# LMI-Based Stability Criteria for Discrete-Time Neural Networks with Multiple Delays

Authors: Hui Xu; Ranchao Wu
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

Discrete neural models are of great importance in numerical simulations and practical implementations. In
the current paper, a discrete model of continuous-time neural networks with variable and distributed delays
is investigated. By Lyapunov stability theory and techniques such as linear matrix inequalities, sufficient
conditions guaranteeing the existence and global exponential stability of the unique equilibrium point are
obtained. Introduction of LMIs enables one to take into consideration the sign of connection weights. To show
the effectiveness of the method, an illustrative example, along with numerical simulation, is presented.


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## Body

## 1. Introduction

During the past decades, various types of neural networks have been proposed and investigated intensively, since they play important roles and have found successful applications in fields such as pattern recognition, signal and image processing, nonlinear optimization problems, and parallel computation. The dynamical behaviors in neural models, such as the existence and their asymptotic stability of equilibria, periodic solutions, bifurcations, and chaos, have been the most active areas of research and have been extensively explored over the past years [1–22].Due to the finite transmission speed of signals among neurons, time delays in interactions between neurons frequently happen and will cause complex dynamics in neural networks [6]; so it is necessary to introduce time delays into the neural models. So far, discrete, time-varying, and distributed delays have been, respectively, introduced to describe the dynamics of neural networks, and various sufficient conditions ensuring the stability have been given.Note that in numerical simulation and practical implementations, discretization of continuous-time models is necessary and of great importance. On the other hand, the dynamics of discrete-time neural networks could be quite different from those of continuous versions and will display much more complicated behaviors. So it is of great theoretical and practical significance to study the dynamics of discrete neural models. For discrete models, such as discrete Hopfield, bidirectional associate memory, and Cohen-Grossberg neural networks, several authors [1, 7–22] have studied the existence and exponential stability of equilibria and periodic solutions.In this paper, a discrete model with both variable and distributed time delays is introduced. By Lyapunov stability theory and linear matrix inequality (LMI) technique, sufficient conditions ensuring the existence and globally exponential stability of a unique equilibrium point are obtained. To show the effectiveness of our results, an illustrative example along with numerical simulation is presented. To our best knowledge, such general models have been seldom touched upon in the existing literatures. As we see, the obtained conditions are easy to verify. Furthermore, introduction of LMIs enables us to take into consideration the sign of connection weights. In contrast, sufficient conditions, for instance, in [7–12], depend on the absolute values of connection weights. That will ignore the differences between neuronal excitatory and inhibitory effects.


## 2. Preliminaries

SetZ to be the set of integers and Z+ the set of nonnegative integers; let N(a,b) represent the set of integers between a and b with a≤b,a,b∈Z, namely, N(a,b)={a,a+1,…,b}.Consider the discrete-time neural networks with both variable and distributed delays:(1)xi(n+1)=aixi(n)+∑j=1m‍bijfj(xj(n-k(n)))+∑j=1m‍cij∑p=1∞‍𝒦j(p)gj(xj(n-p))+Ii,
with initial values
(2)xi(l)=ϕi(l),l∈N(-∞,0),
where x(n)=col(x1(n),…,xm(n))∈Rm,xi(n) are the states of the ith neuron at time n; ai∈(0,1) represents the rate with which the ith neuron resets its potential when isolated from others; bij and cij weigh the strengths of the jth unit on the ith unit; fj and gj are the nonlinear activation functions of the neurons; k(n) denotes the transmission delay along the axon of the jth unit; 𝒦j(p)≥0 is the delay kernel; Ii is the external input on the ith neuron at time n; the initial value functions ϕi(l) are bounded N(-∞,0),i=1,…,m.To investigate stability of system (1), make further assumptions:(H1)
suppose thatbij,cij, Ii∈R, for i=1,…,m, and 0≤k(n)≤k for n∈N(0,∞), with k being constant;(H2)
suppose that𝒦j(p)≥0, ∑p=1∞‍𝒦j(p)=1, and ∑p=1∞‍𝒦j(p)vp<+∞, for some v>1, all i,j=1,…,m;(H3)
assume that functionsfj and gj are bounded and satisfy
(3)fj-≤fj(ξ)-fj(η)ξ-η≤fj+,gj-≤gj(ξ)-gj(η)ξ-η≤gj+,for any ξ,η∈R, where fj+,fj-,gj+, and gj- are some constants and can be positive, negative, or zero, j=1,…,m. So they are less restrictive than sigmoid activation functions and Lipschitz-type ones.For anyϕ=(ϕ1,…,ϕm), a solution of systems (1) and (2) is a vector-valued function x:Z+→Rm satisfying system (1) and initial conditions (2) for n∈Z+. In this paper, it is always assumed that neural model (1) admits a solution represented by x(n,ϕ) or simply x(n). Since the activation functions fj,gj are bounded, it is not difficult to check that system (1) has at least one equilibrium point by Brouwer's fixed point theorem. So with loss of generality, assume that fj(0)=gj(0)=0; that is, x=0 is an equilibrium point. Throughout the paper, denote
(4)∥x(n)∥2=∑i=1m‍|xi(n)|2,kM=supn{k(n)},km=infn{k(n)},F1=diag{f1+f1-,…,fm+fm-},F2=diag{f1++f1-,…,fm++fm-},G1=diag{g1+g1-,…,gm+gm-},G2=diag{g1++g1-,…,gm++gm-},A=diag{a1,…,am},B=(bij)m×m,C=(cij)m×m,f=(f1,…,fm)T,g=(g1,…,gm)T,(5)𝒦=diag(𝒦1,…,𝒦m).
System (1) can be rewritten into the form
(6)x(n+1)=Ax(n)+Bf(x(n-k(n)))+C∑p=1∞‍𝒦(p)g(x(n-p)).Definition 1.
The equilibrium pointx=0 of system (1) is globally exponentially stable if there exist constants η>1 and C*>0 such that for any solution x(n,ϕ) of system (1) with initial conditions ϕ, it holds that
(7)∥x(n,ϕ)∥2≤C*η-nsupl∈N(-∞,0)∥ϕ(l)∥2,∀n∈Z+.


