# LMI-Based Stability Criterion of Impulsive T-S Fuzzy Dynamic Equations via Fixed Point Theory

Authors: Ruofeng Rao; Zhilin Pu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By formulating a contraction mapping and the matrix exponential function, the authors apply linear matrix inequality (LMI) technique to investigate and obtain the LMI-based stability criterion of a class of time-delay Takagi-Sugeno (T-S) fuzzy differential equations. To the best of our knowledge, it is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. It is worth mentioning that LMI methods have high efficiency and other advantages in largescale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. At the end of this paper, a numerical example is presented to illustrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

In this paper, we consider a class of delayed impulsive differential equations, which admits some biomathematics, physics, and engineering backgrounds, including the famous cellular neural networks proposed by Chua and Yang in 1988 [1, 2]. In practice, both time delays and impulse are unavoidable and may cause undesirable dynamic network behaviors such as oscillation and instability. So the stability analysis for delayed impulsive neural networks has become a topic of great theoretic and practical importance in recent years [3–25]. However, the research skill of the above literature is mainly based on Lyapunov theory. And there are many difficulties in applications of corresponding theory to the specific problems [26–32]. Recently, Burton and other authors have applied fixed point theory to investigate the stability of deterministic systems and obtained some more applicable results [11, 26–42]. For example, in [11], the authors used Leray-Schauders fixed point theorem to obtain the stability criteria of neural networks. Besides, the contraction-mapping theory is also an important fixed point theory in studying the stability of dynamics equations (see, e.g., [11, 33, 39–42]). On the other hand, fuzzy logic theory has shown to be an appealing and efficient approach to dealing with the analysis and synthesis problems for complex nonlinear system [20–25]. In practice, the fuzzy model is far more important than stochastic model. 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. To the best of our knowledge, few authors have used the fixed point theorem to study the stability of Takagi-Sugeno fuzzy differential equations with impulses. In addition, the LMI-based stability criterion of neural networks has never been investigated or obtained via any of the fixed point theories. Such a situation motivates our present study. Motivated by the above related literature [3–9, 11, 26–42], we will not only apply the fixed point theory to study the impulsive Takagi-Sugeno fuzzy dynamics equations but also try to obtain the LMI-based stability criterion by applying the contraction-mapping theory. To the best of our knowledge, it is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. It is worth mentioning that LMI methods have high efficiency and other advantages in large-scale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. In the end of this paper, a numerical example is presented to illustrate the effectiveness of the proposed methods. Finally, a conclusion is given in the final chapter.


