# Contraction Mapping Theory and Approach to LMI-Based Stability Criteria of T-S Fuzzy Impulsive Time-Delays Integrodifferential Equations

Authors: Ruofeng Rao; Shouming Zhong
Journal: Journal of Function Spaces
 (2016)

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## Abstract

In this paper, Banach fixed point theorem is employed to derive LMI-based exponential stability of impulsive Takagi-Sugeno (T-S) fuzzy integrodifferential equations, originated from Cohen-Grossberg Neural Networks (CGNNs). As far as we know, Banach fixed point theorem is rarely employed to derive LMI criteria for T-S fuzzy CGNNs, and this inspires our present work. It is worth mentioning that the conditions on the behavior functions are weaker than those of existing results, and the formulated contraction mapping and fixed point technique are different from those of previous literature. Even a corollary of our main result improves one of existing main results due to extending linear function to nonlinear function. Besides, the LMI-based criteria are programmable for computer MATLAB LMI toolbox. Moreover, an analytical table and a numerical example are presented to illustrate the advantage, feasibility, and effectiveness of the proposed methods.


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## Body

## 1. Introduction

In this paper, we consider a class of integrodifferential equations, which is originated from Cohen-Grossberg Neural Networks (CGNNs). CGNNs model was proposed originally by Cohen-Grossberg in 1983 [1]. Since then, the stability analysis of CGNNs has attracted extensive attentions owing to its wide applications. Lyapunov function method is a common technique which derived stability criteria of various time-delays dynamic systems [2–13]. However, any method has its limitations. Fixed point technique may be one of common replacements. Brouwer fixed point theorem and Schauder fixed point theorem were always applied to stability analysis of various CGNNs models [14–17]. Of course, Banach fixed point theorem was also employed to derive the stability criteria of various CGNNs models [17–22]. But for most of existing literature, either using the same inverse function translates CGNNs model into another model similar as follows (see, e.g., [17, 19–21]), (1)xi′t=-biu-1xit+∑i=1nbijtfju-1xjt+cijfju-1xjt-τt,or the stability criteria are too complex [22, Theorem  3.1], which cannot be programmable for computer software while practical engineering is often involved in large-scale computing. Of course, both methods and results of [17, 19–22] are good and referential. But, in order to innovate, we have to find another way. In [18], a LMI-based stability criterion was given for the following CGNNs: (2)dxjtdt=-ajxjtbjxjt-∑k=1ncjkfkxkt+djkgkxkt-τjt,t⩾0,t≠ti,Δxjti=xjti+-xjti=ρjixjti,i=1,2,…,xjθ=ϕjθ,-τ⩽θ⩽0.Theorem A (see [18, Theorem  4]).
If (H1)–(H4) are satisfied and there exists a positive constantα<1 such that the following LMI condition holds, (3)CF¯A~+DG¯A~+H+μΓ+CFA¯+DGA¯⩽αΓA¯,then System (2) is globally exponentially stable, where the four conditions are proposed as follows.(H1)
For anyj∈N, there exist constants F¯j>0, G¯j>0, Fj>0, and Gj>0 such that (4)fjr⩽F¯j,gjr⩽G¯j,fjr-fjs⩽Fjr-s,gjr-gjs⩽Gjr-s,∀r,s∈R.(H2)
For anyj∈N, aj(·) is differentiable, and there exists a constant a~j such that (5)0<ajr⩽a¯j,aj′r⩽a~j,r∈R.(H3)
There exist nonnegative constantsh~ji such that (6)ρjir-ρjis⩽h~jir-s,∀r,s∈R,j∈N,i=1,2,….(H4)
Forj∈N, there exist Υj(t) and a constant γj>0 such that (7)Υjt⩾γj,∀t⩾0,ajrbjr=Υjtr,r∈R.Remark 1.
Denotingχi(s)=ai(s)bi(s), we know from [18, (H4)] that χi(s) can only be a linear function on s. So it is the main objective of this paper to make up the deficiency. Below, we try to make the condition of χi(s) better so that χi(s) may be a nonlinear function on s.
All the good results and methods in existing literature, particularly in [17, 19–22], inspired our current work. Of course, we cannot imitate (1) again, and the condition [18, (H4)] should be replaced by a new suitable condition. Below, we shall propose a new condition which is better than [18, (H4)]. Due to the changes of conditions, we have to formulate new contraction mapping to derive new LMI-based exponential stability criterion for Takagi-Sugeno (T-S) fuzzy impulsive CGNNs with discrete and distributed delays. This is one of main innovations of this paper. In addition, we shall admit weaker condition on the behavior functions than that of existing literature (e.g., [17, 19–22]) published from 2007 to 2016. Moreover, a corollary of our main result is better than [18, Theorem  3.1] due to the above reasons.Remark 2.
Due to the weaker condition on the behavior functions, both our results and methods are novelty (see below “Remark13” and “Table 1” for details).Table 1
Comparing our Theorem8 and Corollary 10 with other existing criteria on CGNNs derived by fixed point theorems.
Fixed point theorems Continuity ofbi Differentiability ofbi Programmability of criteria Our Theorem8 (Corollary 10) Contraction mapping theorem Unnecessary Unnecessary Yes, LMI-based [22, Theorem  3.1] Contraction mapping theorem Yes, necessary [22, (A2)] No [19, Theorem  1] Contraction mapping theorem Yes, necessary [19, (A2)] No [20, Theorems  1 and  2] Contraction mapping theorem Yes, necessary Yes, necessary No [17, Theorems  1–3] Brouwer fixed point theorem Yes, necessary [17, (H3)] No [21, Theorem  3.1] Contraction mapping theorem Yes, necessary Yes, necessary No Conditions [19, (A2)], [22, (A2)], and [17, (H3)] are similar to bis-bit/s-t⩾constant.For convenience’s sake, we introduce the following standard notations.(i)
L=(lij)n×n> 0 (<0) is a positive (negative) definite matrix; that is, yTLy> 0 (<0) for any 0≠y∈Rn.(ii)
L=(lij)n×n⩾ 0 (⩽0) is a semipositive (seminegative) definite matrix; that is, yTLy⩾ 0 (⩽0) for any y∈Rn.(iii)
L∈[L∗,L∗] implies that lij∗⩽lij⩽lij∗ for all i,j with L=(lij)n×n, L∗=(lij∗)n×n, and L∗=(lij∗)n×n.(iv)
L1⩾L2(L1⩽L2) means matrix (L1-L2) is a semipositive (seminegative) definite matrix.(v)
L1>L2(L1<L2) means matrix (L1-L2) is a positive (negative) definite matrix.(vi)
λmax(Φ),λmin(Φ) denotes the largest and smallest eigenvalue of matrix Φ, respectively.(vii)
Denote|L|=(|lij|)n×n for any matrix L=(lij)n×n.(viii)
|u|=(|u1|,|u2|,…,|un|)T for any vector u=(u1,u2,…,un)T∈Rn.(ix)
u⩽ (⩾) v implies that ui⩽ (⩾) vi, ∀i, and u< (>) v implies that ui< (>) vi,∀i, for any vectors u=(u1,u2,…,un)T∈Rn and v=(v1,v2,…,vn)T∈Rn.(x)
I is identity matrix with compatible dimension.


