# Global Exponential Robust Stability of High-Order Hopfield Neural Networks with S-Type Distributed Time Delays

Authors: Haiyong Zheng; Bin Wu; Tengda Wei; Linshan Wang; Yangfan Wang
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

By employing differential inequality technique and Lyapunov functional method, some criteria of global exponential robust stability for the high-order neural networks with S-type distributed time delays are established, which are easy to be verified with a wider adaptive scope.


---

## Body

## 1. Introduction

Neural networks and their various generalizations have been successfully employed in many areas such as pattern recognition, cognitive modeling, adaptive control, and combinatorial optimization [1–7]. Hopfield neural networks (HNNs), as some forms of recurrent artificial neural networks, have been widely studied in recent years [8–12]. The earlier HNNs model proposed by Hopfield [13, 14] was based on the theory of analog circuit consisting of capacitors, resistors, and amplifiers and can be formulated as a system of ordinary equations. Time delays are inevitable in the interactions of neurons in biological and artificial neural networks. The existence of delays is frequently a source of instability for neural networks [9, 10, 15–19].Over the past few decades, the stability of HNNs with time delays has attracted considerable attention in the literature [20–23]. One of the most investigated problems in the study of HNNs is global exponential stability of the equilibrium point. An equilibrium point of HNNs isglobally exponentially stable, if the domain of attraction of the equilibrium point is the whole space and the convergence is in real time.It is worth noting that although the signal propagation is sometimes instantaneous and can be modeled with discrete delays, it may also be distributed during a certain time period so that the distributed delays should be incorporated in the model [24]. Discrete delays and distributed delays attract the attention of many scholars and have been widely studied [17, 19, 25]. To the best of our knowledge, the stability problem for system with both discrete and distributed delays has been a challenging issue, mainly due to the mathematical difficulties in dealing with discrete and distributed delays simultaneously. In 2002, Wang and Xu [26] presented a new neural network model with S-type distributed time delays and demonstrated that S-type distributed time delays include discrete or continuously distributed time delays, but it is not true in the opposite way. In the following years, S-type distributed time delayed neural network models have raised great interest [12, 27–29].Compared with traditional Hopfield neural networks, the high-order Hopfield type neural networks (HOHNNs) [11, 12, 30–34] have the advantages of stronger approximation properties, faster convergence rate, greater storage capacity, and higher fault tolerance. Therefore, it is of considerable interest to explore the theoretical foundations and practical applications of HOHNNs.Motivated by the aforementioned discussion, we studied the problem of global exponential robust stability of HOHNNs with S-type distributed time delays. By employing differential inequality technique and a new Lyapunov functional method, some criteria for the global exponential robust stability of the high-order neural networks with S-type distributed time delays have been established, which are easy to be verified with a wider adaptive scope. Meanwhile, the systems in [10, 12, 26, 31] are some special cases of the HOHNNs with S-type distributed time delays.