## 2. Preliminaries

Let us consider the following delayed differential equations:(1)dx(t)dt=-Bx(t)+Cf(x(t))dx(t)dt=+Dg(x(t-τ(t))),t⩾0,t≠tk,
equipped with the impulsive condition
(2)x(tk+)-x(tk)=ρ(x(tk)),k=1,2,…,
and the initial condition
(3)x(θ)=ϕ(θ),θ∈[-τ,0],
where x(t)=(x1(t),x2(t),…,xn(t))T∈Rn,ϕ(θ)∈C[[-τ,0],Rn]. Functions f(x)=(f1(x1(t)),f2(x2(t)),…,fn(xn(t)))T∈Rn, g(x(t-τ(t)))=(g1(x1(t-τ(t))),g2(x2(t-τ(t))),…,gn(xn(t-τ(t))))T∈Rn, ρ(x(t))=(ρ1(x1(t)),ρ2(x2(t)),…,ρn(xn(t)))T∈Rn, and time delays 0⩽τ(t)⩽τ for all i=1,2,…,n. 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 limits of x(t) at time tk, respectively. We always assume x(tk-)=x(tk), for all k=1,2,…. Similarly as in [33], we assume in this paper that f(0)=g(0)=ρ(0)=0∈Rn.Constant matrixB=diag(b1,b2,…,bn) is a positive definite diagonal matrix, and both C=(cij)n×n and D=(dij)n×n are matrices with n×n dimension. It is well gknown that the above equation admits its practical implications. For example, it can serve as a model of impulsive cellular neural networks with time-varying delays. The parameter cij denotes the connection weight of the jth neuron on the ith neuron at time t. And the parameter dij represents the connection strength of the jth neuron on the ith neuron at time t-τ(t). The constant bi represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and external inputs. fj(xj(t)) is the activation function of the jth neuron at time t, and gj(xj(t-τ(t))) represents the activation function of the jth neuron at time t-τ(t).Below, we describe the T-S fuzzy mathematical model with time delay as follows.Fuzzy Rulej:IFω1(t) is μj1 and …ωs(t) is μjs,  THEN
(4)dx(t)dt=-Bx(t)+Cjf(x(t))+Djg(x(t-τ(t))),11111111111111111t⩾0,t≠tk,k=1,2,…x(tk+)-x(tk)=ρ(x(tk)),k=1,2,…x(θ)=ϕ(θ),θ∈[-τ,0],
where ωk(t)(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 the IF-THEN rules, and s is the number of the premise variables. By the way of a standard fuzzy inference method, the system (4) is inferred as follows:
(5)dx(t)dt=-Bx(t)+∑j=1r‍hj(ω(t))[Cjf(x(t))+Djg(x(t-τ(t)))],11111111111111111111111t⩾0,t≠tk,k=1,2,…,x(tk+)-x(tk)=ρ(x(tk)),k=1,2,…,x(θ)=ϕ(θ),θ∈[-τ,0],
where ω(t)=[ω1(t),ω2(t),…,ωs(t)],hj(ω(t))=wj/∑k=1rwk(w(t)),wj(ω(t)):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 each IF-THEN rule, satisfying hj(ω(t))⩾0 and ∑j=1rhj(ω(t))=1.For convenience's sake, we introduce the following standard notations similarly as [10, (iii)–(X)]:
(6)Q=(qij)n×n>0(<0),Q=(qij)n×n⩾0(⩽0),Q1⩾Q2(Q1⩽Q2),Q1>Q2(Q1<Q2),λminΦ,theidentitymatrixIandthesymmetricterms*.
In addition, we denote |C|=(|cij|)n×n for any matrix C=(cij)n×n and |v|=(|v1|,|v2|,…,|vn|)T for any v=(v1,v2,…,vn)T∈Rn. Denote the finite set 𝒩={1,2,…,n}.Throughout this paper, we assume(A1)
fj is locally Lipschitz continuous, and there exists a positive constant Fj>0 such that |fj′(r)|⩽Fj for all r∈R at which fj is differentiable;(A2)
gj is locally Lipschitz continuous, and there exists a positive constant Gj>0 such that |gj′(r)|⩽Gj for all r∈R at which gj is differentiable;(A3)
ρj is locally Lipschitz continuous, and there exists a positive constant Hj>0 such that |ρj′(r)|⩽Hj for all r∈R at which ρj is differentiable.Lemma 1 (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
(7)f(y)-f(x)=𝔴(y-x),
where [x,y] denotes the segment connecting x and y.Remark 2.
From Lemma1, (A1)–(A3), and [10, equation (27)], we can similarly derive
(8)|f(x)-f(y)|⩽F|x-y|,x,y∈Rn,|g(x)-g(y)|⩽G|x-y|,x,y∈Rn,|ρ(x)-ρ(y)|⩽H|x-y|,x,y∈Rn,
where matrices F=diag(F1,F2,…,Fn), G=diag(G1,G2,…,Gn), and H=diag(H1,H2,…,Hn).Remark 3.
In many previous literature,fi,gi(i∈𝒩) are always assumed to be globally Lipschtiz continuous. However, the most common function fi(r)=r2 is not globally Lipschtiz continuous in R1. Note that we extend the functions f,g from global Lipschtiz continuous functions to locally Lipschtiz continuous functions. Obviously, fi(r)=r2 is a local Lipschitz continuous function.
Similarly as is [11, Definition 2.2], the exponential stability is defined as follows.Definition 4.
Dynamic equation (5) is said to be 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
(9)∥x(t;s,ϕ)∥⩽be-at,∀t>0,
where the norm ∥x(t)∥=(∑i=1n|xi(t)|2)1/2.


