# LMI-Based Approach for Exponential Robust Stability of High-Order Hopfield Neural Networks with Time-Varying Delays

Authors: Yangfan Wang; Linshan Wang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper studies the problems of global exponential robust stability of high-order hopfield neural networks with time-varying delays. By employing a new Lyapunov-Krasovskii functional and linear matrix inequality, some criteria of global exponential robust stability for the high-order neural networks are established, which are easily verifiable and have a wider adaptive.


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## Body

## 1. Introduction

Hopfield neural networks (HNNs) with time delays and their various generalization have been successfully employed in many areas such as pattern recognition, associate memory, and combinatorial optimization (see [1–11]). Recently, the dynamics of high-order Hopfield neural networks (HOHNNs) with time delays have been considerable attention (see [12–18]), due to the neural networks have stronger approximation properties, faster convergence rate, greater storage capacity, and higher fault tolerance than lower order neural networks (see [12]). So, the stability of HOHNNs with time delays should be a focused topic of theoretical as wells as practical importance. This paper studies the problems of global exponential robust stability of high-order hopfield neural networks with time-varying delays. This paper is also an extension of our previous work [19]. By employing a new Lyapunov-Krasovskii functional and linear matrix inequality, some criteria of global exponential robust stability for the high-order neural networks are established, which are easily verifiable and have a wider adaptive.


## 2. Model Description and Preliminaries

We consider the following HOHNNs with time-varying delays:(2.1)dui(t)dt=-aiui(t)+∑j=1nWijfj(uj(t))+∑j=1nTijgj(uj(t-τ(t)))+∑j=1n∑l=1nTijlgj(uj(t-τ(t)))gl(ul(t-τ(t)))+Vi,ui(t0+θ)=ξi(θ),-τ0≤θ≤0,i=1,…,n,0≤τ(t)≤τ0,a̲i≤ai,W̲ij≤Wij≤W̅ij,T̲ij≤Tij≤T̅ij,T̲ijl≤Tijl≤T̅ijl,V̲i≤Vi≤V̅i,i,j,l=1,…,n,
where u(t)=(u1(t),…,un(t))T∈Rn, ai>0, Wij, fj, gj, Vi have the same meanings as those in [13], Tij, Tijl are the first- and second-order synaptic weights of the system (2.1) (see [12]).In this paper, the superscript"T" presents the transpose.We assume throughout that the neuron activation functionsfj(uj), gj(uj), j=1,…,n, satisfy the following conditions:
(2.2)(H1):|gj(uj)|≤Mj,0≤|fj(uj)-fj(vj)||uj-vj|≤l0j,0≤|gj(uj)-gj(vj)||uj-vj|≤l1j,∀uj≠vj,uj,vj,l0j,l1j∈R.
From (2.2), we know(2.3)|f(u1)-f(u2)|≤|L0(u1-u2)|,|g(u1)-g(u2)|≤|L1(u1-u2)|,∀u1,u2∈Rn,
where  ui=[ui1,…,uin]T∈Rn,  f(ui)=[f1(ui1),f2(ui2),…,fn(uin)]T,g(ui)=[g1(ui1),g2(ui2),…,gn(uin)]T, i=1,2. L0=diag(l01,l02,…,l0n), and L1=diag(l11,l12,…,l1n).If there is an equilibrium pointu*=[u1*,…,un*]T of system (2.1), we can rewrite system (2.1) as the following equivalent form:(2.4)d(ui(t)-ui*)dt=-ai(ui(t)-ui*)+∑j=1nWij(fj(uj(t))-fj(uj*))+∑j=1n(Tij+∑l=1n(Tijl+Tilj)ζl)(gj(uj(t-τ(t)))-gj(uj*))ui(t0+θ)=ξi(θ),-τ0≤θ≤0,i=1,…,n.0≤τ(t)≤τ0,a̲i≤ai,W̲ij≤Wij≤W̅ij,T̲ij≤Tij≤T̅ij,T̲ijl≤Tijl≤T̅ijl,i,j,l=1,…,n,
where ζl=(1/2)(gl(ul(t-τ(t)))+gl(ul*)) and |ζl|≤Ml.We easily obtain that system (2.4) is equivalent to system (2.1) (see [13]).The notations in this paper are quite standard:(i)
(2.5)||u(t)||=(∑i=1n|ui(t)|2)1/2,where |ui(t)| denotes Euclid's norm.(ii)
A=(aij)n×n>0(<0): a positive (negative) definite matrix, that is, xTAx>0(<0) for any x∈Rn.(iii)
A=(aij)n×n≥0: a semipositive definite matrix, that is, xTAx≥0 for any x∈Rn.(iv)
A≥B (resp., A>B): this means A-B is a semi-positive definite matrix (resp., positive definite).(v)
I: identity matrix with compatible dimension.(vi)
I∘=(1,1,…,1)T.(vii)
λmax(A) (resp., λmin(A)) means the largest (resp., smallest) eigenvalue of the matrix A.(viii)
C([-τ0,0],Rn) denotes a set of continuous functions.Letu(t,ξ)=[u1(t,ξ),…,un(t,ξ)]T denote the solution u(t,ξ) to system (2.1) from the initial data u(t0+θ,ξ)=ξ(θ) on -τ0≤θ≤0 in ξ(θ)∈C([-τ0,0],Rn).Definition 2.1.
The equilibrium pointu* of system (2.1) is called globally exponentially robustly stable on norm ∥·∥, if for any ξ(θ)∈C([-τ0,0],Rn) there exist scalars, J>0 and α>0 such that the solution u(t,ξ) to system (2.1) with the initial condition u(t0+θ,ξ)=ξ(θ) on -τ0≤θ≤0 satisfies
(2.6)‖u(t,ξ)-u*‖2≤Je-αtsup⁡-τ0≤θ≤0‖ξ(θ)-u*‖2.Lemma 2.2 (see [20]).
The LMI(2.7)[Q(t)S(t)ST(t)R(t)]>0,
where Q(t)=QT(t), R(t)=RT(t), and S(t) depend on t, is equivalent to any one of the following conditions:
(2.8)(K1),R(t)>0,Q(t)-S(t)R-1(t)ST(t)>0;(K2),Q(t)>0,R(t)-ST(t)Q-1(t)S(t)>0.Lemma 2.3 (see [20]).
Letx∈Rn, y∈Rn, and ε>0. Then, one has(2.9)xTy+yTx≤εxTx+ε-1yTy.