## 2. Model Description and Preliminaries

We consider the following HOHNNs with S-type distributed time delays:(1)dxi(t)dt=-aixi(t)+∑j=1n‍bijfj(xj(t))+∑k=1r‍∑j=1n‍wij(k)gj(∫-∞0‍dsηjk(s)xj(t+s))+∑k=1r‍∑j=1n‍∑l=1n‍wijl(k)gj(∫-∞0‍dsηjk(s)xj(t+s))glhhhhhhhhhhhh×(∫-∞0‍dsηlk(s)xl(t+s))+Iihlhhhhhhhhhhhhhh0<a_i≤ai,b_ij≤bij≤b¯ij,hhhhhhw_ij(k)≤wij(k)≤w¯ij(k),w_ijl(k)≤wijl(k)≤w¯ijl(k),hhhhhhhhhhhhhhhhhhhhhhhhhhhhI_i≤Ii≤I¯i,hhhhhhhhhhhhhhhi,j,l=1,…,n;k=1,…,r,
where xi(t), ai, bij, fj, gj, and Ii have the same meanings as those in [28], wij(k), wijl(k) are the first- and second-order synaptic weights of the system (1) (see [12]), and ∫-∞0‍dsηlk(s)xl(t+s) are Lebesgue-Stieltjes integrable, where ηjk(s) and ηlk(s) are nondecreasing bounded variation functions which satisfy
(2)0<∫-∞0‍dsηjk(s)=ξjk.In this paper the superscript “T” presents the transpose and C((-∞,0],Rn) denotes a set of continuous bounded functions.For system (1), the initial condition is
(3)xi(s)=φi(s),s∈(-∞,0],hhhφi(s)∈C((-∞,0],Rn),hhhhhhhhhhhhhi=1,…,n.If there is an equilibrium pointx*=[x1*,…,xn*]T of system (1) with conditions (3), we can rewrite system (1) as the following equivalent form:
(4)d(xi(t)-xi*)dt=-ai(xi(t)-xi*)+∑j=1n‍bij(fj(xj(t))-fj(xj*))+∑k=1r‍∑j=1n‍wij(k)(∫-∞0‍gj(dsηjk(s)xj(t+s))hhhhhhhhhhh-gj(∫-∞0‍dsηjk(s)xj*))+∑k=1r‍∑j=1n‍∑l=1n‍wijl(k)(gj(∫-∞0‍dsηjk(s)xj(t+s))glhhhhhhhhhhhhhhh×(∫-∞0‍dsηlk(s)xl(t+s))hhhhhhhhhhhhhhh-gj(∫-∞0‍dsηjk(s)xj*)hhhhhhhhhhhhhhh×gl(∫-∞0‍dsηlk(s)xl*)),hhhhhhhhhhhhhlh0<a_i≤ai,b_ij≤bij≤b¯ij,hhhhlhw_ij(k)≤wij(k)≤w¯ij(k),w_ijl(k)≤wijl(k)≤w¯ijl(k),hhhhlhhhhhhhhhhhhhhhhhhhhhhhlI_i≤Ii≤I¯i,hhlhlhhhhhhhhhhli,j,l=1,…,n;k=1,…,r.From [12], we know that the following system (5) is equivalent to system (4):
(5)d(xi(t)-xi*)dt=-ai(xi(t)-xi*)+∑j=1n‍bij(fj(xj(t))-fj(xj*))+∑k=1r‍∑j=1n‍(wij(k)+∑l=1n‍(wijl(k)+wilj(k))ζlk)hhhhhhhhhh×(gj(∫-∞0‍dsηjk(s)xj(t+s))hhhhhhhlhhhhhl-gj(∫-∞0‍dsηjk(s)xj*)),hhhhhhhhhhhhh0<a_i≤ai,b_ij≤bij≤b¯ij,hhhhw_ij(k)≤wij(k)≤w¯ij(k),w_ijl(k)≤wijl(k)≤w¯ijl(k),hhhhhhhhhhhhhhhhhhhhhhhhhI_i≤Ii≤I¯i,hhhhhhhhhhhhi,j,l=1,…,n;k=1,…,r,
where ζlk=(1/2)((gl(∫-∞0‍dsηlk(s)xl(t+s))+gl(∫-∞0‍dsηlk(s)xl*))).Definition 1.
The equilibrium pointx*=[x1*,x2*,…,xn*]T of system (1) is called globally exponentially robustly stable, if, for any φi(s)∈C((-∞,0],R), there exist scalars α>0 and Π>0 such that the solution x(t)=[x1(t),…,xn(t)]T to system (1) with initial condition xi(s)=φi(s)∈C((-∞,0],R), i=1,…,n, satisfies
(6)|xi(t)-xi*|≤Πe-αt,t≥0,i=1,…,n.LetA=diag⁡(a_1,…,a_n), σ=diag⁡(σ1,…,σn), ω=diag⁡(ω1,…,ωn), M=[M1,M2,…,Mn]T, Γ=diag⁡(M,M,…,M), B+=(b~ij)n×n=(max⁡{|b_ij|,|b¯ij|})n×n, Wk=(wij(k))n×n, Wk+=(w~ij(k))n×n = (max⁡{|w_ij(k)|,|w¯ij(k)|})n×n, Wik = (wijl(k))n×n, (Wik)T = (wilj(k))n×n, Wik+ = (w~ijl(k))n×n = (max⁡{|w_ijl(k)|,|w¯ijl(k)|})n×n, WHk+ = (W1k++(W1k+)T,W2k++(W2k+)T,…,Wnk++(Wnk+)T)n2×nT, I~i = max⁡{|I_i|,|I¯i|},X∘Y = (xijyij)n×n be Hadamard product of two matrices,
(7)ξk=(ξ1k⋯ξnk⋮⋱⋮ξ1k⋯ξnk)n×n(k=1,…,r).We assume throughout that the neuron activation functionsfj(uj), gj(uj), j=1,…,n, satisfy the following conditions.(H1) Consider
(8)|gj(uj)|≤Mj,0≤|fj(uj)-fj(vj)||uj-vj|≤σj,hhhhhhhh0≤|gj(uj)-gj(vj)||uj-vj|≤ωj,hhhhhhhhhhhhhhhforuj≠vj,uj,vj,σj,ωj∈R.(H2) Consider
(9)C=A-B+σ-∑k=1r‍((Wk++ΓTWHk+)∘ξk)ωisanM-matrix.(H3) Consider
(10)0<∫-∞0‍dsηjk(s)≤1.