## 3. Main Result

Before giving the main result of this paper, we need to define the matrix exponential function as follows.Definition 5.
For a diagonal constants matrixB=diag(b1,b2,…,bn), we denote the matrix exponential function eBt=diag(eb1t,eb2t,…,ebnt) for all t∈R.From the above definition of the matrix exponential function, we are not difficult to obtain the following lemma.Lemma 6.
LetB be a diagonal constants matrix, and let eBt be the matrix exponential function of B. Then, we have(1)
(d/dt)eBt=BeBt,t∈R,(2)
(d/dt)(eBtη)=BeBtη,t∈R,where η=(η1,η2,…,ηn)T∈Rn, and each ηi∈R(i=1,2,…,n) is a constant.In addition, we need to define the rule on vectors inRn as follows.Definition 7.
v
⩽
w if vi-wi⩽0 for all i∈𝒩, where v=(v1,v2,…,vn)T∈Rn,w=(w1,w2,…,wn)T∈Rn.
Now, we present the main result of this paper as follows.Theorem 8.
Assume that there exists a positive constantδ such that infk=1,2,…(tk+1-tk)⩾δ. In addition, there exists a constant 0<λ<1 such that
(10)B-1∑j=1r‍(|Cj|F+|Dj|G)+1δB-1H+H<λI,
where B-1 is the inverse matrix of B. Then, the impulsive fuzzy dynamic equation (5) is exponentially stable in the mean square.Proof.
To apply the fixed point theory, we firstly define a complete metric spaceΩ as follows.
LetΩ be the space consisting of functions q(t):[-τ,∞)→Rn, satisfying that(a)
q(t) is continuous on t≠tk(k=1,2,…),(b)
limt→tk-q(t) and limt→tk+q(t) exist, and q(tk-)=q(tk) for all k=1,2,…,(c)
q(t)=ϕ(t) for t∈[-τ,0],(d)
eβtq(t)→0∈Rn as t→∞, where β>0 is a positive constant, satisfying β<λminB.It is not difficult to verify that the above-mentioned space Ω is a complete metric space if it is equipped with the following metric:
(11)dist(q¯,q~)=maxi=1,2,…,n(supt⩾-τ|q¯i(t)-q~i(t)|),
where q¯=q¯(t)=(q¯1(t),q¯2(t),…,q¯n(t))T∈Ω, and q~=q~(t)=(q~1(t),q~2(t),…,q~n(t))T∈Ω.
Remark  9. Here, we consider the above-defined metric, which is different from those of some previous related literature (see, e.g., [33]) so that the LMI-based stability criterion in this paper may be obtained expediently.
Next, we formulate and define a contraction mappingP:Ω→Ω, which may be divided into three steps.
Step  1. Formulating the mapping.
Letx(t) be a solution of the fuzzy equation (5).
Then, fort⩾0,t≠tk, we have
(12)dx(t)dt(eBtx(t))=BeBtx(t)+eBtdx(t)dtdx(t)dt(eBtx(t))=eBt∑j=1r‍hj(ω(t))[(x(t-τ(t)))Cjf(x(t))(eBtx(t))=eBthj(ω(t))+Djg(x(t-τ(t)))].
Further, we get by the integral nature
(13)x(t)=e-Bt{∫0t‍eBs∑j=1rhj(ω(s))[(x(s-τ(s)))Cjf(x(s))1111111111111111111111+Djg(x(s-τ(s)))]dse-Bt11+η∫0t‍eBs∑j=1r‍},t⩾0,
where η∈Rn is the vector to be determined.
From (2) and (3) or x(0)=ϕ(0), it is not difficult to conclude η=ϕ(0)+∑0<ti<teBtiρ(x(ti)). And, hence,
(14)x(t)=e-Bt{∫0t‍eBs∑j=1r‍hj(ω(s))11111111111111×[Cjf(x(s))+Djg(x(s-τ(s)))]ds111111+ϕ(0)+∑0<ti<t‍eBtiρ(x(ti))∑j=1r‍},t⩾0.
On the other hand, it follows from (14) that
