# Stability in Switched Cohen-Grossberg Neural Networks with Mixed Time Delays and Non-Lipschitz Activation Functions

Authors: Huaiqin Wu; Guohua Xu; Chongyang Wu; Ning Li; Kewang Wang; Qiangqiang Guo
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

The stability for the switched Cohen-Grossberg neural networks
with mixed time delays andα-inverse Hölder activation functions is investigated
under the switching rule with the average dwell time property. By applying multiple Lyapunov-Krasovskii functional approach and linear matrix inequality (LMI) technique, a delay-dependent sufficient criterion is achieved to ensure such switched
neural networks to be globally exponentially stable in terms of LMIs, and the exponential decay estimation is explicitly developed for the states too. Two illustrative examples are given to demonstrate the validity of the theoretical results.


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## Body

## 1. Introduction

In the past few decades, there has been increasing interest in different classes of neural networks such as Hopfield, Cellular, Cohen-Grossberg, and bidirectional associative neural networks due to their potential applications in many areas such as classification, signal and image processing, parallel computing, associate memories, optimization, and cryptography [1–5]. In the design of practical neural networks, the qualitative analysis of neural network dynamics plays an important role. To solve problems of optimization, neural control, and signal processing, neural networks have to be designed in such a way that, for a given external input, they exhibit only one globally asymptotically/exponentially stable equilibrium point. Hence, much effort has been made in the stability of neural networks, and a number of sufficient conditions have been proposed to guarantee the global asymptotic/exponential stability for neural networks with or without delays in recent years, see, for example, [6–26] and the references therein.Recently, by combing the theories of the switched systems and neural networks, several classes of mathematics models of switched neural networks have been established. As a special class of switched systems, switched neural networks, whose individual subsystems are a set of neural networks, have found applications in fields of high speed signal processing, artificial intelligence, and gene selection in a DNA microarray analysis [27–30].Stability issues of switched neural networks have received great attention of researchers so far [31–39]. In [31], based on the Lyapunov-Krasovskii method and LMI approach, some sufficient conditions were derived for global robust exponential stability of a class of switched Hopfield neural networks with time-varying delay under uncertainty. In [32], by combining Cohen-Grossberg neural networks with an arbitrary switching rule, the mathematical model of a class of switched Cohen-Grossberg neural networks with mixed time varying delays was established, and the robust stability for such switched Cohen-Grossberg neural networks was analyzed. In [33], by employing nonlinear measure and LMI techniques, some new sufficient conditions were obtained to ensure global robust asymptotical stability and global robust exponential stability of the unique equilibrium for a class of switched recurrent neural networks with time-varying delay. In [34], authors investigated a large class of switched recurrent neural networks with time-varying structured uncertainties and time-varying delay, some delay-dependent robust periodicity criteria guaranteeing the existence, uniqueness, and global asymptotic stability of periodic solution for all admissible parametric uncertainties were devised by employing free weighting matrices and LMIs. In [35], a new class of switched interval neural networks with discrete and distributed time-varying delays of neural type was developed, and a delay-dependent sufficient criterion was also obtained in terms of LMIs which guarantee the global robust exponential stability for the proposed switched interval neural networks. It should be pointed out that results in [31–35] focused on the stability of switched neural networks under arbitrary switching rule by using common Lyapunov function method. However, common Lyapunov function method requires all the subsystems of the switched system to share a positive definite radially unbounded common Lyapunov function. Generally, this requirement is difficult to achieve.In past few years, much attention has been paid to making use of the dwell time approach to deal with the analysis and synthesis of switched neural networks [36–39]. It should be pointed out that the average dwell time method is regarded as an important and attractive method to find a suitable switching signal to guarantee switched system stability or improve other performance and has been widely applied to investigate the analysis and synthesis for switched system with or without time-delay, see for example, [40–44]. Very recently, in [36], based on multiple Lyapunov functions method and LMI techniques, the authors presented some sufficient conditions in terms of LMIs which guarantee the robust exponential stability for uncertain switched Cohen-Grossberg neural networks with interval time-varying delay and distributed time-varying delay under the switching rule with the average dwell time property. In [37], by using the average dwell time method, the delay-dependent sufficient conditions were derived towards the robust exponential stability for a class of discrete-time switched Hopfield neural networks with time delay. In [38], by applying a new Lyapunov-Krasovskii functional and the average dwell time method, a delay-range-dependent exponential stability criteria and decay estimation are presented in terms of LMIs for switched Hopfield neural networks.It should be noted that, all the results reported in [31–39] are concerned with switched neural networks with Lipschitz activation functions. To the best of our knowledge, very little attention has been paid to the problem of delay-dependent stability for switched neural networks without Lipschitz activation functions, which motivates the work of this paper.In this paper, our aim is to study the delay-dependent exponential stability problem for a class of switched Cohen-Grossberg neural networks with mixed time delays andα-inverse Hölder activation functions. Here, it should be pointed out that α-inverse Hölder activation functions are a class of non-Lipschitz functions. By applying Brouwer degree properties, LMI technique and constructing a novel Lyapunov-Krasovskii functional, the existence, uniqueness and global exponential stability of equilibrium point are proved for Cohen-Grossberg neural networks with mixed time delays and α-inverse Hölder activation functions. By means of the multiple Lyapunov-Krasovskii functional and the average dwell time approach, a delay-dependent sufficient condition in terms of LMIs is presented to ensure to the considered switched neural networks to be globally exponentially stable, and a explicit expression for the exponential decay estimation is also obtained for the states. Two illustrative examples are given to demonstrate the validity of the theoretical results.The rest of this paper is organized as follows. In Section2, the model formulation and some preliminaries are given. In Section 3, the existence, uniqueness, and global exponential stability of equilibrium point are proved for Cohen-Grossberg neural networks with mixed time delays and α-inverse Hölder activation functions. In Section 4, the global exponential stability criterion and state decay estimation are presented for the switched Cohen-Grossberg neural networks with mixed time delays and α-inverse Hölder activation functions. In Section 5, two numerical examples are presented to demonstrate the validity of the proposed results. Some conclusions are made in Section 6.Notations 1.
Throughout this paper,R denotes the set of real numbers, Rn denotes the n-dimensional Euclidean space, Rm×n denotes the set of all m×n real matrices. For any matrix A, AT denotes the transpose of A. A-1 denotes the inverse of A. If A is a real symmetric matrix, A>0(A<0) means that A is positive definite (negative definite). Given the column vectors x=(x1,…,xn)T,y=(y1,…,yn)T∈Rn, xTy=∑i=1nxiyi. ∥x∥=(∑i=1nxi2)1/2. ẋ(t) denotes the derivative of x(t), * represents the symmetric form of matrix.


