# Fixed Points and Exponential Stability for Impulsive Time-Delays BAM Neural Networks via LMI Approach and Contraction Mapping Principle

Authors: Ruofeng Rao; Zhilin Pu; Shouming Zhong; Xinggui Li
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

The fixed point technique has been employed in the stability analysis of time-delays bidirectional associative memory (BAM) neural networks with impulse. By formulating a contraction mapping in a product space, a new LMI-based exponential stability criterion was derived. Lately, fixed point methods have educed various good results inspiring this work, but those criteria cannot be programmed by a computer. In this paper, LMI conditions of the obtained result can be applicable to computer Matlab LMI toolbox which meets the need of the large-scale calculation in real engineering. Moreover, a numerical example and a comparable table are presented to illustrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

Bidirectional associative memory (BAM) neural networks model was originally introduced by Kosko [1, 2]: (1)x˙i=-aixit+∑j=1pwjigjyjt+Ii,i=1,2,…,n,y˙j=-bjyjt+∑i=1nvijgixit+Jj,j=1,2,…,n.Thanks to its generalization of the single-layer autoassociative Hebbian correlation to two-layer pattern-matched heteroassociative circuits, widespread applications have been found in various areas, such as automatic control, signal and image processing, pattern recognition, artificial intelligence, parallel computation, and optimization problems. Often, a stable equilibrium of BAM neural networks is the important precondition of the successful applications. There are many factors influencing the stability of neural networks, in which pulse and time delay are usually the main reasons [3–8]. So, the stability analysis of impulse or delays system has become a hot topic. All the time, the Lyapunov functional method is employed to deduce stability criteria [9–20]. But every method may have its limitations. During the recent decades, other techniques have been developed to investigate the stability, in which the fixed point method is always one of those alternatives [21–27]. For example, in 2015, Zhou utilized Brouwer’s fixed point theorem to prove the existence and uniqueness of equilibrium of the hybrid BAM neural networks with proportional delays and finally constructed appropriate delay differential inequalities to derive the stability of equilibrium [28]. In [29], Banach fixed point theorem was applied to show the existence of the unique equilibrium of BAM neural networks with time-varying delays in the leakage terms, and then the Lyapunov functional method was for demonstrating the global exponential stability. Different from [28, 29], we shall use Banach fixed point theorem deriving straightway the stability criterion of impulsive time-delay BAM neural networks, in which LMI conditions facilitate computer programming. Finally, a numerical example is presented to illustrate the effectiveness of the proposed methods.For the sake of convenience, we introduce the following standard notations [18]:(i)
L=(lij)n×n>0(<0): a positive (negative) definite matrix; that is, yTLy>0(<0) for any 0≠y∈Rn.(ii)
L=(lij)n×n⩾0(⩽0): a semipositive (seminegative) definite matrix; that is, yTLy⩾0(⩽0) for any y∈Rn.(iii)
L1⩾L2(L1⩽L2): this means matrix (L1-L2) is a semipositive (seminegative) definite matrix.(iv)
L1>L2(L1<L2): this means matrix (L1-L2) is a positive (negative) definite matrix.(v)
λmax(Φ) and λmin(Φ) denote the largest and the smallest eigenvalue of matrix Φ, respectively.(vi)
Denote|L|=(|lij|)n×n for any matrix L=(lij)n×n.(vii)
|u|=(|u1|,|u2|,…,|un|)T for any vector u=(u1,u2,…,un)T∈Rn.(viii)
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.(ix)
I is identity matrix with compatible dimension.Remark 1.
Different from the methods of [28, 29], it will be the first time to utilize contraction mapping principle to infer directly the LMI-based stability criterion of BAM neural networks, convenient for computer programming. Recently, there have been a lot of good results and methods [21–29] enlightening our current work. In this paper, we shall propose the LMI-based criterion, novel against the existing results, published from 2013 to 2016 (see Remark 10 and Table 1).Table 1
Comparing Theorem6 with other existing results.
Types of equations Fixed point methods LMI-based? Theorem6 Impulsive BAM neural networks Contraction mapping principle Yes [28, Th. 2–4] BAM neural networks Brouwer’s fixed theorem No [29, Th. 2.1, Th. 3.1] BAM neural networks Contraction mapping principle No [22, Th. 1.2] Neural networks Contraction mapping principle No [23, Th. 3.1] Cellular neural networks Krasnoselskii fixed point theorem No [21, Th. 2.1] Neutral differential equations Contraction mapping principle No [30, Th. 1.1-1.2] Neural networks Schauder fixed point theorem No