## 3. Main Results

Let(3.1)M=[M1,M2,…,Mn]T,Γ=diag⁡(M,M,…,M),ζ=[ζ1,ζ2,…,ζn]T,Π=diag⁡(ζ,ζ,…,ζ),A̲=diag⁡(a̲i),W=(Wij)n×n,W+=(Wij)n×n,W̅=(max⁡{|W̲ij|,|W̅ij|})n×n,W̅T=(max⁡{|W̲ji|,|W̅ji|})n×n,T=(Tij)n×n,T+=(Tij)n×n,T̅=(max⁡{|T̲ij|,|T̅ij|})n×n,T̅T=(max⁡{|T̲ji|,|T̅ji|})n×n,Ti=(Tijl)n×n,TiT=(Tilj)n×n,Ti+=(Tijl)n×n,T̅i=(max⁡{|T̲ijl|,|T̅ijl|})n×n,T̅iT=(max⁡{|T̲ilj|,|T̅ilj|})n×nTH=(T1+T1T,T2+T2T,…,Tn+TnT)n2×nT,TH+=(T1++(T1T)+,T2++(T2T)+,…,Tn++(TnT)+)n2×nT,T̅H=(T̅1+T̅1T,T̅2+T̅2T,…,T̅n+T̅nT)n2×nT,WΠ=T+ΠTTH,W̅Π=T̅+ΠTT̅H.For the purpose of simplicity, we rewrite the system (2.4) as the following vector form:(3.2)dydt=-Ay(t)+WF0(y(t))+(T+ΠTTH)F1(y(t-τ(t)))y(t0+θ)=φ(θ),-τ0≤θ≤0,0≤τ(t)≤τ0,a̲i≤ai,W̲ij≤Wij≤W̅ij,T̲ij≤Tij≤T̅ij,T̲ijl≤Tijl≤T̅ijl,i,j,l=1,…,n,
where y=y(t)=u(t,ξ(θ))-u* for any ξ(θ)∈C([-τ0,0],Rn), F0(y(t))=f(y(t)+u*)-f(u*), f(y(t)+u*)=[f1(y1(t)+u1*),f2(y2(t)+u2*),…,fn(yn(t)+un*)]T,F1(y(t-τ(t)))=g(y(t-τ(t))+u*)-g(u*), g(y(t-τ(t))+u*)=[g1(y1(t-τ(t))+u1*),g2(y2(t-τ(t))+u2*),…,gn(yn(t-τ(t))+un*)]T,φ(θ)=ξ(θ)-u*,ξ(θ)=(ξ1(θ),…,ξn(θ))T.Theorem 3.1.
Given a positive definite matrixQ=L1TL1>0. The equilibrium of system (2.1) is globally exponentially robustly stable on norm ∥·∥ for any τ(t) satisfying τ̇(t)≤η<1, if system (2.1) satisfies (H1) and
(3.3)(H2):C=A-W+L0-WΓ+L1is M-matrix,
where WΓ+=T++ΓTTH+, and
(3.4)(H3):[ΛL0βW̅TβW̅ΠTL0T-I00βW̅0-I0βW̅Π00-(1-η)I]≤0,
where W̅Π=T̅+ΠTT̅H, and Λ=β(αI-2A̲)+eατ0Q, β>0.Proof of Theorem3.1.
Let(3.5)hi(ui,Vi)=aiui-∑j=1nWijfj(uj)-∑j=1nTijgj(uj)-∑j=1n∑l=1nTijlgj(uj)gl(ul)+Vi=0,i=1,…,n.
It is obvious that the solutions to (3.5) are the equilibrium points of system (2.1).
Let us define homotopic mapping as follows:(3.6)H(u,λ)=(H1(u1,λ),…,Hn(un,λ))T,
where
(3.7)Hi(ui,λ)=λhi(ui,Vi)+(1-λ)ui,λ∈[0,1].By homotopy invariance theorem (see [21]), topological degree theory (see [22]), (H1)-(H2), and the proof which is similar to Theorem 3.1 in [23], we can conclude that (3.5) has at least one solution. That is, system (2.1) has at least an equilibrium point.