## 2. Neural Network Model and Preliminaries

The Cohen-Grossberg neural networks with mixed time delays can be described by the following differential equation system(2.1)ẋ(t)=α(x(t))[-β(x(t))+W0g(x(t))+W1g(x(t-τ))+W2∫t-τtg(x(s))ds+J],
where x(t)=(x1(t),…,xn(t))T is the vector of neuron states at time t; α(x(t))=diag⁡(α1(x1(t)),…,αn(xn(t))) represents the amplification function; β(x(t))=diag⁡(β1(x1(t)),…,βn(xn(t))) is the behaved function; g(x(t))=(g1(x1(t)),…,gn(xn(t)))T is called the neuron activation function; Wi,i=0,1,2, are the connection weight matrices; τ>0 denotes the discrete and distributed time delay; J=(J1,…,Jn)T denotes the external input. The initial value associated with (2.1) is assumed to be x(s)=φ(s), and φ(s) is a continuous function on [-τ,0].In the following, some definitions and lemmas, which play important roles in the proof of our theorems below, are introduced.Definition 2.1.
The equilibrium pointx* of the neural networks (2.1) is said to be globally exponentially stable, if there exist scalarsη>0, 𝒯>0, and δ>0,such that
(2.2)∥x(t)-x*∥≤ηe-δt,t≥T,
where x(t) is the solution of the system (2.1) with the initial value x(s)=φ(s), and s∈[-τ,0], δ is called the exponential convergence rate.Definition 2.2 (see [23, 24]).
A continuous function𝒢:R→R,is said to be anα-inverse Hölder function, if(i)
𝒢 is a monotonic nondecreasing function,(ii)
for anyρ∈R, there exist constants qρ>0 and rρ>0 which are correlated with ρ, satisfying
(2.3)|G(θ)-G(ρ)|≥qρ|θ-ρ|α,∀|θ-ρ|≤rρ,
where α>0 is a constant.The class ofα-inverse Hölder functions is denoted by ℐℋ(α). There are a great number of functions which belong to ℐℋ(α). For example, 𝒢(θ)=arctanθ,g(θ)=θ3+θ∈ℐℋ(1), and 𝒢(θ)=θ3∈ℐℋ(3).Remark 2.3.
If a continuous function𝒢:R→R satisfies that
(2.4)|G(θ)-G(ρ)|≤l|θ-ρ|,∀θ,ρ∈R,
where ℓ>0 is a constant, then 𝒢 is said to be a Lipschitz-continuous function. When α=1 and qρ is independent on ρ, 1-inverse Hölder functions are called inverse Lipschitz functions. It is easy to see that α-inverse Hölder functions are a class of non-Lipschitz functions.Let functionℱ:Rn→Rn be locally Lipschitz continuous. According to Rademacher's theorem [45], ℱ is differentiable almost everywhere. Let Dℱ denote the set of those points where ℱ is differentiable, then, ℱ̇(x) is the Jacobian of ℱ at x∈Dℱ and the set Dℱ is dense in Rn. The generalized Jacobian of a locally Lipschitz function is defined as follows.Definition 2.4.
For anyx∈Rn, the generalized Jacobian∂ℱ(x)of a locally Lipschitz continuous  functionℱ:Rn→Rnis a set of matrices defined by
(2.5)∂F(x)=co⁡{W∣thereexistsasequence{xk}⊂DFwithlim⁡xk→xḞ(xk)=W},
where co⁡(·) denotes the convex hull of a set.The generalized Jacobian is a natural generalization of the Jacobian for continuously differentiable functions, at those pointsx, where ℱ is continuously differentiable, ∂ℱ(x) reduces to a single matrix which is Jacobian ℱ̇(x) of ℱ.Definition 2.5.
For any switching signalσ(t)and any finite constantsT1,T2satisfyingT2>T1≥0,letNσ(T1,T2)denote the switching times on the time interval(T1,T2). IfNσ(T1,T2)≤N0+(T2-T1)/Taholds forTa>0,N0≥0,thenTais said to be the average dwell time.Lemma 2.6 (see [23, 24]).
If𝒢(θ)∈ℐℋ(α), then for any  ρ0∈R, one has
(2.6)∫ρ0+∞[G(θ)-G(ρ0)]dθ=∫ρ0-∞[G(θ)-G(ρ0)]dθ=+∞.Lemma 2.7 (see [23, 24]).
If𝒢(θ)∈ℐℋ(α) and  𝒢(0)=0, then there exist constants  q0>0 and  r0>0, such that
(2.7)|G(θ)|≥q0|θ|α,∀|θ|≤r0.
Moreover,
(2.8)|G(θ)|≥q0r0α,∀|θ|≥r0.Lemma 2.8 (see [25]).
Letℱ:Rn→Rn be locally Lipschitz continuous. For any given  x,y∈Rn, there exists an element  𝒲 in the union  ∪z∈[x,y]∂ℱ(z) such that
(2.9)F(y)-F(x)=W(y-x),
where [x,y] denotes the segment connecting x and y.LetΩ be a nonempty, bounded, and open subset of Rn. The closure and boundary of Ω are denoted by Ω¯ and ∂Ω, respectively.Lemma 2.9 (see [46]).
(1) Let  ℋ:[0,1]×Ω¯→Rn be a continuous mapping. If  p∈̅ℋ(λ,∂Ω)forallλ∈[0,1], then Brouwer degree  deg⁡(ℋ(λ,·),Ω,p) is constant  (∀λ∈[0,1]). In this case, one has  deg⁡(ℋ(0,·),Ω,p)=deg⁡(ℋ(1,·),Ω,p).
(2) Let ℌ:Ω¯→Rn be a continuous mapping. If deg⁡(ℌ,Ω,p)≠0, then the equation ℌ(x)=p has at least a solution in Ω.Lemma 2.10.
Letx,y∈Rn and  G>0, then
(2.10)2xTy≤xTGx+yTG-1y.Lemma 2.11 (Schur complement).
Given constant symmetric matricesΣ1,Σ2,Σ3 where  Σ1=Σ1T and  0<Σ2=Σ2T, then  Σ1+Σ3TΣ2-1Σ3<0 if and only if
(2.11)[Σ1Σ3TΣ3-Σ2]<0,or[-Σ2Σ3TΣ3Σ1]<0.Lemma 2.12 (Jensen's inequality).
For any constant matrixΩ∈Rn×n,andΩ=ΩT>0,scalarγ>0, vector function  ω:[0,γ]→Rn, such that the integrations concerned are well defined, then
(2.12)1γ(∫0γω(s)ds)TΩ(∫0γω(s)ds)≤∫0γω(s)TΩω(s)ds.To give our main results in the next sections, we need to present the following assumptions.(H1)
αi(s) is continuous, and 0<α̲i<αi(s)<α¯i for all s∈R, i=1,2,…,n.(H2)
βi(s) is locally Lipschitz continuous, and there exists a constant βi>0 such that β̇i(s)≥βi for all s∈R at which βi(s) is differentiable, i=1,2,…,n.(H3)
gi(s)∈ℐℋ(α), i=1,2,…,n.