## 2. Preliminaries

Consider the following delayed differential equations:(2)dxtdt=-Axt+Cfyt-τt,t∈0,+∞,t≠tk,k=1,2,…,dytdt=-Byt+Dgxt-ht,t∈0,+∞,t≠tk,k=1,2,…,xtk+-xtk=ρxtk,ytk+-ytk=ρytk,k=1,2,…,xs=ξs,ys=ηs,s∈-τ,0,where x(t)=x1t,x2t,…,xntT, y(t)=(y1t,y2t,…,ynt)T∈Rn, and ξs,ηs∈C(-τ,0,Rn). Here, C-τ,0,Rn represents the space of continuous functions from [-τ,0] to Rn. Active functions fxt-τt=(f1x1t-τt,f2x2t-τt,…,fn(xn(t-τ(t))))T, g(x(t-h(t)))=(g1(x1(t-h(t))),g2(x2(t-h(t))),…,gn(xn(t-h(t))))T∈Rn, impulsive function ρ(x(t))=(ρ1(x1(t)),ρ2x2t,…,ρn(xn(t)))T∈Rn, and time delays 0⩽τ(t),h(t)⩽τ. 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. We always assume x(tk-)=x(tk), for all k=1,2,…. Similar to [23], we assume in this paper that f(0)=g(0)=ρ(0)=0∈Rn. Constant matrices A=diaga1,a2,…,an, B=diagb1,b2,…,bn are positive definite diagonal matrices, and both C=(cij)n×n and D=(dij)n×n are matrices with n×n dimension.Throughout this paper, we assume thatF=diag(F1,F2,…,Fn), G=diag(G1,G2,…,Gn), and H=diag(H1,H2,…,Hn) are diagonal matrices, satisfying(A1)
fx-fy⩽Fx-y, x,y∈Rn.(A2)
gx-gy⩽Gx-y, x,y∈Rn.(A3)
ρx-ρy⩽Hx-y, x,y∈Rn.Definition 2.
Dynamic equations (2) are said to be globally exponentially stable if, for any initial condition ξ(s),η(s)∈C-τ,0,Rn, there exist a pair of positive constants a and b such that (3)xt;s,ξ,ηyt;s,ξ,η⩽be-at,∀t>0,where the norm x(t)y(t)=∑i=1nxit2+∑i=1nyit21/2, and x=x1,…,xn, y=(y1,…,yn)∈Rn.Definition 3.
For a diagonal constants matrixB=diag(b1,b2,…,bn), one denotes the matrix exponential function eBt=diag(eb1t,eb2t,…,ebnt) for all t∈R.Lemma 4 (see [31] contraction mapping theorem).
LetP be a contraction operator on a complete metric space Θ; then, there exists a unique point θ∈Θ for which P(θ)=θ.Lemma 5.
LetB be a diagonal constants matrix and eBt be the matrix exponential function of B. Then, one has(1)
d/dteBt=BeBt, t∈R;(2)
d/dt(eBtα)=BeBtα, t∈R,
whereα=α1,α2,…,αnT∈Rn and each αi∈R(i=1,2,…,n) is a constant.