## 3. Exponential Stability of Equilibrium Points

By Lyapunov stability theory and LMI technique, the global exponential stability of the equilibrium point is established. Clearly, ifx=0 is exponentially stable, the equilibrium point is unique. Now we will investigate the exponential stability of the origin.Theorem 2.
Suppose that (H1)–(H3) hold and further there exist a numberv≥η>1, positive definite matrix P, Σ=diag{e1,…,em}, and semipositive diagonal matrices U=diag{u1,…,um}, V=diag{v1,…,vm}, and  S=diag{s1,…,sm}, such that
(8)W=[Φ110APB+UF20VG2APC*-2SF10SF200**Φ33000***Φ440BTPC****αΣ-2V0*****ηCTPC-Σ]<0,
where
(9)Φ11=ηAPA-P-2UF1-2VG1,Φ33=(kM-km+1)Σ-2U,Φ44=ηBTPB-η-kMΣ-2S,α=diag{α1,…,αm},αj=∑p=1∞‍𝒦j(p)ηp.
Then the origin of system (1) is exponentially stable.Proof.
Define a Lyapunov functionalV(n)=V(x1,…,xm)(n) as follows:
(10)V(n)=V1(n)+V2(n)+V3(n)+V4(n),
where
(11)V1(n)=ηnxT(n)Px(n),V2(n)=∑s=n-k(n)n-1‍ηsfT(x(s))Σf(x(s)),V3(n)=∑r=-kM+2-km+1‍∑s=n+r-1n-1‍ηsfT(x(s))Σf(x(s)),(12)V4(n)=∑j=1m‍∑p=1∞‍𝒦j(p)∑s=n-pn-1‍ηs+pgj2(xj(s))ej.
To investigate the exponential stability of the origin, it is necessary to calculate the difference ΔV(n)=V(n+1)-V(n) along the trajectory of (6). From (6), we have
(13)ΔV1(n)=ηn+1xT(n+1)Px(n+1)-ηnxT(x)Px(n)=ηn+1[(x(n-k(n)))xT(n)APAx(n)+2xT(n)APBf=ηn+1.b×(x(n-k(n)))+2xT(n)APCϕ(x(n))=ηn+1.b+fT(x(n-k(n)))BTPBf(x(n-k(n)))=ηn+1.b+2fT(x(n-k(n)))BTPCϕ(x(n))=ηn+1.b+ϕ(x(n))TCTPCϕ(x(n))(x(n-k(n)))]-ηnxT(n)Px(n),
where ϕ(x(n))=∑p=1∞‍𝒦(p)g(x(n-p)). Since km≤k(n)≤kM, one obtains
(14)ΔV2(n)=∑s=n+1-k(n+1)n‍ηsfT(x(s))Σf(x(s))-∑s=n-k(n)n-1‍ηsfT(x(s))Σf(x(s))=ηnfT(x(n))Σf(x(n))-ηn-k(n)fT(x(n-k(n)))Σf×(x(n-k(n)))+∑s=n+1-kmn-1‍ηsfT(x(s))Σf(x(s))+∑s=n+1-k(n+1)n-km‍ηsfT(x(s))Σf(x(s))-∑s=n+1-k(n)n-1‍ηsfT(x(s))Σf(x(s))≤ηnfT(x(n))Σf(x(n))-ηn-k(n)fT(x(n-k(n)))Σf×(x(n-k(n)))+∑s=n+1-kMn-km‍ηsfT(x(s))Σf(x(s)),ΔV3(n)=(kM-km)ηnfT(x(n))Σf(x(n))-∑s=n+1-kMn-km‍ηsfT(x(s))Σf(x(s)),ΔV4(n)=∑j=1m‍∑p=1∞‍𝒦j(p)[(x(n-p))ηn+pgj2(x(n))ΔV4(n)=∑j=1m‍∑p=1∞‍b-ηngj2(x(n-p))]ej=ηn∑j=1m‍(∑p=1∞‍𝒦j(p)ηp)gj2(x(n))ej-ηn∑j=1m‍∑p=1∞‍𝒦j(p)gj2(x(n-p))ej=ηngT(x(n))αΣg(x(n))-ηn∑j=1m‍(∑p=1∞‍𝒦j(p))∑p=1∞‍𝒦j(p)gj2(x(n-p))ej≤ηn[gT(x(n))αΣg(x(n))≤ηnb.b-ϕT(x(n))Σϕ(x(n))].
Therefore, we have
(15)ΔV(n)=ΔV1(n)+ΔV2(n)+ΔV3(n)≤ηn[(x(n))TxT(n)(ηAPA-P)x(n)≤ηnb+2xT(n)APBf(x(n-k(n)))≤ηnb+2xT(n)APCϕ(x(n))≤ηnb+(kM-km+1)fT≤ηnb×(x(n))Σf(x(n))≤ηnb+fT(x(n-k(n)))(ηBTPB-η-kMΣ)≤ηnb×f(x(n-k(n)))≤ηnb+2fT(x(n-k(n)))BTPCϕ(x(n))≤ηnb+gT(x(n))αΣg(x(n))≤ηnb+ϕ(x(n))T(ηCTPC-Σ)ϕ(x(n))].
From (H3), one has(16)(fj(xj(n))-fj+xj(n))(fj(xj(n))-fj-xj(n))≤0,(gj(xj(n))-gj+xj(n))(gj(xj(n))-gj-xj(n))≤0,bbbbbbbbbbbbbbbbbbbb..bbbbbbbbbb.j=1,2,…,m.
Then for U=diag{u1,…,um}≥0, V=diag{v1,…,vm}≥0, and S=diag{s1,…,sm}≥0, one has
(17)ΔV(n)≤ΔV(n)-2∑j=1m‍uj(fj(xj(n))-fj+xj(n))×(fj(xj(n))-fj-xj(n))-2∑j=1m‍vj(gj(xj(n))-gj+xj(n))×(gj(xj(n))-gj-xj(n))-2∑j=1m‍sj(fj(xj(n-k(n)))-fj+xj(n-k(n)))×(fj(xj(n-k(n)))-fj-xj(n-k(n)))≤ξTWξ,
where ξ=(xT(n),xT(n-k(n)),fT(x(n)),fT(x(n-k(n))),gT(x(n)),ϕ(x(n)))T. From W<0, it follows that ΔV(n)<0. Note that
(18)V(n)≥λm(P)ηn∥x(n)∥2,V(0)≤xT(0)Px(0)+(kM-km+1)×∑s=-kM-1‍ηsfT(x(s))Σf(x(s))+∑j=1m‍ej∑p=1∞‍𝒦j(p)∑s=-p-1‍ηs+pgj2(xj(s))≤[λM(P)+δ1L(kM-km+1+δ2)]sups∈N(-∞,0)∥x(s)∥2,
where λm(P),λM(P) are the minimum and maximum eigenvalues of matrix P, respectively, δ1=1/(η-1), δ2=∑j=1m‍ejαj, and L=max{|fj+|,|fj-|,|gj+|,|gj-|}, so one has
(19)∥x(n,ϕ)∥2≤C*η-nsupl∈N(-∞,0)∥ϕ(l)∥2,n∈Z+,
where C*=λm-1(P)[λM(P)+δ1L(kM-km+1+δ2)]. This implies that the equilibrium solution x=0 of system (1) is globally exponentially stable. The proof is completed.Remark 3.
By employing LMI (8), the signs of bij, cij, that is, the differences between neural excitatory and inhibitory interaction, are taken into consideration.Remark 4.
Ifη=1, the equilibrium point 0 of system (1) is said to be globally stable.