## 3. Exponential Stability of the Cohen-Grossberg Neural Network

In this section, the Cohen-Grossberg neural network (2.1) is considered the main results on the existence and stability of equilibrium point of the neural network (2.1) will be presented in the following theorem.Theorem 3.1.
Under the assumptions(H1)–(H3), if there exist two positive definite matrices  S,T, two positive definite diagonal matrices  M,  P and a scalar  γ>0 such that
(3.1)[γPA̲-1-PB-BPPW0+γMA̲-1-MBPW1PW2W0TP+γA̲-1M-BMMW0+W0TM+S+τTMW1MW2W1TPW1TM-e-γτS0W2TPW2TM0-e-γττT]<0
is satisfied, where A̲=diag⁡(α̲1,α̲2,…,α̲n), B=diag⁡(β1,β2,…,βn), M=diag⁡(m1,m2,…,mn), P=diag⁡(p1,p2,…,pn), then the Cohen-Grossberg neural network (2.1) has one unique equilibrium point which is globally exponentially stable.Proof.
We should prove this theorem in three steps.
Step 1.
In this step, we will prove the existence of the equilibrium point.
Letℌ(x)=β(x)-(W0+W1+τW2)g(x)-J. By the assumption (H1), x*∈Rn is an equilibrium point of the system (2.1) if and only if ℌ(x*)=0. Rewrite ℌ(x) as
(3.2)H(x)=β̃(x)-(W0+W1+τW2)g̃(x)+H(0),
where β̃(x)=β(x)-β(0),g̃(x)=g(x)-g(0). Obviously, g̃i(0)=0, g̃i∈ℐℋ(α) and xigĩ(xi)>0(xi≠0). By Lemma 2.8 and the assumption (H2), it can follow that β̃(x)=β(x)-β(0)=B̃x,B̃∈∪z∈[0,x]∂β(z), where B̃=diag⁡(β̃1,β̃2,…,β̃n) with β̃i≥βi,i=1,2,…,n.
LetΩℛ={x∈Rn:∥x∥<ℛ},ℛ>0. Define the mappingℋ:[0,1]×Ω¯→Rnas
(3.3)H(λ,x)=B̃x-λ(W0+W1+τW2)g̃(x)+λH(0),
where Ω¯ℛ={x∈Rn:∥x∥≤ℛ}.
From (3.1) and the definition of negative definite matrix, we can obtain
(3.4)[MW0+W0TM+S+τTMW1MW2W1TM-e-γτS0W2TM0-e-γττT]<0.
By Lemma 2.11, (3.4) is equivalent to
(3.5)MW0+W0TM+S+eγτMW1S-1W1TM+τeγτMW2T-1W2TM+τT<0.
By means of Lemma2.10, we have
(3.6)g̃(x)TMW1g̃(x)≤12[g̃(x)TMW1S-1W1TMg̃(x)+g̃(x)TSg̃(x)],g̃(x)TMW2g̃(x)≤12[g̃(x)TMW2T-1W2TMg̃(x)+g̃(x)TTg̃(x)].
By using (3.5) and (3.6), we have
(3.7)g̃(x)TMH(λ,x)=g̃(x)TMB̃x+λg̃(x)TMH(0)-λg̃(x)TM(W0+W1+τW2)g̃(x)=g̃(x)TM(B̃x+λH(0))-λg̃(x)TM(W0+W1+τW2)g̃(x)≥g̃(x)TM(B̃x+λH(0))-λ2[g̃(x)T(MW0+W0TM+S+MW1S-1W1TM+τMW2T-1W2TM+τT)g̃(x)]≥g̃(x)TM(B̃x+λH(0))≥∑i=1n[miβ̃i|g̃i(xi)||xi|-|g̃i(xi)||(MH(0))i|]=∑i=1nmiβ̃i|g̃i(xi)|[|xi|-|(MH(0))i|miβ̃i],
where (Mℌ(0))i(i=1,2,…,n) denotes the ith element of vector (Mℌ(0)).
By virtue of Lemma2.7, there exist constants qi0>0 and ri0>0, i=1,2,…,n, such that
(3.8)|g̃i(xi)|≥qi0ri0α,∀|xi|≥ri0,i=1,2,…,n.
Leta=max⁡1≤i≤n(|(Mℌ(0))i|/miβ̃i), 𝒩k={n1,…,nk}⊂{1,2,…,n},forallk<n. Define Ω𝒩k={x:|xi|≤a,i∈𝒩k,x∈Rk}, g̃𝒩k(x)=∑i∈𝒩kmiβ̃i|g̃i(xi)|[|xi|-a]. Noting that Ω𝒩k is a compact subset of Rk, and g̃𝒩k is continuous on Ω𝒩k, it follows that g̃𝒩k can reach its the minimum min⁡x∈Ω𝒩kg̃𝒩k(x) on Ω𝒩k.