## 3. Main Result

Theorem 6.
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 (4)CF+1δH+AH-λA<0,DG+1δH+BH-λB<0,and then the impulsive fuzzy dynamic equations (2) are globally exponentially stable, where δ=infk=1,2,…tk+1-tk>0.Proof.
To apply the fixed point theory, we firstly define a complete metric spaceΩ=Ω1×Ω2 as follows.
LetΩi(i=1,2) be the space consisting of functions qi(t):[-τ,∞)→Rn, satisfying the following:(a)
qi(t) is continuous on t∈0,+∞∖{tk}k=1∞.(b)
q1(t)=ξ(t), q2(t)=η(t), for t∈[-τ,0].(c)
limt→tk-qi(t)=qi(tk), and limt→tk+qi(t) exists, for all k=1,2,….(d)
eγtqi(t)→0∈Rn as t→∞, where γ>0 is a positive constant, satisfying γ<min{λminA,λminB}.
It is not difficult to verify that the product spaceΩ is a complete metric space if it is equipped with the following metric: (5)distq¯,q~=maxi=1,2,…,2n-1,2nsupt⩾-τq¯it-q~it,where (6)q¯=q¯t=q¯1tq¯2t=q¯1t,q¯2t,…,q¯2ntT∈Ω,q~=q~t=q~1tq~2t=q~1t,…,q~2ntT∈Ω,and q¯i∈Ωi, q~i∈Ωi, i=1,2.
Next, we are to construct a contraction mappingP:Ω→Ω, which may be divided into three steps.
Step  1 (formulating the mapping). Let (xT(t),yT(t))T=(x1(t),x2(t),…,xn(t),y1(t),…,yn(t))T be a solution of system (2).
Then, fort⩾0, t≠tk, we have (7)dxtdteAtxt=AeAtxt+eAtdxtdt=eAtCfyt-τt.Further, we get by the integral property (8)xt=e-At∫0teAsCfys-τsds+χ,t⩾0,where χ∈Rn is the vector to be determined.
Next, we claim thatχ=ξ(0)+∑0<tk<teAtkρxtk, and (9)xt=e-Atξ0+e-At∫0teAsCfys-τsds+∑0<tk<teAtkρxtk,t⩾0.
Indeed, on the one hand, we can conclude from (9) that (10)eAtxt=ξ0+∫0teAsCfys-τsds+∑0<tk<teAtkρxtk. For t⩾0, t≠tk, taking the time derivative of both sides leads to (11)eAtdxtdt+AeAtxt=ddteAtxt=eAtCfyt-τt,
or(12)dxtdt+Axt=Cfyt-τt,which is the first equation of system (2).
Moreover, ast→tj-, we can gain by (9) (13)xtj-=limε→0+xtj-ε=xtj,j=1,2,…,xtj+=limε→0+xtj+ε=xtj+ρxtj,j=1,2,….
On the other hand, multiplying both sides of the first equation of system (2) by eAt yields (14)eAtdxtdt+AeAtxt=eAtCfyt-τt,t⩾0,t≠tk.Moreover, integrating from tk-1+ε to t∈(tk-1,tk) gives (15)eAtxt=eAtk-1+εxtk-1+ε+∫tk-1+εteAsCfys-τsds,which yields, after letting ε→0+,(16)eAtxt=eAtk-1xtk-1++∫tk-1teAsCfys-τsds,t∈tk-1,tk.Throughout this paper, we assume that ε is a sufficient small positive number. Now, taking t=tk-ε in the above equation yields (17)eAtk-εxtk-ε=eAtk-1xtk-1++∫tk-1tk-εeAsCfys-τsds,which yields by letting ε→0+(18)eAtkxtk=eAtk-1xtk-1++∫tk-1tkeAsCfys-τsds.So, we have actually got (19)eAtxt=eAtk-1xtk-1++∫tk-1teAsCfys-τsds=eAtk-1xtk-1+∫tk-1teAsCfys-τsds+eAtk-1ρxtk-1,for all t∈(tk-1,tk], k=1,2,…. Furthermore, (19) generates (20)eAtk-1xtk-1=eAtk-2xtk-2+∫tk-2tk-1eAsCfys-τsds+eAtk-2ρxtk-2,⋮eAt2xt2=eAt1xt1+∫t1t2eAsCfys-τsds+eAt1ρxt1,eAt1xt1=ξ0+∫0t1eAsCfys-τsds.Making a synthesis of the above equations results in (9).
Similarly, we can obtain(21)yt=e-Btη0+e-Bt∫0teBsDgxs-hsds+∑0<tk<teBtkρytk,t⩾0.
So, we may define the mappingP on the space Ω as follows: (22)Pxtyt=e-Atξ0+e-At∫0teAsCfys-τsds+∑0<tk<teAtkρxtke-Btη0+e-Bt∫0teBsDgxs-hsds+∑0<tk<teBtkρytk,fort∈0,+∞,(23)Pxtyt=ξtηt,fort∈-τ,0.Step  2. We claim that Pxtyt∈Ω for any xtyt∈Ω. In other words, we need to prove that Pxtyt must satisfy conditions (a)–(d) of Ω.
Indeed,P(·) satisfies condition (b) due to (23).