(15)x(tk)=e-Btk{∫0tk‍eBs∑j=1r‍hj(ω(s))x(tk)=e-Btk111×[(x(s-τ(s)))Cjf(x(s))11111111111111111+Djg(x(s-τ(s)))]ds1111111+ϕ(0)+∑0<ti<tk‍eBtiρ(x(ti))∑j=1r‍},x(tk+ε)=e-Btk+ε{∫0tk+ε‍eBs∑j=1r‍hj(ω(s))x(tk+ε)=e-Btk+ε111×[(x(s-τ(s)))Cjf(x(s))11111111111111111111+Djg(x(s-τ(s)))]ds111111111+ϕ(0)+∑0<ti<tk+ε‍eBtiρ(x(ti))∑j=1r‍}.
Letε→0+; then we can derive from the two equations above that
(16)limε→0+x(tk+ε)-x(tk)=x(tk+)-x(tk)=ρ(x(tk)).
So, we may define the mapping P on the space Ω as follows:
(17)Px(t)=e-Btϕ(0)+e-Bt∫0t‍eBs∑j=1r‍hj(ω(s))=e-Btϕ(0)+e-Bt∫0t‍eBs1×[(x(s-τ(s)))Cjf(x(s))111111111111111111111111+Djg(x(s-τ(s)))]ds=+e-Bt∑0<ti<t‍eBtiρ(x(ti)),
on t⩾0, and Px(s)=ϕ(s) on s∈[-τ,0].
Step  2. We claim that Px(t)∈Ω for any x(t)∈Ω. That is, Px(t) satisfies the conditions (a)–(d) of Ω.
Indeed, sincePx(s)=ϕ(s) on s∈[-τ,0], the condition (c) is satisfied. It is obvious from (17) that Px(t) is continuous on t≠tk and t⩾0. And then the condition (a) is satisfied.
Next, we verify the condition (b).
Indeed, for any giventk, we can get from (17)
(18)Px(tk+ε)-Px(tk)=π1+π2+π3,
where
(19)π1=e-B(tk+ε)ϕ(0)-e-B(tk)ϕ(0),π2=e-B(tk+ε)∫0tk+ε‍eBs∑j=1r‍hj(ω(s))π2=111111111111111×[(x(s-τ(s)))Cjf(x(s))1111111111111111111111+Djg(x(s-τ(s)))]dsπ2=-e-Btk∫0tk‍eBs∑j=1r‍hj(ω(s))π2=1111111111111×[(x(s-τ(s)))Cjf(x(s))11111111111111111111+Djg(x(s-τ(s)))]ds,π3=e-B(tk+ε)∑0<ti<tk+ε‍eBtiρ(x(ti))π3=-e-Btk∑0<ti<tk‍eBtiρ(x(ti)).
Obviously,π1→0 and π2→0 as ε→0. In addition, letting ε→0-, we have π3→0. Letting ε→0+, we get by (18)
(20)Px(tk+)-Px(tk)=limε→0+π3=ρ(x(tk)).
So the condition (b) is satisfied.
Finally, it is followed from (17) that
(21)eβtPx(t)=eβte-Btϕ(0)eβtPx(t)=+eβte-Bt∫0t‍eBs∑j=1r‍hj(ω(s))eβtPx(t)=111111111111×[(x(s-τ(s)))Cjf(x(s))1111111111111111111111111+Djg(x(s-τ(s)))]dseβtPx(t)=+eβte-Bt∑0<ti<t‍eBtiρ(x(ti)).
Obviously, eβte-Btϕ(0)→0∈Rn as t→∞.
On the other hand, it follows fromeβtx(t)→0 that, for any given ε>0, there exists a corresponding constant t* such that |eβtx(t)|<εu for all t⩾t*, where
(22)u=(1,1,…,1)T∈Rn.
In addition, (A3) derives
(23)|eβte-Bt∑0<ti<t‍eBtiρ(x(ti))|⩽eβte-Bt∑0<ti<t‍eBtiH|x(ti)|⩽eβte-Bt(∑0<ti⩽t*‍eBtiH|x(ti)|+∑t*<ti<t‍eBtiH|x(ti)|).
Obviously,
(24)eβte-Bt∑0<ti⩽t*‍eBtiH|x(ti)|=e(βI-B)t∑0<ti⩽t*‍eBtiH|x(ti)|⟶0∈Rn,t⟶∞.
Besides,(25)eβte-Bt∑t*<ti<t‍eBtiH|x(ti)|=e(βI-B)t∑t*<ti<t‍e(B-βI)tiH(eβti|x(ti)|)⩽εe(βI-B)t∑t*<ti<t‍e(B-βI)tiHu=(εe(β-b1)t∑t*<ti<t‍e(b1-β)tiH1εe(β-b2)t∑t*<ti<t‍e(b2-β)tiH2⋮εe(β-bn)t∑t*<ti<t‍e(bn-β)tiHn).
To proveeβte-Bt∑t*<ti<teBtiH|x(ti)|→0∈Rn, we only need to prove
(26)εe(β-b1)t∑t*<ti<t‍e(b1-β)tiH1⟶0,t⟶∞,
for the other cases can be similarly proved.
Indeed, we may as well assumetk<t⩽tk+1 and tm-1<t*⩽tm. Then, we have
(27)εe(β-b1)t∑t*<ti<t‍e(b1-β)tiH1=εδH1e(β-b1)t∑tm⩽ti⩽tk‍δe(b1-β)ti⩽εδH1e(β-b1)t(δe(b1-β)tk+∑tm⩽ti⩽tk-1‍(ti+1-ti)e(b1-β)ti)⩽εδH1e(β-b1)t(δe(b1-β)tk+∫t*t‍e(b1-β)sds)⩽εH1e-(b1-β)(t-tk)+H1δ(b1-β)ε,
which implies that (26) holds. And, hence, eβte-Bt∑t*<ti<teBtiH|x(ti)|→0∈Rn,t→∞.
Below, we only need to prove(28)eβte-Bt∫0t‍eBs∑j=1r‍hj(ω(s))1111111111×[(x(s-τ(s)))Cjf(x(s))1111111111111111+Djg(x(s-τ(s)))]ds⟶0,111111111111111111111111111111t⟶∞.