Letr0=max⁡1≤i≤nri0, l=min⁡1≤i≤n{miβ̃iqi0ri0α}, ℳ𝒩k=min⁡x∈Ω𝒩kg̃𝒩k(x), and ℳ=min⁡{ℳ𝒩k:𝒩k⊂{1,2,…,n}}. Set ℛ>max⁡{n(a-ℳ/l),nr0} and x∈∂ΩR, then there exist two index sets 𝒩 and 𝒩¯, such that
(3.9)|xi|≤a,i∈N,|xi|>a,i∈N¯,
where 𝒩⋃𝒩¯={1,2,…,n}. Furthermore, there exists an index i¯0 in 𝒩¯ such that
(3.10)|xi¯0|≥Rn≥max⁡{a,r0}.
By (3.8) and (3.10), for any x∈∂ΩR and λ∈[0,1],(3.11)(g̃(x))TMH(λ,x)≥∑i∈Nmiβ̃i|g̃i(xi)|[|xi|-a]+∑i∈N¯miβ̃i|g̃i(xi)|[|xi|-a]≥M+mi¯0β̃i¯0|g̃i¯0(xi¯0)|[|xi¯0|-a]≥M+mi¯0β̃i¯0qi¯0ri¯0α[|xi¯0|-a]≥mi¯0β̃i¯0qi¯0ri¯0α[|xi¯0|-a+Ml]≥mi¯0β̃i¯0qi¯0ri¯0α[Rn-a+Ml]>0.
Hence, this implies that ℋ(λ,x)≠0 for any x∈∂ΩR and λ∈[0,1]. By applying Lemma 2.9 (1), it follows
(3.12)deg⁡(H(0,x),ΩR,0)=deg⁡(H(1,x),ΩR,0),
that is, deg⁡(ℌ(x),Ωℛ,0)=deg⁡(B̃x,Ωℛ,0)=sgn⁡|B̃|≠0, where |B̃| is the determinant of B̃. By Lemma 2.9 (2), ℌ(x)=0 has at least one solution in Ωℛ, that is, the system (2.1) has at least an equilibrium point.
Step 2.
In this step, the proof of the uniqueness of the equilibrium point by the method of contradiction will be given.
Assume thatx1* and x2* are two different equilibrium points of the system (2.1), then
(3.13)β(x1*)-β(x2*)=(W0+W1+τW2)[g(x1*)-g(x2*)].
From Lemma 2.8, it can follow that
(3.14)β(x1*)-β(x2*)=B̂(x1*-x2*),
where B̂∈∪z∈[x1*,x2*]∂β(z), B̂=diag⁡(β̂1,β̂2,…,β̂n) with β̂i≥βi,i=1,2,…,n.
By means of (3.5), (3.6), (3.13), and (3.14), it follows that
(3.15)0<[g(x1*)-g(x2*)]TMB̂(x1*-x2*)=[g(x1*)-g(x2*)]TM(W0+W1+τW2)[g(x1*)-g(x2*)]≤12[g(x1*)-g(x2*)]T×(MW0+W0TM+S+MW1S-1W1TM+τMW2T-1W2TM+τT)[g(x1*)-g(x2*)]<0.
This is a contradiction. Hence, x1*=x2*. This shows that the equilibrium point of the system (2.1) is unique.
Step 3.
In this step, we will prove that the system (2.1) is globally exponentially stable.
LetF(x,t)=α(x)[-β(x)+W0g(x)+W1g(xτ)+W2∫t-τtg(x(s))ds+J],where xτ(t)=x(t-τ). Since gi∈ℐℋ(α), α(x),β(x) are continuous functions, F(x,t) are continuous and locally bounded. Hence, we can obtain the existence of the local solution of the system (2.1) with initial value x(t)=φ(t),t∈[-τ,0] on [0,t*(φ)), where t*(φ)∈(0,+∞) or t*(φ)=+∞, and [0,t*(φ)) is the maximal right-side existence interval of the local solution.
Letx* be the unique equilibrium point of the system (2.1). Make a transformation u(t)=x(t)-x*, then system (2.1) is transformed into
(3.16)u̇(t)=α̅(u(t))[-β̅(u(t))+W0g̅(u(t))+W1g̅(u(t-τ))+W2∫t-τtg̅(u(s))ds],
whereα̅(u(t))=diag⁡(α̅1(u1(t)),α̅2(u2(t)),…,α̅n(un(t))),β̅(u(t))=(β̅1(u1(t)),β̅2(u2(t)),…,β̅n(un(t)))T,g̅(u(t))=(g̅1(u1(t)),g̅2(u2(t)),…,g̅n(un(t)))T,α̅i(ui(t))=αi(ui(t)+xi*),β̅i(ui(t))=βi(ui(t)+xi*)-βi(xi*),g̅i(ui(t))=gi(ui(t)+xi*)-gi(xi*),i=1,2,…,n.Similarly to (3.14), from Lemma 2.8, we have
(3.17)β̅(u)=β(u+x*)-β(x*)=B̅u,