## 2. Preliminaries

Cohen-Grossberg Neural Networks (CGNNs) with discrete and distributed delays have been investigated in many papers [23–28]. Consider the following impulsive CGNNs with discrete and distributed delays,(8)dxtdt=-AxtBxt-Cfxt-Dgxt-τt-E∫t-δttηxsds,t⩾0,t≠tk,xtk+-xtk-=ρxtk,k=1,2,…with the initial condition (9)xs=ϕs,s∈-τ,0,where x=(x1,x2,…,xn)∈Rn with xi(t) being the state variables of the ith neuron at time t. The neuron active functions f(x)=(f1(x1(t)),f2(x2(t)),…,fn(xn(t)))T∈Rn for any given t∈R. A(x(t))=diag(a1(x1(t)),a2(x2(t)),…,an(xn(t))) is n-dimension diagonal matrix with ai(xi(t)) representing an amplification function, and B(x(t))=(b1(x1(t)),…,bn(xn(t)))T∈Rn with bj(xj(t)) being an appropriately behavior function. C=(cij)n×n, D=(dij)n×n, and E=(eij)n×n are the connection weight matrices with cij,dij,eij representing the strengths of connectivity between cells i and j at time t. Vector functions f(x(t))=(f1(x1(t)),…,fn(xn(t)))T,g(x(t))=(g1(x1(t-τ(t))),…,gn(xn(t-τ(t))))T, and η(x(s))=(η1(x1(s)),…,ηn(xn(s)))T with activation functions fj(·),gj(·),ηj(·) telling how the jth neuron reacts to the input. τ(t) is discrete delay, and δ(t) is distributed time-delay. τ is a positive constant with δ(t),τ(t)∈[0,τ].ρ(x(t))=(ρ1(x1(t)),…,ρn(xn(t)))T denotes the impulsive function, and the fixed impulsive moments tk(k=1,2,…) satisfy 0<t1<t2<⋯ with limk→+∞tk=+∞.x(tk+) and x(tk-) stand for the right-hand and left-hand limit of x(t) at time tk, respectively. Further, suppose that (10)xtk-=limt→tk-⁡xt=xtk,ytk-=limt→tk-⁡yt=ytk,k=1,2,….In addition, we assume that ϕ(s)∈C([-τ,0];R), where C([-τ,0];R) is the space of all the continuous functions defined on [-τ,0].Since practice has shown that fuzzy logic theory is an efficient approach to deal with the analysis and synthesis problems for complex nonlinear system, the fuzzy model is far more important than stochastic model [29–31]. Among various kinds of fuzzy methods, Takagi-Sugeno (T-S) fuzzy models provide a successful method to describe certain complex nonlinear systems using some local linear subsystems.Below, we describe the T-S fuzzy mathematical model with time-delay as follows.Fuzzy Rule j . IF ω 1 ( t ) is μj1,…,ωs∗(t) is μjs∗THEN(11)dxtdt=-AxtBxt-Cjfxt-Djgxt-τt-Ej∫t-δttηxsds,t⩾0,t≠tk,xtk+-xtk-=ρxtk,k=1,2,…,where ωkt(k=1,2,…,s∗) is the premise variable, μjk(j=1,2,…,r;k=1,2,…,s∗) is the fuzzy set that is characterized by membership function, r is the number of theIF-THEN rules, and s∗ is the number of the premise variables. By way of a standard fuzzy inference method, System (11) is inferred as follows:(12)dxtdt=-AxtBxt-∑j=1r∗hjωtCjfxt+Djgxt-τt+Ej∫t-δttηxsds,t⩾0,t≠tk,xtk+-xtk-=ρxtk,k=1,2,…,where ω(t)=[ω1(t),ω2(t),…,ωs(t)], hjωt=wjωt/∑k=1rwkωt, and wjωt:Rs∗→0,1(j=1,2,…,r∗) is the membership function of the system with respect to the fuzzy rule j. hj can be regarded as the normalized weight of eachIF-THEN rule, satisfying hj(ω(t))⩾0 and ∑j=1r∗hj(ω(t))=1.Throughout this paper, we assume thatB(0)=f(0)=g(0)=η(0)=0∈Rn and the following.(A1)
