[0,20], kij(s)=0 for s∈[20,+∞). The system parameters are as follows:
(58)D1=(-100-1),A=(2-0.1-43.2),α=β=(-1.3-0.2-0.2-4.2),Γ=Θ=(-0.5-0.55-2.5).
We can choose σ0=0.8 and L1=L2=1 such that kij∈𝕃e and (A1) holds, respectively. Obviously, (A2) holds with ηk=e0.15, k∈ℕ.
Choosing the initial valueϕ(s)=(3,-6)T for s∈(-∞,0], the drive system (57) possesses a chaotic behavior as shown in Figure 1.Figure 1
Chaos behavior of drive system.Case 1.
The response system without random noise is given by(59)dyi(t)dt=-ciyi+∑j=12‍aijfj(yj)+⋀j=12αijfj(yj(t-1))+⋁j=12βijfj(yj(t-1))+⋀j=12γij∫0+∞kij(s)fj(yj(t-s))ds+⋁j=12θij∫0+∞kij(s)fj×(yj(t-s))ds+Ui(t),t≠tk,Δyi(tk)=yi(tk+)-yi(tk-)=-Jkyi(tk-).
The control gain matrices M and N can be chosen as
(60)M=(-10.600-20.6),N=(e-20e-2010e-205e-20).
By simple calculation, we get that
(61)D^=(-61.514.4-5)
is a nonsingular ℳ-matrix, which implies that (A6) holds. Moreover, we can choose λ=0.2 and z=(1,3.5)T∈ℳD^ such that (A7) holds. Therefore, by Theorem 10, the drive-response systems (57) and (59) are globally exponentially synchronized. The simulation result with ψ(s)=(1.3,1.8)T is shown in Figure 2.Figure 2
Synchronization of (57) and (59).Remark 12.
It is worth noting that the impulsive perturbationsJk>2, which are not a satisfied condition (H2) in [31]. That is to say, even in the absence of the unbounded distributed delays, the results in [31] still cannot be applied to the synchronization problem of (57) and (59).Case 2.
Consider the response system with random noise as follows:(62)dyi(t)=[-ciyi+∑j=12‍aijfj(yj)+⋀j=12αijfj(yj(t-1))+⋁j=12βijfj(yj(t-1))+⋀j=12γij∫0+∞kij(s)fj(yj(t-s))ds+⋁j=12θij∫0+∞kij(s)×fj(yj(t-s))ds+Ui(t)⋁j=12θij]dt+∑j=12‍σij((t-τij(t))t,xj(t)-yj(t),xj(t-τij(t))-yj(t-τij(t)))dwj(t),t≠tk,Δyi(tk)=yi(tk+)-yi(tk-)=-Jkyi(tk-),
and the noise intensity matrix is
(63)σ=(-x1(t)0.5x1(t-1)x2(t)-0.5x2(t-1)).
Clearly, we can choose
(64)C=(1001),D=(0.25000.25)
such that (A3) holds.
Forp=2, let control gain matrices be
(65)M=(-8.47500-25.925),N=(e-20e-2010e-205e-20).
It is easy to deduce that(66)D~=(-51.514.4-5)
is a nonsingular ℳ-matrix, which implies that (A4) holds. Meanwhile, we can choose λ=0.1 and z=(1,3)T∈ℳD~ such that (A5) holds. Hence, by Theorem 7, the drive-response systems (57) and (62) are globally exponentially synchronized in mean square. The simulation result based on Euler-Maruyama method is illustrated in Figure 3.Figure 3
Synchronization of (57) and (62).Remark 13.
The schemes proposed in [25–29] cannot solve the synchronization problem of (57) and (62) due to the impulsive perturbations and random noise. Besides this, the distributed delays make those methods in [30–32] cannot be applied to synchronization problem of (57) and (62).


## 5. Conclusion

In this paper, we investigate the synchronization problem of IFCNNs with mixed delays. Based on the properties of nonsingularℳ-matrix and some stochastic analysis approaches, some useful synchronization criteria under both impulse and random noise are obtained. The methods used in this paper are novel and can be extended to many other types of neural networks. These problems will be considered in near future.



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