where B̅∈∪z∈[x*,u+x*]∂β(z), B̅=diag⁡(β̅1,β̅2,…,β̅n) with β̅i≥βi,i=1,2,…,n.
Consider the following Lyapunov-Krasovskii functional candidate(3.18)V(t)=V1(t)+V2(t)+V3(t)+V4(t),
where
(3.19)V1(t)=2eγt∑i=1npi∫0ui(t)sα̅i(s)ds,V2(t)=2eγt∑i=1nmi∫0ui(t)g̅i(s)α̅i(s)ds,V3(t)=∫t-τtg̅(u(θ))TSg̅(u(θ))eγθdθ,V4(t)=∫-τ0∫t+θteγsg̅(u(s))TTg̅(u(s))dsdθ.
Calculating the time derivative of V(t) along the trajectories of the system (3.16) on [0,t*(φ)) with (3.17), by the assumption (H1) and Lemma 2.12, we have
(3.20)V̇1(t)≤γeγtu(t)TPA̲-1u(t)+2eγtu(t)TP×[-B̅u(t)+W0g̅(u(t))+W1g̅(u(t-τ(t)))+W2∫t-τ(t)tg̅(u(s))ds]≤eγtu(t)T(γPA̲-1-2PB)u(t)+2eγtu(t)TPW0g̅(u(t))+2eγtu(t)TPW1g̅(u(t-τ))+2eγtu(t)TPW2∫t-τtg̅(u(s))ds,V̇2(t)≤2γeγtu(t)TMA̲-1g̅(u(t))+2eγtg̅(u(t))TM×[-B̅u(t)+W0g̅(u(t))+W1g̅(u(t-τ(t)))+W2∫t-τ(t)tg̅(u(s))ds]≤2eγtu(t)T(γMA̲-1-MB)g̅(u(t))+2eγtg̅(u(t))TMW0g̅(u(t))+2eγtg̅(u(t))TMW1g̅(u(t-τ))+2eγtg̅(u(t))TMW2∫t-τtg̅(u(s))ds,V̇3(t)=eγtg̅(u(t))TSg̅(u(t))-eγte-γτg̅(u(t-τ))TSg̅(u(t-τ)),V̇4(t)≤τeγtg̅(u(t))TTg̅(u(t))-∫t-τteγsg̅(u(s))TTg̅(u(s))ds≤τeγtg̅(u(t))TTg̅(u(t))-eγ(t-τ)∫t-τtg̅(u(s))TTg̅(u(s))ds≤τeγtg̅(u(t))TTg̅(u(t))-eγ(t-τ)1τ(∫t-τtg̅(u(s))Tds)T(∫t-τtg̅(u(s))ds).
Let ξ(t)=[u(t)T,g̅(u(t))T,g̅(u(t-τ))T,(∫t-τtg̅(u(s))ds)T]T. By (3.17) and (3.20), we can obtain
(3.21)V̇(t)≤eγτξT(t)[γPA̲-1-PB-BPPW0+γMA̲-1-MBPW1PW2W0TP+γA̲-1M-BMMW0+W0TM+S+τTMW1MW2W1TPW1TM-e-γτS0W2TPW2TM0-e-γττT]ξ(t)<0,foranyξT(t)≠0.
This implies V(t)≤V(0), t∈[0,t*(φ)). Furthermore, from (3.18), it follows that
(3.22)2∑i=1nmiα̅i∫0ui(t)g̅i(s)ds≤2∑i=1nmi∫0ui(t)g̅i(s)α̅i(s)ds≤V(0)e-γt≤V(0).
By (3.22) and Lemma 2.6, it is easy to derive that ui(t),i=1,2,…,n, are bounded on [0,t*). By virtue of the continuous theorem [47], t*(φ)=+∞.
Sinceg̅i(s)∈ℐℋ(α),g̅i(0)=0, by Lemma 2.7, there exist constants q̅i0>0 and r̅i0>0, such that
(3.23)|g̅i(s)|≥q̅i0|s|α,∀|s|≤r̅i0,i=1,2,…,n.
Moreover, from (3.22), we can get that
(3.24)lim⁡t→+∞ui(t)=0,i=1,2,…,n.
Thus there exists a scalar 𝒯>0, when t≥𝒯, ui(t)∈[-r̅0,r̅0], where r̅0=min⁡1≤i≤nr̅i0. Let m=min⁡1≤i≤nmi, α̅=max⁡1≤i≤nα̅i, and q̅0=min⁡1≤i≤nq̅i0. From (3.22) and (3.23), we have
(3.25)V(0)2e-γt≥∑i=1nmiα̅i∫0ui(t)g̅i(s)ds≥∑i=1nmiα̅i∫0|ui(t)|q̅i0|s|αds≥∑i=1nmiα̅i∫0|ui(t)|q̅0|s|αds≥mq̅0α̅(α+1){max⁡1≤i≤n|ui(t)|}α+1,t≥T.
That is, when t≥𝒯,
(3.26)max⁡1≤i≤n|ui(t)|≤[α̅(α+1)V(0)2mq̅0]1/(1+α)e-(γ/(1+α))t,
where
(3.27)V(0)=2∑i=1npi∫0φi(0)-xi*sα̅i(s)ds+2∑i=1nmi∫0φi(0)-xi*g̅i(s)α̅i(s)ds+∫-τ0g̅(φ(θ)-x*)TSg̅(φ(θ)-x*)eγθdθ+∫-τ0∫θ0eγsg̅(φ(s)-x*)TTg̅(φ(s)-x*)dsdθ.
Let η=n[α̅(α+1)V(0)/2mq̅0]1/(1+α). By (3.26), ∥x(t)-x*∥≤ηe-(γ/(1+α))t, for all t≥𝒯. This shows that the equilibrium point of the system (2.1) is globally exponentially stable. This completes the proof of Theorem 3.1.