fi,gi,ηi,ρi all are Lipschitz continuous functions with Lipschitz constants Fi,Gi,Qi,Hi, respectively. In addition, there are positive constants F∗,G∗,Q∗ such that (13)fis⩽F∗,gis⩽G∗,ηis⩽Q∗.(A2)
There is positive definite diagonal matrixA¯ such that A(x(t))∈[0,A¯], and ai(s) is Lipschitz continuous with Lipschitz constant a~i, for all s∈R, i=1,2,…,n, where Axt=diag⁡a1x1t,a2x2t,…,anxnt is n-dimension diagonal matrix.(A3)
For eachi, there exists a constant mi>0 such that the function (ai(s)bi(s)-mis) is Lipschitz continuous with the Lipschitz constant li.Remark 3.
There are a large number of functionsai(·),bi(·) satisfying (A2) and (A3).
For example, we denote(14)φs=arctan⁡ss,s≠0,1,s=0.Obviously, φ(s) is continuous for all s∈R so that maxs∈Rφ(s) exists, and maxs∈Rφ(s)⩾φ(0)>0. Let mi=1+maxs∈Rφ(s), and then the constant mi>1. Next, let (15)ais=mi+φs,bis=s,and then (16)aisbis=mis+sφs,0<1⩽ais⩽2mi,bi0=0.Hence, (17)aisbis-mis=arctan⁡s,s≠0,0,s=0,and it is easy to prove that the above function (ai(s)bi(s)-mis) is differentiable, and aisbis-mis′⩽1 for all s∈R. So we can set li=1 such that (18)aisbis-mis-aitbit-mit⩽sups∈R⁡aisbis-mis′s-t⩽lis-t.In addition, we can prove that ai(s)=mi+φ(s) is also Lipschitz continuous. Indeed, (19)ais-ait=φs-φt.Below, we only need to prove φ(s) is Lipschitz continuous. Firstly, we claim that φ(s) is differentiable for all s∈R. In fact, we can get by L’Hospital’s rule and s=tan⁡r(20)φ′0=lims→0⁡φs-1s-0=lims→0⁡arctan⁡s/s-1s=lims→0⁡arctan⁡ss′=lims→0⁡s/1+s2-arctan⁡ss2=limr→0⁡tan⁡r/1+tan2⁡r-rtan2⁡r=limr→0⁡1/2sin⁡2r-rr2=limr→0⁡cos⁡2r-12r=limr→0⁡-2sin⁡2r2=0.
Hence,φ(s) is differentiable for all s∈R, and sups∈R⁡φ′s=sups≠0⁡arctan⁡s/s′. Further, Lagrangian differential mean value theorem derives (21)φs-φt⩽s-tsups≠0⁡arctan⁡ss′.Owing to (22)lims→0⁡arctan⁡ss′=0,lims→∞⁡arctan⁡ss′=lims→∞⁡s/1+s2-arctan⁡ss2=0,we can conclude that the continuous function arctan⁡s/s′ is bounded in all s≠0, which implies that sups≠0⁡arctan⁡s/s′ is a positive constant, and hence both φ(s) and ai(s) are Lipschitz continuous for all s∈R. This has proved that ai(s) and bi(s) defined as (15) satisfy (A2) and (A3).Remark 4.
Letχi(s)=ai(s)bi(s); we know that condition (A3) admits that χi(s) is a nonlinear function on s while [18, (H4)] derives that χi(s) can only be a linear function on s.Remark 5.
Since (H4) is replaced with (A3), the methods of [18] cannot be applied to this paper, and we have to formulate new contraction mapping, different from [18].Definition 6.
Impulsive fuzzy CGNNs (12) with initial condition (9) are said to be globally exponentially stable if, for any initial condition ϕ(s)∈C([-τ,0],Rn), there exists a pair of positive constants a and b such that (23)xt;s,ϕ⩽be-at,∀t>0,where the norm xt=∑i=1nxit21/2.Lemma 7 (contraction mapping theorem, see [32]).
LettingP be a contraction operator on a complete metric space Θ, then there exists a unique point θ∈Θ for which P(θ)=θ.