## 3. Main Results

Theorem 2.
The equilibrium of system (1) is globally exponentially robustly stable, if system (1) satisfies (H1), (H2), and (H3).Proof.
Part 1: Existence of the Equilibrium Point. Let
(11)hi(xi,Ii)=aixi-∑j=1n‍bijfj(xj)-∑k=1r‍∑j=1n‍wij(k)gj(ξjkxj)-∑k=1r‍∑j=1n‍∑l=1n‍wijl(k)gj(ξjkxj)gl(ξlkxk)-Ii=0,hhhhhhhhhhhhhhhhhhhhhhhlhi=1,…,n.
It is obvious that the solutions to (11) are the equilibrium points of system (1).
Let us define homotopic mapping as follows:(12)H(x,λ)=(H1(x1,λ),…,Hn(xn,λ))T,
where
(13)Hi(xi,λ)=λhi(xi,Ii)+(1-λ)xi,λ∈[0,1].
By homotopy invariance theorem (see [35]), topological degree theory (see [36]), (H2), and the proof, which is similar to Theorem 1 in [28], we can conclude that (13) has at least one solution.
That is, system (1) has at least an equilibrium point.
Part 2: Global Existence of the Solutions to System (1). Since C=A-B+σ-∑‍k=1r((Wk++ΓTWHK+)∘ξk)ω is an M-matrix, there exists a constant vector L∈Rn such that CL>0 (see [15]); that is,
(14)Ji=a_iLi-∑j=1n‍(b~ijσj+∑k=1r‍(w~ij(k)+∑l=1n‍Ml(w~ijl(k)+w~ilj(k)))hhhhhhl×ξjkωj∑j=1n‍)Lj>0.
Supposex(t)=[x1(t),…,xn(t)]T is a solution to system (1) and also satisfies the initial condition xi(s)=φi(s)∈C((-∞,0],Rn), s∈(-∞,0], i=1,…,n.
Let us choose a positive constantN such that
(15)N>max⁡(hhhhhhhhhhhhh+∑k=1r‍∑j=1n‍∑l=1n‍w~ijl(k)|gj(0)||gl(0)|+I~i)1Lisup⁡-∞<s≤0|φi(s)|,hhhhhhhhhJi-1(∑j=1n‍b~ij|fj(0)|+∑k=1r‍∑j=1n‍w~ij(k)|gj(0)|hhhhhhhhhhhhh+∑k=1r‍∑j=1n‍∑l=1n‍w~ijl(k)|gj(0)||gl(0)|+I~i)).
From (15) we know that
(16)|xi(t)|<NLi,i=1,…,n,t∈(-∞,0].
Then, we will show that(17)|xi(t)|<NLi,i=1,…,n,t∈(-∞,+∞).
If (17) is not true, there must be some positive integer i0 and t0>0, such that
(18)|xi0(t)|<NLi0,t∈(-∞,t0),|xi0(t0)|=NLi0.
From(H1), (15), and (18), we know
(19)d|xi0(t)|dt|t=t0≤-a_i0|xi0(t0)|+∑j=1n‍b~ij|fj(xj)|+∑k=1r‍∑j=1n‍w~ij(k)|gj(∫-∞0‍dsηjk(s)xj(t0+s))|+∑k=1r‍∑j=1n‍∑l=1n‍w~ijl(k)hhhhhhhhhl×|gj(∫-∞0‍dsηjk(s)xj(t0+s))glhhhhhhhhhhhl×(∫-∞0‍dsηlk(s)xl(t0+s))|+I~i≤-a_i0|xi0(t0)|+∑j=1n‍b~ij(σj|xj(t0)|+|fj(0)|)+∑k=1r‍∑j=1n‍w~ij(k)(ωj|∫-∞0‍dsηjk(s)xj(t0+s)|hhhhhhhhhhlhhh+|gj(0)|ωj|∫-∞0‍dsηjk(s)xj(t0+s)|)+∑k=1r‍∑j=1n‍∑l=1n‍w~ijl(k)(Mlωj|∫-∞0‍dsηjk(s)xj(t0+s)|hhhhhhhhhhhhhhhl+Mjωl|∫-∞0‍dsηlk(s)xl(t0+s)|hhhhhhhhhhhhhhhl+|gj(0)||gl(0)||∫-∞0‍dsηjk(s)xj(t0+s)|)+I~i≤-a_i0|xi0(t0)|+∑j=1n‍b~ij(σj|xj(t0)|+|fj(0)|)+∑k=1r‍∑j=1n‍w~ij(k)(ωj|∫-∞0‍dsηjk(s)xj(t0+s)|+|gj(0)|)+∑k=1r‍∑j=1n‍∑l=1n‍(w~ijl(k)+w~ilj(k))hhhhhhhhhhh×(Mlωj|∫-∞0‍dsηjk(s)xj(t0+s)|hhhhhhhhhhhhhhh+|gj(0)||gl(0)|(Mlωj|∫-∞0‍dsηjk(s)xj(t0+s)|)+I~i≤[(∑j=1n‍b~ijσj-a_i0Lihhh+(∑j=1n‍b~ijσj+∑k=1r‍∑j=1n‍(w~ij(k)+∑l=1n‍(w~ijl(k)+w~ilj(k)))hhhhhhh×ξjkωj(∑j=1n‍b~ijσj)Lj]N+(∑j=1n‍b~ij|fj(0)|+∑k=1r‍∑j=1n‍w~ij(k)|gj(0)|hhhhh+∑k=1r‍∑j=1n‍∑l=1n‍w~ijl(k)|gj(0)||gl(0)|+I~i)<0.
So(20)|xi0(t0)|<|xi0(t)|<NLi0,t∈(-∞,t0],
which leads by contradiction to (18).
Hence, (17) holds. That is, the solutions to system (1) are bounded. So the solutions to system (1) are of global existence.
Part 3: Global Exponential Stability of System (1). From (H2), we know that there exists constant pi>0, such that
(21)pia_i-∑j=1n‍pj(∑k=1r‍(w~ij(k)+∑l=1n‍Ml(w~ijl(k)+w~ilj(k)))b~ijσjhhhhhhhhlhl+∑k=1r‍(w~ij(k)+∑l=1n‍Ml(w~ijl(k)+w~ilj(k)))ξjkωj)>0.
So, from(H3), we can choose a constant α>0 sufficiently small, such that
(22)pi(a_i-α)-∑k=1r‍∑j=1n‍pj(b~ijσj+(w~ij(k)+∑l=1n‍Ml(w~ijl(k)+w~ilj(k)))ξjkωj)×|∫-∞0‍e-αsdsηjk(s)|>0.
Letφ=[φ1,…,φn]T∈C((-∞,0],Rn), and
(23)∥φ-x*∥=max⁡i=1,…,nsup⁡-∞<s≤0pi-1|φi(t)-xi*|>0.
Then, we will show that there existsΠ>0 such that
(24)|xi-xi*|<Πe-αt,t≥0,i=1,…,n.
Define a Lyapunov functional by(25)Vi(t)=|xi(t)-xi*|eαt,i=1,…,n.
ItsDini derivative reads
(26)D+(Vi(t))≤α|xi(t)-xi*|eαt-ai|xi(t)-xi*|eαt+[∑j=1n‍bij(|fj(xj(t))-fj(xj*)|)hhhhh+∑k=1r‍∑j=1n‍(wij(k)+∑l=1n‍(wijl(k)(t)+wilj(k)(t))ζl)hhhhh×|gj(∫-∞0‍dsηjk(s)xj(t+s))hhhhhhhl-gj(∫-∞0‍dsηjk(s)xj*)|[∑j=1n‍bij(|fj(xj(t))-fj(xj*)|)]eαt≤pi(-a_i+α)pi-1|xi(t)-xi*|eαt+∑j=1n‍pjσjb~ijpj-1|xj(t)-xj*|eαt+∑k=1r‍∑j=1n‍pj(w~ij(k)+∑l=1n‍(w~ijl(k)+w~ilj(k))Ml)ωjpj-1×∫-∞0‍|xj(t+s)-xj*|eα(t+s)dsηjk(s)×|∫-∞0‍e-αsdsηjk(s)|.
IfT>1, we have
(27)pi-1|xi(t)-xi*|eαt<T∥φ-x*∥,hhhhhhhhhhhht≤0,i=1,…,n.
Then, we will prove that(28)pi-1Vi(t)=pj-1|x(t)-xi*|eαt<T∥φ-x*∥,hhhhhhhhhhhhhhhhhhht>0,i=1,…,n.
If (28) is not true, there exists i0∈{1,2,…,n} and t0>0 such that
(29)pi0-1Vi0(t0)=T∥φ-x*∥,pi0-1Vi0(t)<T∥φ-x*∥,t<t0,pj-1Vj(t)<T∥φ-x*∥,∀t≤t0,hhhhhhhhhhj=1,2,…,n,j≠i0.
From (29), we have