## 4. Stability of the Switched Cohen-Grossberg Neural Network

The switched Cohen-Grossberg neural networks with mixed time delays consist of a set of Cohen-Grossberg neural networks with mixed time delays and a switching rule. Each of the Cohen-Grossberg neural networks with mixed time delays is regarded as an individual subsystem. The operation mode of the switched neural networks is determined by the switching rule. In the following, we will develop the switched Cohen-Grossberg neural networks model with mixed time delays.Suppose thatx* is the unique equilibrium point of the system (2.1). Similar to (3.16), make a transformation u(t)=x(t)-x*, then system (2.1) is transformed into (4.1)u̇(t)=α̅(u(t))[-β̅(u(t))+W0g̅(u(t))+W1g̅(u(t-τ))+W2∫t-τtg̅(u(s))ds].The switched Cohen-Grossberg neural networks with mixed time delays can be described as(4.2)u̇(t)=α̅(u(t))[-β̅(u(t))+W0σ(t)g̅(u(t))+W1σ(t)g̅(u(t-τ))+W2σ(t)∫t-τtg̅(u(s))ds],
where σ(t):[0,+∞)→Γ={1,2,…,N} is the switching signal, which is a piecewise constant function of time. This means that the matrices (W0σ(t),W1σ(t),W2σ(t),Jσ(t)) are allowed to take values, at an arbitrary time, in the finite set {(W01,W11,W21,J1),(W02,W12,W22,J2),…,(W0N,W1N,W2N,JN)}. The initial value associated with (4.3) is assumed to be u(s)=φ̃(s), φ̃(s) is a continuous function on [-τ,0].In this paper, it is assumed that the switching ruleσ is not known a priori and its instantaneous value is available in real time. Corresponding to the switching signal σ(t), we have the switching sequence {xt0;(i0,t0),…,(ik,tk),…,∣ik∈Γ,k=0,1,…}, which means that the ikth subsystem is activated when t∈[tk,tk+1).In the following, we will consider the switched Cohen-Grossberg neural networks with mixed time delays in (4.2). The average dwell time approach will be used to derive the exponential stability of the network.Theorem 4.1.
Under the assumptions(H1)–(H3), if there exist positive definite matrices  Si,Ti, positive definite diagonal matrices  Mi,  Pi and scalars  γ>0,μ≥1 such that
(4.3)[γPA̲-1-PiBi-BiPiPiW0i+γMiA̲-1-MiBiPiW1iPiW2iW0iTPi+γA̲-1Mi-BiMiMiW0i+W0iTMi+Si+τTiMiW1iMiW2iW1iTPiW1iTMi-e-γτSi0W2iTPiW2iTMi0-e-γττTi]<0,(4.4)Pi≤μPj,Mi≤μMj,Si≤μSj,Ti≤μTj,∀i,j∈Γ,
where Mi=diag⁡(mi1,mi2,…,min), Pi=diag⁡(pi1,pi2,…,pin), then the switched Cohen-Grossberg neural network (4.2) is globally exponentially stable for any switching signal with average dwell time satisfying
(4.5)Ta>Ta*=(1+α)ln⁡μγ.
Moreover, an estimate of the state decay for the system (4.2) is given by