### 3.1. Part 2- Globally Exponentially Stable

Define a Lyapunov-Krasovskii functional candidate by(3.8)V(y(t))=βeαtyT(t)y(t)+eατ0∫t-τ(t)teαθyT(θ)Qy(θ)dθ>0,
where Q=QT>0.From system (3.2), its Dini derivative can be calculated as(3.9)D+V(y(t))=eαt[αβyT(t)y(t)+yT(t)(-βAT-βA)y(t)+β(F0T(y(t))WTy(t)+yT(t)WF0(y(t)))+βF1T(y(t-τ(t)))(T+ΠTTH)Ty(t)+yT(t)(T+ΠTTH)F1(y(t-τ(t)))]+eατ0[eαtyT(t)Qy(t)-eα(t-τ(t))yT(t-τ(t))Qy(t-τ(t))(1-τ̇(t))].Since0≤τ(t)≤τ0,  τ̇(t)≤η<1, and Q=L1TL1>0, we have(3.10)-eατ0eα(t-τ(t))yT(t-τ(t))Qy(t-τ(t))(1-τ̇(t))≤-eαtyT(t-τ(t))Qy(t-τ(t))(1-η),From Lemma2.3 and (2.3), we have(3.11)eαtβ(F0T(y(t))WTy(t)+yT(t)WF0(y(t)))≤eαtβ(βyT(t)WWTy(t)+1βF0T(y(t))F0(y(t)))≤eαtyT(t)(β2WWT+L0TL0)y(t)≤eαtyT(t)(β2W̅W̅T+L0TL0)y(t),eαtβ(F1T(y(t-τ(t)))(T+ΠTTH)Ty(t)+yT(t)(T+ΠTTH)F1(y(t-τ(t))))≤eαtβ((1-ηβ)-1yT(t)WΠWΠTy(t)+1-ηβF1T(y(t-τ(t)))F1(y(t-τ(t))))≤eαt(β21-ηyT(t)WΠWΠTy(t)+(1-η)(yT(t-τ(t))Qy(t-τ(t))))≤eαt(β21-ηyT(t)W̅ΠW̅ΠTy(t)+(1-η)(yT(t-τ(t))Qy(t-τ(t)))).In view ofQ>0,  τ̇(t)≤η<1, (3.8)–(3.11), and (H3), it follows from Lemma 2.2 that (3.12)D+V(y(t))≤eαtyT(t)(11-ηβ(αI-2A)+eατ0Q+β2(W̅W̅T+11-ηW̅ΠW̅ΠT)+L0TL0)y(t)≤0.From (3.12), we have(3.13)V(y(t))≤V(y(t0)).From (3.9) and (3.13), we can know that the solution y(t) to system (3.2) satisfies(3.14)eαtβ‖y(t)‖2=eαt(βyT(t)y(t))≤V(y(t))≤V(y(t0))≤(β+1α(eατ0-1)λmax⁡(Q))sup⁡-τ0≤θ≤0‖φ(θ)‖.So,(3.15)‖u(t,ξ)-u*‖2≤Je-αtsup⁡-τ0≤θ≤0‖ξ(θ)-u*‖2,
where J=1+1/αβ(eατ0-1)λmax(Q), for any ξ(θ)∈C([-τ0,0],Rn).If there exists another equilibriumu**=[u1**,u2**,…,un**]T of system (2.1), we have |ui*-ui**|≤|ui(t,ξ)-ui*|+|ui(t,ξ)-ui**|→0,  t→∞,  i=1,…,n.From the above proof, the system (2.1) has a unique equilibrium point u*, which is globally exponentially robustly stable. Theorem 3.1 is proved.Remark 3.2.
Whenai=a̲i>0, b̲ij=bij=b̅ij, w̲ij=wij=w̅ij, w̲ijl(k)=wijl(k)=w̅ijl(k), V̲i=Vi=V̅i, the system (2.1) becomes to system in [12, 14]. So, the system in [12, 14] is special case of system (2.1).