(30)0≤dVi0(t)dt|t=t0≤pi0(-a_i0+α)pi0-1|xi0(t0)-xi0*|eαt0+∑j=1n‍pjσj|bi0j|pj-1|xj(t0)-xj*|eαt0+∑k=1r‍∑j=1n‍pj(|wi0j(k)|∑l=1n(|wi0jl(k)|+|wi0lj(k)|)Mlhhhhhhhhhl+∑l=1n(|wi0jl(k)|+|wi0lj(k)|)Ml)ωjpj-1hhhhhhhhhhlh×|∫-∞0‍e-αsdsηjk(s)|hhhhhhhhhhlh×∫-∞0‍|xj(t0+s)-xj*|hhhhhhhhhhhh×eα(t0+s)dsηjk(s).
Becausepi0-1Vi0(t0)=T∥φ-x*∥ and pj-1Vj(t)≤T∥φ-x*∥, for all t≤t0, j=1,…,n, we can obtain
(31)dVi0(t)dt|t=t0≤{|∫-∞0‍e-αsdsηjk(s)|pi0(-a_i0+α)hhh+∑j=1n‍pj((w~i0j(k)+∑l=1n‍Ml(w~i0jl(k)+w~i0lj(k)))b~i0jσjhhhhhhhhhh+∑k=1r‍(w~i0j(k)+∑l=1n‍Ml(w~i0jl(k)+w~i0lj(k)))ξjkωj)hhh×|∫-∞0‍e-αsdsηjk(s)|}T∥φ-x*∥<0.
It is obvious that (31) is in contradiction to (30). Hence, (28) holds. That is,
(32)pi-1|xi(t)-x*|<T∥φ-x*∥e-αt,t>0,i=1,…,n.
So(33)|xi(t)-x*|<Πe-αt,t>0,i=1,…,n,
where Π=max⁡1≤i≤n{pi}T∥φ-x*∥>0.
If there exists another equilibriumx** of system (1), we have |xi*-xi**|≤|xi(t)-xi*|+|xi(t)-xi**|→0, t→∞, i=1,…,n.
From above proof, the system (1) has a unique equilibrium point x*, which is globally exponentially robustly stable.
The proof of Theorem2 is completed.Remark 3.
The system (1) includes system with discrete time delays and with continuously distributed delays. Conversely, it is not true.
Whenwijlk=0 and
(34)ηjk(s)={∑k=1r‍ηjk,s=τ1=0,∑k=2r‍ηjk,τ2≤s<0,∑k=3r‍ηjk,τ3≤s<τ2,j=1,…,n,⋯ηjr,τr≤s<τr-1,0,-∞<s<τr,
system (1) becomes a HNNs model with discrete time delays
(35)dxi(t)dt=-aixi(t)+∑j=1n‍bijfj(xj(t))+∑k=1r‍∑j=1n‍wij(k)gj(ηjkxj(t-τk))+Ii,hhhlhhhh0<a_i≤ai,b_ij≤bij≤b¯ij,hhlhw_ij(k)≤wij(k)≤w¯ij(k),I_i≤Ii≤I¯i,hhhhllhhi,j,l=1,…,n;k=1,…,r.
Whenwijlk=0 and ηjk(s)∈C1(-∞,0], j=1,…,n, k=1,…,r, the value of the synaptic connectivity from neuron j to i is a continuous function on (∞,0], which means that time delays influence the network continuously, and system (1) belongs to a HNNs model with continuous time delays:
(36)dxi(t)dt=-aixi(t)+∑j=1n‍bijfj(xj(t))+∑k=1r‍∑j=1n‍wij(k)gj(∫-∞0‍ηjk′(s)xj(t+s)ds)+Ii,hhhhhhhhhhhhhhlh0<a_i≤ai,b_ij≤bij≤b¯ij,hhhhhhhhhhhhhw_ij(k)≤wij(k)≤w¯ij(k),I_i≤Ii≤I¯i,hhhhhhhhhlhhhhhhi,j,l=1,…,n;k=1,…,r.
So system (1) is widely representative.Remark 4.
Whenai=a_i>0, b_ij=bij=b¯ij, w_ij=wij=w¯ij, w_ijl(k)=wijl(k)=w¯ijl(k), and I_i=Ii=I¯i,  system (1) becomes the system (1) in [12]. So the systems in [10, 12, 26, 31] are also the special cases of system (1) (see [12]).