(4.6)‖u(t)‖≤n{e-γt0α̅(α+1)2mq̅0}1/(1+α)max⁡i∈ΓVi(t0)e-(γ/(1+α)-(ln⁡μ)/Ta)(t-t0),t>t0.Proof.
Consider the multiple Lyapunov-Krasovskii functional candidate(4.7)Vσ(t)(t)=V1σ(t)(t)+V2σ(t)(t)+V3σ(t)(t)+V4σ(t)(t),
where
(4.8)V1σ(t)(t)=2eγt∑j=1npσ(t)j∫0uj(t)sα̅j(s)ds,V2σ(t)(t)=2eγt∑j=1nmσ(t)j∫0uj(t)g̅j(s)α̅j(s)ds,V3σ(t)(t)=∫t-τtg̅(u(θ))TSσ(t)g̅(u(θ))eγθdθ,V4σ(t)(t)=∫-τ0∫t+θteγsg̅(u(s))TTσ(t)g̅(u(s))dsdθ.
When t∈[tk,tk+1), the ikth subsystem is activated. Arguing as in the proof Theorem 3.1, we can get V̇σ(t)(t)≤0. Thus, Vσ(t)(t)≤Vσ(tk)(tk). In the light of (4.4) and (4.7), it follows that Vσ(tk)(tk)≤μVσ(tk-)(tk-). Therefore, when t∈[tk,tk+1), we have
(4.9)Vσ(t)(t)≤Vσ(tk)(tk)≤μVσ(tk-)(tk-)≤μVσ(tk-1)(tk-1)≤μ2Vσ(tk-1-)(tk-1-)≤μ2Vσ(tk-2)(tk-2)≤⋯≤μkVσ(t0)(t0).
By (4.9); there exist constants 𝒯>t0, when t≥𝒯, we have
(4.10)Vσ(t0)(t0)2μke-γt≥∑j=1nmσ(t)jα̅j∫0uj(t)g̅i(s)ds≥∑j=1nmσ(t)jα̅j∫0|ui(t)|q̅i0|s|αds≥∑j=1nmσ(t)jα̅j∫0|ui(t)|q̅0|s|αds≥mq̅0α̅(α+1){max⁡1≤i≤n|ui(t)|}α+1,
where m=min⁡j∈Γ,1≤i≤nmji, α̅=max⁡1≤i≤nα̅i and q̅0=min⁡1≤i≤nq̅i0. Hence,
(4.11)max⁡1≤i≤n|ui(t)|≤{α̅(α+1)Vσ(t0)(t0)2mq̅0}1/(1+α)μke-(γ/(1+α))t.
This implies that
(4.12)∥u(t)∥≤n{α̅(α+1)Vσ(t0)(t0)2mq̅0}1/(1+α)μke-(γ/(1+α))t.
Due to k≤(t-t0)/Ta and Ta>Ta*=(1+α)((ln⁡μ)/γ), then we can get
(4.13)‖u(t)‖≤n{e-γt0α̅(α+1)2mq̅0}1/(1+α)max⁡i∈ΓVi(t0)μke-(γ/(1+α))(t-t0)=n{e-γt0α̅(α+1)2mq̅0}1/(1+α)max⁡i∈ΓVi(t0)ekln⁡μe-(γ/(1+α))(t-t0)≤n{e-γt0α̅(α+1)2mq̅0}1/(1+α)max⁡i∈ΓVi(t0)e-(γ/(1+α)-(ln⁡μ)/Ta)(t-t0),t>t0.
This implies that the switched Cohen-Grossberg neural network (4.2) is globally exponentially stable. The proof is completed.Remark 4.2.
It is clear that, according to Theorem4.1, the delay-dependent stability of the considered neural networks is dependent on μ for given γ. If μ=1, we have from (4.5) that the switching signal can be arbitrary, and (4.4) reduces to Pi≤Pj,Mi≤Mj,Si≤Sj,Ti≤Tj,foralli,j∈Γ, which implies Pi=Pj,Mi=Mj,Si=Sj,Ti=Tj,foralli,j∈Γ, which means that it requires a common Lyapunov functional for all subsystems. If μ→+∞, we get from (4.5) that there is no switching, that is, switching signal will have a great dwelltime on the average.Remark 4.3.
In the existing literature [2, 3, 17, 22, 27–29, 34–38], the activation functions of switched neural networks are required to be Lipschitz continuous. However, in this paper, the activation function is inverse Hölder function. It is obvious the results of this paper are different from the results in the above literatures. Hence, the work in this paper is an extension of the scope of the current investigation in this field.