## 4. Numerical Example

Next, an illustrative example is given to show the effectiveness of the obtained results. Consider the discrete-time neural model (6) with parameters:
(20)A=(0.5000.6),B=(0.05-0.1-0.10.15),C=(0.1-0.10.050.15),f1(u)=tanh(2u),f2(u)=tanh(-2u),g1(u)=g2(u)=arctanu,𝒦1(p)=𝒦2(p)=34p,k(n)=4+2(-1)n,
then it is not difficult to see that F1=G1=0, F2=diag{2,-2}, G2=diag{1,1}, kM=6, and  km=2. Take η=2, then by solving LMI (8), it has feasible solutions that are Σ=S=V=I and
(21)P=(19.8208-1.8309-1.830917.0882),U=(2.5213002.5710),
so from Theorems 2, this system admits a unique equilibrium 0, with all other solutions converging to it exponentially as n→∞ see Figures 1 and 2.Figure 1
Trajectories ofx1(n) versus n.Figure 2
Trajectories ofx2(n) versus n.


## 5. Conclusions

In the current paper, a class of discrete-time neural networks with both variable and distributed delays has been studied. Using Lyapunov stability and LMI technique, the existence and global exponential stability of the unique equilibrium point have been established. The obtained results are easy to verify, so they will be of practical use for applying discrete neural models.



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