In fact, it follows fromeβtx(t)→0 that, for any given ε>0, there exists a corresponding constant t* such that |eβtx(t)|<εu for all t⩾t*. Then, we get by (A1)
(29)|eβte-Bt∫0t‍eBs∑j=1r‍hj(ω(s))Cjf(x(s))ds|⩽e-(B-βI)t∫0t*‍eBs∑j=1r‍|Cj|F|x(s)|ds+e-(B-βI)t∫t*t‍eBs∑j=1r‍|Cj|F|x(s)|ds.
On the one hand,(30)e-(B-βI)t∫0t*‍eBs∑j=1r‍|Cj|F|x(s)|ds⩽e-(B-βI)t∫0t*‍eBs∑j=1r‍|Cj|F(∑i∈𝒩‍sups∈[0,t*]|xi(s)|)uds⩽(∑i=1m‍wi)e-(B-βI)t∫0t*‍eBsuds⟶0∈Rn,t⟶∞,
where w=∑j=1r|Cj|F(∑i∈𝒩sups∈[0,t*]|xi(s)|)u=(w1,w2,…,wn)T∈Rn.
On the other hand,(31)e-(B-βI)t∫t*t‍eBs∑j=1r‍|Cj|F|x(s)|ds⩽εe-(B-βI)t∫t*t‍e(B-βI)sw*ds⩽ε(∑i=1n‍wi*)(1b1-β,1b2-β,…,1bn-β)T,
where w*=∑j=1r|Cj|Fu=(w1*,w2*,…,wn*)T∈Rn.
Now we can conclude from (29)–(31) that
(32)eβte-Bt∫0t‍eBs∑j=1r‍hj(ω(s))×[Cjf(x(s))]ds⟶0∈Rn,t⟶∞.
Similarly, we can prove
(33)eβte-Bt∫0t‍eBs∑j=1r‍hj(ω(s))×[Djg(x(s-τ(s)))]ds⟶0∈Rn,t⟶∞.
Indeed, we can similarly define the corresponding constantT* for any given ε>0, satisfying |eβtx(t)|<εu for all t⩾T*. Then, we have
(34)|eβte-Bt∫0t‍eBs∑j=1r‍hj(ω(s))[Djg(x(s-τ(s)))]ds|⩽eβte-Bt(∫0T*‍eBs∑j=1r‍|Dj|G|x(s-τ(s))|ds⩽eβte-Bt+∫T*t‍eBs∑j=1r‍|Dj|G|x(s-τ(s))|ds).
Similarly, we can prove
(35)eβte-Bt∫0T*‍eBs∑j=1r‍|Dj|G|x(s-τ(s))|ds⩽(∑i=1m‍w¯i)e-(B-βI)t∫0T*‍eBsuds⟶0∈Rn,t⟶∞,
where w¯=∑j=1r|Dj|G(∑i∈𝒩sups∈[-τ,T*]|xi(s)|)u=(w¯1,w¯2,…,w¯n)T∈Rn.
On the other hand,(36)eβte-Bt∫T*t‍eBs∑j=1r‍|Dj|G|x(s-τ(s))|ds⩽εeβτe-(B-βI)t∫T*t‍e(B-βI)sw¯*ds⩽ε(∑i∈𝒩‍w¯i*)eβτ(1b1-β,1b2-β,…,1bn-β)T,
where w¯*=∑j=1r|Dj|Gu=(w¯1*,w¯2*,…,w¯n*)T∈Rn.
From (34)–(36), we can conclude that (33) holds. Hence, the condition (d) is satisfied.
Step  3. Below, we only need to prove that P is a contraction mapping.
Indeed, for anyx=x(t)=(x1(t),x2(t),…,xn(t))T,y=y(t)=(y1(t),y2(t),…,yn(t))T∈Ω, we estimate |Px(t)-Py(t)|⩽K1+K2+K3, where
(37)K1=e-Bt∫0t‍eBs∑j=1r‍|Cj||f(x(s))-f(y(s))|dsK1⩽e-Bt∫0t‍eBs∑j=1r‍|Cj|F|x(s)-y(s)|ds,K2=e-Bt∫0t‍eBs∑j=1r‍hj(ω(s))|Dj|K11111111111×|g(x(s-τ(s)))-g(y(s-τ(s)))|ds,K3=e-Bt∑0<ti<t‍eBti|ρ(x(ti))-ρ(y(ti))|.
From mathematical analysis and computation, we can derive(38)K1=e-Bt∫0t‍eBs∑j=1r‍|Cj||f(x(s))-f(y(s))|dsK1⩽e-Bt∫0t‍eBs∑j=1r‍|Cj|F|x(s)-y(s)|dsK1⩽dist(x,y)e-Bt∫0t‍eBs∑j=1r‍|Cj|FudsK1⩽(B-1∑j=1r‍|Cj|F)dist(x,y)u.
Similarly, we have(39)K2=e-Bt∫0t‍eBs∑j=1r‍hj(ω(s))|Dj||g(x(s-τ(s)))K2=e-BteeeeeeeeBshj(ω(s))|Dj|-g(y(s-τ(s)))|dsK2⩽(B-1∑j=1r‍|Dj|G)dist(x,y)u,K3⩽e-Bt∑0<ti<t‍eBtiH|(x(ti))-(y(ti))|K3⩽1δdist(x,y)e-Bt(∑t1⩽ti⩽tk-1‍eBti(ti+1-ti)HuK3⩽1δdist(x,y)e-Bt11+eBtkδHu∑t1⩽ti⩽tk-1‍)K3⩽1δdist(x,y)e-Bt(∫0t‍eBsHuds+eBtδHu)K3⩽dist(x,y)(1δB-1H+H)u.
Combining the above three inequalities results in(40)|Px(t)-Py(t)|⩽(B-1∑j=1r‍(|Cj|F+|Dj|G)+1δB-1H+H)udist(x,y)<λIudist(x,y),
and hence
(41)dist(P(x),P(y))⩽λdist(x,y).
Therefore,P:Ω→Ω is a contraction mapping such that there exists the fixed point x(t) of P in Ω, which implies that x(t) is the solution for the the impulsive fuzzy dynamic equation (5), satisfying eβt∥x(t)∥→0 as t→∞. So the proof is completed.