Besides, letε be a real number; we have (24)Pxt+εyt+ε=e-At+εξ0+e-At+ε∫0t+εeAsCfys-τsds+∑0<tk<t+εeAtkρxtke-Bt+εη0+e-Bt+ε∫0t+εeBsDgxs-hsds+∑0<tk<t+εeBtkρytk.In (24), letting ε→0 brings about (25)limε→0⁡Pxt+εyt+ε=Pxtyt,∀t≠tk,t⩾0,which implies that Pxtyt is continuous on t⩾0, t≠tk. Here, the convergences are under the metric defined on the space Ω. Below, all the convergences of vector functions are in this sense. Thus, condition (a) is satisfied.
Furthermore, lettingt=tj, j=1,2,…, in (24), we have (26)limε→0-Pxtj+εytj+ε=Pxtjytj,limε→0+Pxtj+εytj+ε=Pxtjytj+ρxtjρytj,which implies that condition (c) is also satisfied.
Next, condition (d) is satisfied if only(27)eγte-Atξ0+e-At∫0teAsCfys-τsds+∑0<tk<teAtkρxtke-Btη0+e-Bt∫0teBsDgxs-hsds+∑0<tk<teBtkρytk⟶00∈R2n,t⟶+∞.Indeed, obviously, eγte-Atξ(0)→0∈Rn and eγte-Btη(0)→0∈Rn as t→∞.
Below, we may firstly prove(28)eγte-At∫0teAsCfys-τsds⟶0∈Rn,t⟶∞.
In fact, it follows fromeγtx(t)→0 that, for any given ε>0, there exists a corresponding constant t∗>τ such that (29)eγtxt+eγtyt<εμ,∀t⩾t∗,whereμ=1,1,…,1T∈Rn.Next, we get by (A1) (30)eγte-At∫0teAsCfys-τsds⩽e-A-γIt∫0t∗eAsCFys-τsds+e-A-γIt∫t∗teAsCFys-τsds.
On the one hand,(31)e-A-γIt∫0t∗eAsCFys-τsds⩽t∗e-A-γIteAt∗CFmaxi⁡sups∈-τ,t∗⁡yisμ⟶0∈Rn,t⟶∞.
On the other hand, obviously, there exists a positive numbera0 such that |C|Fμ⩽a0μ. So, we have(32)e-A-γIt∫t∗teAsCFys-τsds⩽εeγτe-A-γIt∫t∗teA-γIsCFμds⩽εa0eγτe-A-γItea1-γta1-γ0⋯000ea2-γta2-γ0⋯0⋱00⋯0ean-γtan-γμ=εa0eγτ1a1-γ,1a2-γ,…,1an-γT.Combining (30)–(32) yields (28). Similarly, we can prove (33)eγte-Bt∫0teBsDgxs-hsds⟶0∈Rn,t⟶∞.
Next, we need to prove that ift→+∞, (34)eγte-At∑0<tk<teAtkρxtke-Bt∑0<tk<teBtkρytk=eγte-At∑0<tk<t∗eAtkρxtke-Bt∑0<tk<t∗eBtkρytk+eγte-At∑t∗<tk<teAtkρxtke-Bt∑t∗<tk<teBtkρytk⟶0∈R2n.Indeed, it is obvious that (35)eγte-At∑0<tk<t∗eAtkρxtke-Bt∑0<tk<t∗eBtkρytk=eγI-At∑0<tk<t∗eAtkρxtkeγI-Bt∑0<tk<t∗eBtkρytk⟶0∈R2n,t⟶+∞.Below, we shall prove (36)eγte-At∑t∗<tk<teAtkρxtke-Bt∑t∗<tk<teBtkρytk⟶0∈R2n,t⟶+∞.Firstly, we may assume that tm-1<t∗⩽tm and tj<t⩽tj+1 for any given t>t∗. Hence, (37)eγte-At∑t∗<tk<teAtkρxtk⩽eγte-At∑t∗<tk<teAtk-γItkHeγtkxtk⩽εδe-γ-a1tδea1-γtj+1+∫t∗tea1-γsdsH1⋮e-γ-antδean-γtj+1+∫t∗tean-γsdsHn⩽εδe-γ-a1tδea1-γtj+1+1a1-γea1-γtH1⋮e-γ-antδean-γtj+1+1an-γea1-γtHn,which together with the arbitrariness of the positive number ε implies that (38)e-At∑t∗<tk<teAtkρxtk⟶0∈Rn,t⟶+∞. Similarly, we can also get (39)e-Bt∑t∗<tk<teBtkρytk⟶0∈Rn,t⟶+∞.Hence, we have proved (36) and (34). And so we can conclude (27) from (28), (33), and (34). This means that condition (d) is satisfied, too.
Therefore,Pxtyt∈Ω for any xtyt∈Ω.
Step  3. Below, we only need to prove that P is a contraction mapping.
Indeed, for anyxtyt,x¯ty¯t∈Ω, we have (40)Pxtyt-Px¯ty¯t⩽e-At∫0teAsCfys-τs-fy¯s-τsdse-Bt∫0teBsDgxs-hs-gx¯s-hsds+e-At∑0<tk<teAtkρxtk-ρx¯tke-Bt∑0<tk<teBtkρytk-ρy¯tk⩽A-1CFμB-1DGμ+1δe-At∫0teAsds+δeAtHμe-Bt∫0teBsds+δeBtHμdistxtyt,x¯ty¯t⩽A-1CF+1δA-1H+HμB-1DG+1δB-1H+Hμdistxtyt,x¯ty¯t<λμμdistxtyt,x¯ty¯t, and hence (41)distPxtyt,Px¯ty¯t⩽λdistxtyt,x¯ty¯t,where A-1 and B-1 are the inverse matrices of A and B, respectively.
Therefore,P:Ω→Ω is a contraction mapping such that there exists the fixed point xtyt of P in Ω, which implies that xtyt is a solution of the impulsive fuzzy dynamic equations (2), satisfying eγtxtyt→0 as t→+∞. And the proof is completed.Remark 7.
Impulsive BAM neural networks model brings some mathematical difficulties to contraction mapping technique. However, in this paper, we set up the contraction mapping on the complete product space to overcome obstacles.