## 3. Main Results

If (A1)–(A3) hold, we can derive the following main result.Theorem 8.
Impulsive fuzzy CGNNs (12) with initial condition (9) are global exponential stability if there exists a positive number α<1, satisfying the following LMI condition: (24)L+A¯∑j=1r∗CjF+F∗A∑j=1r∗Cj+A¯∑j=1r∗DjG+G∗A∑j=1r∗Dj+τA¯∑j=1r∗EjQ+Q∗A∑j=1r∗Ej+1μH+MH-αM<0, where μ=infk=1,2,…(tk+1-tk)>0, and (25)F=diag⁡F1,F2,…,Fn,G=diag⁡G1,G2,…,Gn,Q=diag⁡Q1,Q2,…,Qn,H=diag⁡H1,H2,…,Hn,A=diag⁡a~1,a~2,…,a~n,M=diag⁡m1,…,mn,L=diag⁡l1,…,ln.Proof.
First of all, we need to formulate integral equations equivalent to (12).
Denote, for convenience,ζi(s)≜ai(s)bi(s)-mis, and then ζxt=ζ1x1t,ζ2x2t,…,ζnxntT. Thereby, we have (26)ζxt=AxtBxt-Mxt.And hence ζ(0)=A(0)B(0)-M·0=0.
Next, we claim that System (12) with initial condition (9) is equivalent to (27)xt=e-Mt∫0teMr-ζxr+Axr∑j=1r∗hjωrCjfxr+Djgxr-τr+Ej∫r-δrrηxsdsdr+e-Mtϕ0+e-Mt∑0<tk<teMtkρxtk,t⩾0,with initial condition (9).
On one hand, we can prove that the solution of System (27) with initial condition (9) is that of System (12) with initial condition (9).
Indeed, we get by (27) (28)eMtxt=∫0teMr-ζxr+Axr∑j=1r∗hjωrCjfxr+Djgxr-τr+Ej∫r-δrrηxsdsdr+ϕ0+∑0<tk<teMtkρxtk,t⩾0,
Differentiating both sides of (28) on t results in that for t⩾0,t≠tk, (29)MeMtxt+eMtdxtdt=eMt-ζxt+Axt∑j=1r∗hjωtCjfxt+Djgxt-τt+Ej∫t-δttηxsds,which generates the first equation of System (12).
Further, lett→ti- in (27) produce x(ti-)=limt→ti-x(t)=x(ti), and let t→ti+ in (27) come to x(ti+)=limt→ti+x(t)=x(ti)+ρ(x(ti)), which derives the second equation of (12).
Thus, we have proved the above claim.
On the other hand, we claim that the solution of System (12) with initial condition (9) is that of System (27) with initial condition (9).
In fact, multiplying both sides of the first equation of System (12) with eMt results in (30)eMtdxtdt+MeMtxt=eMt-ζxt+Axt∑j=1r∗hjωtCjfxt+Djgxt-τt+Ej∫t-δttηxsds,for all t⩾0,t≠tk. Moreover, integrating from tk-1+ε to t∈(tk-1,tk) gives (31)eMtxt=eMtk-1+εxtk-1+ε+∫tk-1+εteMs-ζxs+Axs∑j=1r∗hjωsCjfxs+Djgxs-τs+Ej∫s-δssηxrdrds,which yields, after letting ε→0+, (32)eMtxt=eMtk-1xtk-1++∫tk-1teMs-ζxs+Axs∑j=1r∗hjωsCjfxs+Djgxs-τs+Ej∫s-δssηxrdrds,for all t∈(tk-1,tk).
Throughout this paper, we assume thatε is a sufficiently small positive number. Now, taking t=tk-ε in (32) reaches (33)eMtk-εxtk-ε=eMtk-1xtk-1++∫tk-1tk-εeMs-ζxs+Axs∑j=1r∗hjωsCjfxs+Djgxs-τs+Ej∫s-δssηxrdrds,which yields by letting ε→0+(34)eMtkxtk=eMtk-1xtk-1++∫tk-1tkeMs-ζxs+Axs∑j=1r∗hjωsCjfxs+Djgxs-τs+Ej∫s-δssηxrdrds.Combining (32) and the above equation comes to (35)eMtxt=eMtk-1xtk-1++∫tk-1teMs-ζxs+Axs∑j=1r∗hjωsCjfxs+Djgxs-τs+Ej∫s-δssηxrdrds=eMtk-1xtk-1+∫tk-1teMs-ζxs+Axs∑j=1r∗hjωsCjfxs+Djgxs-τs+Ej∫s-δssηxrdrds+eMtk-1ρxtk-1,for all t∈(tk-1,tk],k=1,2,…. Thereby, we have (36)eMtk-1xtk-1=eMtk-2xtk-2+∫tk-2tk-1eMs-ζxs+Axs∑j=1r∗hjωsCjfxs+Djgxs-τs+Ej∫s-δssηxrdrds+eMtk-2ρxtk-2,⋮eMt2xt2=eMt1xt1+∫t1t2eMs-ζxs+Axs∑j=1r∗hjωsCjfxs+Djgxs-τs+Ej∫s-δssηxrdrds+eMt1ρxt1,eMt1xt1=ϕ0+∫0t1eMs-ζxs+Axs∑j=1r∗hjωsCjfxs+Djgxs-τs+Ej∫s-δssηxrdrds.Synthesizing the above analysis derives System (27). Hence, we have proved that each solution of (12) with initial condition (9) is that of (27) with initial condition (9). Now, we have proved that System (12) with initial condition (9) is really equivalent to integral equations (27) with initial condition (9).
To apply the contraction mapping theorem, we firstly define the complete metric spaceX as follows.
LetX be the space consisting of functions q(t):[-τ,∞)→Rn, satisfying the following:(a)
q(t) is continuous on t∈0,+∞∖tkk=1∞.(b)
q(t)=ϕ(t), for t∈-τ,0.(c)
limt→tk-q(t)=q(tk) and limt→tk+q(t) exists, for all k=1,2,….(d)
eβtq(t)→0∈Rn as t→+∞, where β>0 is a positive constant, satisfying β<λminM.
It is not difficult to verify that the product spaceX is a complete metric space if it is equipped with the following metric: (37)dist⁡q¯,q~=maxi=1,2,…,n⁡supt⩾-τ⁡q¯it-q~it,where q¯=q¯(t)=q¯1t,q¯2t,…,q¯ntT∈X and q~=q~(t)=q~1t,q~2t,…,q~ntT∈X.
Hence, we define the mappingP as follows: (38)Pxt=e-Mt∫0teMr-ζxr+Axr∑j=1r∗hjωrCjfxr+Djgxr-τr+Ej∫r-δrrηxsdsdr+e-Mtϕ0+e-Mt∑0<tk<teMtkρxtk,t⩾0,and P(x(s))=ϕ(s) for all s∈[-τ,0].
Below, we are to prove thatP is contraction mapping from X into X.
We may firstly proveP(X)⊂X. So we need to verify that P(x) satisfies conditions (a)–(d) for all x∈X.
Indeed, forx(t)∈X,P(x(t)) is continuous on t∈[0,+∞)∖tkk=1∞ owing to (38), and P(·) satisfies condition (a). Further, the definition of P implies that P(·) satisfies condition (b). Besides, we can derive from (38) that (39)Pxti-=limε→0+⁡Pxti-ε=Pxti,Pxti+=limε→0+⁡Pxti+ε=Pxti+ρxti,i=1,2,…,which comes to a conclusion that P(·) satisfies condition (c). Remark, the above convergence is under the metric of the metric space X. Below, all the convergence is in this sense. No longer repeat.
Throughout this section, we assume thatε is a sufficiently small positive real number.
Below, we need to prove thatP(·) satisfies condition (d).