## 4. Example

For the sake of simplicity, we consider given one-dimension HOHNNs with S-type distributed time delays as follows:(37)dx1(t)dt=-a1x1(t)+b11f1(x1(t))+∑k=13‍w11(k)g1(∫-∞0‍dsη1k(s)x1(t+s))+∑k=13‍w111(k)g1(∫-∞0‍dsη1k(s)x1(t+s))hhhhhhh×g1(∫-∞0‍dsη1k(s)x1(t+s))+I1hhhhhhhhhhhhhhhhhi,j,l=1;k=1,…,3.In system (37), f1(z)=|z| and g1(z)=tanh(z), z∈R, which satisfy (H1), ΓT=M=M1=1, and σ1=ω1=1.Letη1k(t)=k-1et; then 0<∫-∞0‍dsη1k(s)=ξ1k=k-1≤1, k=1,…,3, which satisfies (H3).The parameters of the system (37) are given as follows:
(38)a_1=3.3≤a1,-1=b_11≤b11≤b¯11=1,0.2=w_11(1)≤w11(1)≤w¯11(1)=0.4,-0.4=w_11(2)≤w11(2)≤w¯11(2)=0.4,-0.1=w_11(3)≤w11(3)≤w¯11(3)=0.2,0=w_111(1)≤w111(1)≤w¯111(1)=0.2,-0.6=w_111(2)≤w111(2)≤w¯111(3)=0.8,-0.5=w_111(3)≤w111(3)≤w¯111(3)=0.5,0=I_1≤I1≤I¯1=1.From(H2) and the above parameters, we can easily obtain that C=3.3-1-[(0.4+0.4)×1+(0.4+1.6)×2-1+(0.2+1)×3-1]=0.1>0 is an M-matrix.Therefore, it follows from Theorem2 that the null solution to system (37) is globally exponentially robustly stable.


## 5. Conclusion

We have investigated the global exponential robust stability of high-order Hopfield neural networks with S-type distributed time delays, which is of theoretical as well as practical importance for the development of neural networks with time delays. The system (1) considered here is more general compared to the systems in literatures [10, 12, 26, 31]. By employing differential inequality technique and Lyapunov functional method, some criteria of global exponential robust stability for the high-order neural networks with S-type distributed time delays are established, which are easily verifiable and have a wider applicable range. The linear matrix inequality (LMI) approach is also widely used to establish the desired sufficient conditions for stability analysis of delayed neural networks [11, 37]. Wen et al. [17] have done some great work in control and filtering problems for neural systems. In future extension, we will do some research in stability of high-order Hopfield neural networks with S-type distributed time delays using LMI method.



---

*Source: 705496-2014-06-26.xml*