## 4. Numerical Example

Example 8.
We equip the impulsive system with the following parameters:(42)A=1.8002.1,B=2001.9,C=-0.20.0100.3,D=0.30.020-0.1,F=0.1000.2,G=0.2000.1,H=0.3000.2.Let δ=1.5. Then, we can use Matlab LMI toolbox to solve LMI conditions (4), obtaining the datum feasible as follows: (43)λ=0.9913.Obviously, 0<λ<1. Thereby, we can conclude from Theorem 6 that the impulsive equations (2) are globally exponentially stable.Remark 9.
Example8 illustrates the effectiveness of LMI-based criterion (Theorem 6). Table 1 presents a comparable result among the related literature, mainly published from 2013 to 2016.Remark 10.
From Table1, we know that there are many existing literatures involving fixed point technique and stability analysis, and a lot of interesting conclusions are derived [21–23, 28–30]. Motivated by some methods of those literatures, we utilized Banach contraction mapping theorem to obtain the LMI-based stability criterion applicable to computer Matlab LMI toolbox. It is well known that computer software can solve large-scale computations in actual engineering, which demonstrates the superiority of the proposed method in this paper to a certain extent.


## 5. Conclusion

Recently, fixed point technique and methods are employed to the stability analysis to BAM neural networks, and some stability criteria are derived (see, e.g., [28, 29]). Some good methods and results of related literature [21–23, 28–30] inspire our current work. Different from existing papers, we utilized Banach fixed point theorem deriving immediately the exponential stability criterion applicable to computer Matlab LMI toolbox. Computer programming is suitable for large-scale computation in practical engineering.



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