Indeed, forx(t)∈X, we have (40)eβtPx=eβtPx⩽U1+U2+U3+U4+U5+U6,where (41)U1≜eβte-Mt∫0teMr-ζxrdr=eβte-Mt∫0teMrζxr-ζ0dr⩽eβte-Mt∫0teMrLxrdr=e-M-βIt∫0teM-βIreβrLxrdr.Owing to condition (d), for x(t)∈X,eβtx(t)→0 as t→+∞. So, for any given ε>0, there exists sufficiently large t∗>τ>0 such that (42)eβtxt⩽εξ,∀t⩾t∗,where the fixed vector (43)ξ≜1,1,…,1T∈Rn.When t>t∗, we have (44)U1⩽e-M-βIt∫0teM-βIreβrLxrdr⩽e-M-βIt∫0t∗eMrL∑i=1nsupr∈-τ,t∗⁡xirξdr+∫t∗teM-βIrLεξdr.It is obvious that (45)e-M-βIt∫0t∗eMrL∑i=1nsupr∈-τ,t∗⁡xirξdr⟶0,t⟶+∞. Besides, (46)e-M-βIt∫t∗teM-βIrdr⩽e-M-βItdiag⁡1m1-β,…,1mn-βeM-βItLεξ=diag⁡1m1-β,…,1mn-βLεξ,which together with arbitrariness implies that (47)e-M-βIt∫t∗teM-βIrdr⟶0,t⟶+∞.Now we have actually proved that U1→0 as t→+∞.
Similarly, we have(48)U2≜eβte-Mt∫0teMrAxr∑j=1r∗hjωrCjfxrdr⩽e-M-βIt∫0t∗eMrA¯∑j=1r∗CjFxrdr+∫t∗teM-βIrA¯∑j=1r∗CjFeβrxrdr.Similarly, we use the methods employed in (44)–(47), obtaining U2→0 as t→+∞.
Besides,(49)U3≜eβte-Mt∫0teMrAxr∑j=1r∗hjωrDjgxr-τrdr⩽e-M-βIt∫0t∗+τeMrA¯∑j=1r∗DjGxr-τrdr+∫t∗+τteM-βIreβτrA¯∑j=1r∗DjGeβr-τrxr-τrdr⩽e-M-βIt∫0t∗+τeMrA¯∑j=1r∗DjGsupr∈-τ,t∗+τ⁡xrdr+eβτ∫t∗+τteM-βIrA¯∑j=1r∗DjGεξdr.Similarly, we use the methods employed in (44)–(47), obtaining U3→0 as t→+∞.
Next,(50)U4≜eβte-Mt∫0teMrAxr∑j=1r∗hjωrEj∫r-δrrηxsdsdr⩽e-M-βIt∫0t∗+τeMrA¯∑j=1r∗Ej∫r-τrQxsdsdr+∫t∗+τteMrA¯∑j=1r∗Ej∫r-τrQxsdsdr⩽e-M-βIt∫0t∗+τeMrA¯∑j=1r∗EjτQ∑i=1nsups∈-τ,t∗+τ⁡xisξdr+∫t∗+τteMrA¯∑j=1r∗Ej∫r-τrQxsdsdr.It is clear that (51)e-M-βIt∫0t∗+τeMrA¯∑j=1r∗EjτQ∑i=1nsups∈-τ,t∗+τ⁡xisξdr⟶0ast⟶+∞.Besides, (52)e-M-βIt∫t∗+τteMrA¯∑j=1r∗Ej∫r-τrQxsdsdr⩽e-M-βIt∫t∗+τteMre-βr-τA¯∑j=1r∗Ej∫r-τrQeβsxsdsdr⩽εdiag⁡1m1-β,…,1mn-βeβτA¯∑j=1r∗EjτQξ.Similarly, we use the methods employed in (44)–(47), obtaining U4→0 as t→+∞.
Obviously,U5≜eβte-Mtϕ0→0 as t→+∞.
Below, we assume thattm<t∗⩽tm+1 and tj<t⩽tj+1 with j=0,1,2,…. Here, t0=0. (53)U6≜eβte-Mt∑0<tk<teMtkρxtk⩽e-M-βIt∑0<tk⩽t∗eMtkHxtk+∑t∗<tk<teMtkHxtk.Obviously, (54)e-M-βIt∑0<tk⩽t∗eMtkHxtk=e-M-βIt∑t1⩽tk⩽tm+1eMtkHxtk⟶0,t⟶+∞.Next, (55)e-M-βIt∑t∗<tk<teMtkHxtk⩽e-M-βIt∑tm+1⩽tk⩽tjeM-βItkHεξ⩽e-M-βIt1μμeM-βItjHεξ+∑tm+1⩽tk⩽tj-1tk+1-tkeM-βItkHεξ⩽e-M-βIt1μμeM-βItjHεξ+∫t∗teM-βIsdsHεξ⩽ε1μμ+diag⁡1m1-β,…,1mn-βHξ,which together with the arbitrariness of ε implies (56)e-M-βIt∑t∗<tk<teMtkHxtk⟶0,t⟶+∞.So we have proved that U6→0 as t→+∞.
Synthesizing the above analysis results ineβtP(x(t))→0 as t→+∞, for all x(t)∈X. Thus, P(·) satisfies all conditions (a)–(d), which derives P(X)⊂X.
Finally, we claim thatP:X→X is contraction mapping.
Indeed, for anyx,y∈X,(57)Px-Py⩽e-Mt∫0teMr-ζxr+Axr∑j=1r∗hjωrCjfxr+Djgxr-τr+Ej∫r-δrrηxsdsdr-∫0teMr-ζyr+Ayr∑j=1r∗hjωrCjfyr+Djgyr-τr+Ej∫r-δrrηysdsdr+e-Mt∑0<tk<teMtkρxtk-∑0<tk<teMtkρytk⩽K1+K2+K3+K4+K5,where (58)K1≜e-Mt∫0teMrζxr-ζyrdr⩽e-Mt∫0teMrLxr-yrdr⩽e-Mt∫0teMrLdrdist⁡x,yξ⩽e-MtM-1eMtLdist⁡x,yξ=M-1Ldist⁡x,yξ,K2≜e-Mt∫0teMrAxr∑j=1r∗hjωrCjfxr-Ayr∑j=1r∗hjωrCjfyrdr⩽e-Mt∫0teMrAxr∑j=1r∗Cjfxr-fyr+Axr-Ayr∑j=1r∗Cjfyrdr⩽e-Mt∫0teMrA¯∑j=1r∗CjFxr-yr+Adiag⁡x1r-y1r,…,xnr-ynr∑j=1r∗CjF∗ξdr⩽e-Mt∫0teMrA¯∑j=1r∗CjFdist⁡x,yξdr+e-Mt∫0teMrAdiag⁡x1r-y1r,…,xnr-ynr∑j=1r∗CjF∗ξdr⩽e-Mt∫0teMrdrA¯∑j=1r∗CjFdist⁡x,yξ+e-Mt∫0teMrAdist⁡x,yI∑j=1r∗CjF∗ξdr⩽M-1A¯∑j=1r∗CjF+F∗A∑j=1r∗Cjdist⁡x,yξ,K3≜e-Mt∫0teMrAxr∑j=1r∗hjωrDjgxr-τr-Ayr∑j=1r∗hjωrDjgyr-τrdr⩽e-Mt∫0teMrAxr∑j=1r∗Djgxr-τr-gyr-τr+Axr-Ayr∑j=1r∗Djgyr-τrdr⩽e-Mt∫0teMrA¯∑j=1r∗DjGxr-τr-yr-τr+Adiag⁡x1r-y1r,…,xnr-ynr∑j=1r∗DjG∗ξdr⩽e-Mt∫0teMrA¯∑j=1r∗DjGdist⁡x,yξ+Adist⁡x,yI∑j=1r∗DjG∗ξdr⩽e-MtM-1eMtA¯∑j=1r∗DjGdist⁡x,yξ+Adist⁡x,yI∑j=1r∗DjG∗ξ=M-1A¯∑j=1r∗DjG+G∗A∑j=1r∗Djdist⁡x,yξ,K4≜e-Mt∫0teMrAxr∑j=1r∗hjωrEj∫r-δrrηxsds-Ayr∑j=1r∗hjωrEj∫r-δrrηysds)dr⩽e-Mt∫0teMrAxr∑j=1r∗Ej∫r-δrrηxs-ηysds+Axr-Ayr∑j=1r∗Ej∫r-δrrηysdsdr⩽e-Mt∫0teMrA¯∑j=1r∗Ej∫r-τrQxs-ysds+Adiag⁡x1r-y1r,…,xnr-ynr∑j=1r∗Ej∫r-τrQ∗ξdsdr⩽M-1τA¯∑j=1r∗EjQ+Q∗A∑j=1r∗Ejdist⁡x,yξ,K5≜e-Mt∑0<tk<teMtkρxtk-∑0<tk<teMtkρytk⩽e-Mt∑0<tk<teMtkρxtk-ρytk⩽e-Mt∑0<tk<teMtkHxtk-ytk⩽dist⁡x,ye-Mt∑0<tk<teMtkHξ⩽1μdist⁡x,ye-Mt∑t1⩽tk⩽tj-1tk+1-tkeMtk+μeMtjHξ⩽1μdist⁡x,ye-Mt∫0teMsds+μeMtHξ⩽1μM-1H+Hdistx,yξ,where we assume tj<t⩽tj+1 with j=0,1,2,…. Here, t0=0.
Combining the above five inequalities and (24) produces (59)Pxt-Pyt⩽M-1L+M-1A¯CF+F∗AC+M-1A¯DG+G∗AD+M-1τA¯EQ+Q∗AE+1μM-1H+Hdistx,yξ<αdistx,yξ,and hence (60)distPx,Px⩽αdistx,y.Therefore, P:X→X is contraction mapping such that there exists the fixed point x(t) of P in X, which implies that x(t) is the solution for the impulsive fuzzy CGNNs (12) with initial condition (9), satisfying eβtx(t)→0 as t→+∞. So the proof is completed.Remark 9.
From the proof, we know that the formulated contraction mapping is different from that of existing literature involved in the fixed point technique and CGNNs models. One can also understand from Remark1 that we proposed in this paper a really feasible new condition on amplification function ai and behavior function bi. This implies that our result and methods are novelty versus [18, Theorem  4]. Besides, our conditions and result are also different from those of existing literature [17, 19–22] (see below “Table 1” and “Remark 13”).
In case of ignoring fuzzy factors and distributed delay (lettingE=0) in (12), we can get the following impulsive CGNNs with discrete delay(61)dxtdt=-AxtBxt-Cfxt-Dgxt-τt,t⩾0,t≠tk,xtk+-xtk-=ρxtk,k=1,2,….Corollary 10.
Impulsive CGNNs (61) with initial condition (9) are global exponential stability if there is a positive number α<1, satisfying the following LMI condition: (62)L+A¯CF+F∗AC+A¯DG+G∗AD+1μH+MH-αM<0, where μ=infk=1,2,…(tk+1-tk)>0.Remark 11.
As a corollary of our main result, the conditions and conclusion are better than the main result of [18] (see Remark 1).