## 4. Numerical Example

Example 1.
Consider the T-S fuzzy impulsive dynamic equations as follows.
Fuzzy Rule 1:
IFω1(t) is 1/e-2ω1(t), THEN
(42)dx(t)dt=-Bx(t)+C1f(x(t))+D1g(x(t-τ(t))),dx(t)dt=t⩾0,t≠tk,k=1,2,…x(tk+)-x(tk)=ρ(x(tk)),k=1,2,…x(θ)=ϕ(θ),θ∈[-τ,0],
Fuzzy Rule 2:
IFω2(t) is 1-1/e-2ω1(t), THEN
(43)dx(t)dt=-Bx(t)+C2f(x(t))+D2g(x(t-τ(t))),dx(t)dt=111111t⩾0,t≠tk,k=1,2,…x(tk+)-x(tk)=ρ(x(tk)),k=1,2,…x(θ)=ϕ(θ),θ∈[-τ,0],
where
(44)B=(2001.9),C1=(-0.2000.3)=D1,C2=(0.300-0.1)=D2,F=(0.1000.2)=G,H=(0.7000.6).
Let δ=1.5. Then we can use Matlab LMI toolbox to solve the LMI condition (10) and obtain tmin=-0.0046<0 which implies it is feasible (see [10, Remark 29(3)] for details). Further, extracting the datum shows
(45)λ=0.9883,
which means 0<λ<1. Thereby, we can conclude from Theorem 8 that this impulsive fuzzy dynamic equation is exponentially stable in the mean square.


## 5. Conclusion

By formulating a contraction mapping and the matrix exponential function, the author applies linear matrix inequality (LMI) technique to investigate and obtain the LMI-based stability criterion of a class of time-delay Takagi-Sugeno (T-S) fuzzy differential equations. It is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. The LMI methods have high efficiency and other advantages in large-scale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. A numerical example is presented to illustrate the effectiveness of the proposed methods. In the end of this paper, we have to point out that there are still many difficulties in obtaining the LMI-based stability criteria for some other dynamics equations, such as Cohen-Grossberg neural networks (see, e.g., [11, 19]) and other neural networks. These problems remain open and challenging.



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*Source: 261353-2013-09-25.xml*