## 5. Illustrative Examples

In this section, two illustrative examples will be given to check the validity of the results obtained in Theorems3.1 and 4.1.Example 5.1.
Consider a second-order Cohen-Grossberg neural networks with mixed time delays in (2.1) with the following parameters:
(5.1)W0=(-99990.0090.0001-9999),W1=(0.250.030.010.25),W2=(0.050.030.020.45),J=(0,0)T.
Set αi(θ)=2+sin⁡θ, βi(θ)={θ,θ≥0,2θ,θ≤0,. The activation functions are taken as gi(θ)=θ3, i=1,2, and τ=1.
It is easy to check that assumptions(H1)–(H3) hold. βi(θ) is locally Lipschitz; A̲ and B is the second-order identity matrix.
Takeγ=0.3. Solving the LMI in (3.1) by using appropriate LMI solver in the Matlab, the feasible positive definite matrices P,M,S,T could be as
(5.2)P=(0.0017000.0017),M=(23.33160023.3316),S=(3.4137-0.0019-0.00193.4137),T=(3.4137-0.0019-0.00193.4137).
All assumptions of Theorem 3.1 hold. Hence, this neural network has one unique equilibrium point, which is globally exponentially stable.
Figures1 and 2 display the state trajectories of this neural network with initial values φ(t)=(cos⁡t,sin⁡t)T,(0.5cos⁡t,-0.5sin⁡t)T,(-1.5cos⁡t,1.5sin⁡t)T,(2cos⁡t,2sin⁡t)T and (-2.5cos⁡t,2.5sin⁡t)T,t∈[-1,0]. It can be seen that these trajectories converge to the unique equilibrium x*=(0,0)T of the network. This is in accordance with the conclusion of Theorem 3.1.Figure 1
The convergence of the statex1(t) of the network in Example 5.1.Figure 2
The convergence of the statex2(t) of the network in Example 5.1.Example 5.2.
Consider a third-order switched Cohen-Grossberg neural networks with mixed time delays in (4.3) with the switching signal σ(t):[0,+∞)→Γ={1,2} and the following parameters:(5.3)W01=(-832.6513.130.516.01-807.4393.522.033.05-478.954),W02=(-49594.033.310.01-80792.828.036.05-9278),W11=(0.250.160.030.010.320.250.210.320.06),W12=(0.250.160.030.010.324.250.210.320.06),W21=(0.951.040.031.021.031.4520.045.0280.37),W22=(0.050.040.030.560.032.450.040.020.37).
Set αi(θ)=2-cos⁡θ, βi(θ) are the functions in Example 5.1, and the activation functions are taken as gi(θ)=θ3+θ, i=1,2,3, and τ=1.It is easy to check that assumptions(H1)–(H3) hold; A̲ and B is the third-order identity matrix.Takeγ=0.1,μ=1.2. Solving the LMIs in (4.3) and (4.4) by using appropriate LMI solver in the Matlab, the feasible positive definite matrices P1,P2,M1,M2,S1,S2,T1,T2 could be as(5.4)P1=(0.63340000.63340000.6334),M1=(0.04180000.04180000.0418),S1=(0.75000.0182-0.06850.01820.8893-0.0165-0.0685-0.01650.2732),T1=(0.78110.02660.06790.02660.88950.01830.06790.01830.8362),P2=(0.66200000.66200000.6620),M2=(0.03650000.03650000.0365),S2=(0.78550.0191-0.07170.01910.9312-0.0173-0.0717-0.01730.2862),T2=(0.47100.01610.04110.01610.53680.01110.04110.01110.5045).
By using (4.5), it follows that the average dwell time Ta*=3.6461. All the assumptions of Theorem 4.1 hold. Hence, this switched neural network is globally exponentially stable.For numerical simulation, assume that the two subsystems are switched every four seconds. Figures3, 4, and 5 display the state trajectories of this neural network with initial valuesφ̃(t)=(-0.5cos⁡t,0.5sin⁡t,0.5cos⁡t)T,(cos⁡t,-sin⁡t,-cos⁡t)T,(-1.5cos⁡t,1.5sin⁡t,1.5cos⁡t)T,Figure 3
The convergence of the statex1(t) of the network in Example 5.2.Figure 4
The convergence of the statex2(t) of the network in Example 5.2.Figure 5
The convergence of the statex3(t) of the network in Example 5.2.(-2cos⁡t,2sin⁡t,2sin⁡t)Tand (2.5cos⁡t,-2.5sin⁡t,-2.5cos⁡t)T,t∈[-1,0]. It can be seen that these trajectories converge to the unique equilibrium u*=(0,0,0)T of the network. This is in accordance with the conclusion of Theorem 4.1.


## 6. Conclusion

In this paper, the existence, uniqueness, and global stability of the equilibrium point for Cohen-Grossberg neural networks with mixed time delays,α-inverse Hölder neuron activation functions, and nonsmooth behaved functions have been discussed. By applying multiple Lyapunov-Krasovskii functional, a delay-dependent global exponential stability criterion has been obtained in terms of LMIs for the switched Cohen-Grossberg neural networks with mixed time delays and α-inverse Hölder neuron activation functions under the switching rule with the average dwell time property. The results obtained are easily checked and applied in practice engineering.When neuron activation functions are non-Lipschitz functions, it is possible that the neural network system has not the global solution and the equilibrium point. This leads to difficulty in solving the stability problem, particularly exponential stability for the switched neural networks with non-Lipschitz activation functions. In the future, the stability problem for the switched neural networks with other non-Lipschitz activation functions will be expected to be solved.



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*Source: 435402-2012-05-23.xml*