## 4. Numerical Example

Example 1.
Consider the T-S fuzzy impulsive discrete and distributed delays CGNNs model as follows.
Fuzzy Rule  1. IF ω 1 ( t ) is 1/e-2ω1(t),THEN(63)dxtdt=-AxtBxt-C1fxt-D1gxt-τt-E1∫t-δttηxsds,t⩾0,t≠tk,xtk+-xtk-=ρxtk,k=1,2,…,xs=ϕs,s∈-τ,0.Fuzzy Rule  2. IF ω2(t) is 1-1/e-2ω1(t), THEN(64)dxtdt=-AxtBxt-C2fxt-D2gxt-τt-E2∫t-δttηxsds,t⩾0,t≠tk,xtk+-xtk-=ρxtk,k=1,2,…,xs=ϕs,s∈-τ,0,where r∗=2,F∗=0.2897,G∗=0.2679,Q∗=0.2102, (65)D1=0.12130.01100.1312,E1=0.15110.010100.1887,D2=0.181200.0020.1002,E2=0.181100.02100.1077,A¯=0.2111000.1192,C1=0.21130.011100.299,C2=0.181100.00210.1102,G=0.1011000.2102,L=0.1318000.1201,Q=0.1002000.2001,A=0.1812000.1901,F=0.1027000.2103,H=0.4586000.4386,M=2.9013002.8131.Let μ=0.5,τ=11.7, then we can use MATLAB LMI toolbox to solve the LMI condition (24), extracting the feasible datum (66)α=0.9981,which means 0<α<1. Thereby, we can conclude from Theorem 8 that the impulsive fuzzy dynamic equations are globally exponentially stable.Remark 12.
From Example1, the allowable upper bound of time-delays τ=11.7 and the pulse interval μ=0.5 indicate that the range of feasibility is extensive and broad.
Table1 shows the comparison of our Theorem 8 and Corollary 10 with other criteria of CGNNs (derived by fixed point theorems) about the behavior function bj and programmability for computer software.Remark 13.
From Table1, we know that the conditions of our behavior function bj are weaker than those of existing results published from 2007 to 2016. Moreover, our criterion is LMI-based, which is programmable for computer LMI toolbox. In addition, our Corollary 10 is better than [18, Theorem  4], the main result of [18] (see Remark 4 for details).


## 5. Conclusions

In many existing literatures related to Cohen-Grossberg Neural Networks, various fixed point theorems were applied to derive the stability criteria by imitating System (1). However, in this paper, the authors proposed a new condition on behavior functions and formulate new contraction mapping to derive a new LMI-based exponential stability. To a certain extent, this paper improves the previous related results (see Table 1, Remarks 4 and 13). Moreover, numerical example is presented to illustrate the feasibility and effectiveness of the proposed methods.



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