## 3.1. Part 2- Globally Exponentially Stable

Define a Lyapunov-Krasovskii functional candidate by(3.8)V(y(t))=βeαtyT(t)y(t)+eατ0∫t-τ(t)teαθyT(θ)Qy(θ)dθ>0,
where Q=QT>0.From system (3.2), its Dini derivative can be calculated as(3.9)D+V(y(t))=eαt[αβyT(t)y(t)+yT(t)(-βAT-βA)y(t)+β(F0T(y(t))WTy(t)+yT(t)WF0(y(t)))+βF1T(y(t-τ(t)))(T+ΠTTH)Ty(t)+yT(t)(T+ΠTTH)F1(y(t-τ(t)))]+eατ0[eαtyT(t)Qy(t)-eα(t-τ(t))yT(t-τ(t))Qy(t-τ(t))(1-τ̇(t))].Since0≤τ(t)≤τ0,  τ̇(t)≤η<1, and Q=L1TL1>0, we have(3.10)-eατ0eα(t-τ(t))yT(t-τ(t))Qy(t-τ(t))(1-τ̇(t))≤-eαtyT(t-τ(t))Qy(t-τ(t))(1-η),From Lemma2.3 and (2.3), we have(3.11)eαtβ(F0T(y(t))WTy(t)+yT(t)WF0(y(t)))≤eαtβ(βyT(t)WWTy(t)+1βF0T(y(t))F0(y(t)))≤eαtyT(t)(β2WWT+L0TL0)y(t)≤eαtyT(t)(β2W̅W̅T+L0TL0)y(t),eαtβ(F1T(y(t-τ(t)))(T+ΠTTH)Ty(t)+yT(t)(T+ΠTTH)F1(y(t-τ(t))))≤eαtβ((1-ηβ)-1yT(t)WΠWΠTy(t)+1-ηβF1T(y(t-τ(t)))F1(y(t-τ(t))))≤eαt(β21-ηyT(t)WΠWΠTy(t)+(1-η)(yT(t-τ(t))Qy(t-τ(t))))≤eαt(β21-ηyT(t)W̅ΠW̅ΠTy(t)+(1-η)(yT(t-τ(t))Qy(t-τ(t)))).In view ofQ>0,  τ̇(t)≤η<1, (3.8)–(3.11), and (H3), it follows from Lemma 2.2 that (3.12)D+V(y(t))≤eαtyT(t)(11-ηβ(αI-2A)+eατ0Q+β2(W̅W̅T+11-ηW̅ΠW̅ΠT)+L0TL0)y(t)≤0.From (3.12), we have(3.13)V(y(t))≤V(y(t0)).From (3.9) and (3.13), we can know that the solution y(t) to system (3.2) satisfies(3.14)eαtβ‖y(t)‖2=eαt(βyT(t)y(t))≤V(y(t))≤V(y(t0))≤(β+1α(eατ0-1)λmax⁡(Q))sup⁡-τ0≤θ≤0‖φ(θ)‖.So,(3.15)‖u(t,ξ)-u*‖2≤Je-αtsup⁡-τ0≤θ≤0‖ξ(θ)-u*‖2,
where J=1+1/αβ(eατ0-1)λmax(Q), for any ξ(θ)∈C([-τ0,0],Rn).If there exists another equilibriumu**=[u1**,u2**,…,un**]T of system (2.1), we have |ui*-ui**|≤|ui(t,ξ)-ui*|+|ui(t,ξ)-ui**|→0,  t→∞,  i=1,…,n.From the above proof, the system (2.1) has a unique equilibrium point u*, which is globally exponentially robustly stable. Theorem 3.1 is proved.Remark 3.2.
Whenai=a̲i>0, b̲ij=bij=b̅ij, w̲ij=wij=w̅ij, w̲ijl(k)=wijl(k)=w̅ijl(k), V̲i=Vi=V̅i, the system (2.1) becomes to system in [12, 14]. So, the system in [12, 14] is special case of system (2.1).


## 4. Conclusion

We have investigated global exponential robust stability of high-order hopfield neural networks with time-varying delays. By employing a new Lyapunov-Krasovskii functional and linear matrix inequality, some criteria of global exponential robust stability for the neural networks are established, which are easily verifiable.The systems found in the literature are special cases of the system (2.1). So the problem addressed in this paper should be a focused topic of theoretical as well as practical importance.



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