# Less Conservative Stability Criteria for Neutral Type Neural Networks with Mixed Time-Varying Delays

Authors: Kaibo Shi; Hong Zhu; Shouming Zhong; Yong Zeng; Yuping Zhang; Li Liang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper investigates the problem of dependent stability criteria for neutral type neural networks with mixed time-varying delays. Firstly, some new delay-dependent stability results are obtained by employing the more general partitioning approach and generalizing the famous Jensen inequality. Secondly, based on a new type of Lyapunov-Krasovskii functional with the cross terms of variables, less conservative stability criteria
are proposed in terms of linear matrix inequalities (LMIs). Furthermore, it is the first time that the idea of second-order convex combination and the property of quadratic convex function applied to the derivation of neutral type neural networks play an important role in reducing the conservatism of the paper. Finally, four numerical examples are given to show the effectiveness and the advantage of the proposed method.


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## Body

## 1. Introduction

During the last two decades, delayed neural networks have drawn a great deal of attention because of their extensive applications in various scientific and technical areas, such as pattern recognition, power systems, parallel computing, signal processing, finance, associative memories, mechanics of structures, and other scientific areas [1–30]. It is well known that time delay regarded as a major cause of instability and poor performance often appears in many neural networks. Therefore, the stability analysis for delayed neural networks has been investigated extensively in recent few decades. Generally speaking, studying the dynamical behavior of delayed neural networks can be mainly classified into two types: delay-independent stability and delay-dependent stability. As is known to all, delay-dependent stability criteria are less conservative than delay-independent ones when the size of time delay is small.On the other hand, due to the complicated dynamic properties of the neural cells in the real world, there exist many neural network models such as distributed networks, chemical reactors, and heat exchanges that cannot characterize the properties of a neural reaction process precisely. It is natural and important that these systems will contain some information about the derivative of the past state to further describe and model the dynamics of the complex neural reactions. This new type of neural networks is called neutral neural networks or neural networks of neutral type. However, many researchers have focused on the global stability of neural networks of neutral type only with constant time delay in recent years, which is very restrictive. Hence, described with neutral functional differential equations with discrete and distributed delays, these neural networks called neutral type neural networks with mixed time-varying delays have a lot of research on space. The differential expression not only defines the derivative term of the current state but also explains the derivative term of the past state. Furthermore, it is necessary to have some information about the derivative of the past state in the systems to characterize the dynamics of such complex neural reactions. Practically, neutral type phenomenon always appears in studies of automatic control, chemical reactors, distributed networks, dynamic process including steam and water pipes, population ecology, heat exchanges, microwave oscillators, systems of turbojet engine, lossless transmission lines, vibrating masses attached to an elastic bar, and so on. For this reason, there has been a growing research interest in the study of delayed neural networks of neutral type in the recent years. Therefore, some less conservative stability criteria for neutral type neural networks with mixed time-varying delays have been reported in recently [25, 31–35]. Many methods have been proposed in these results to reduce the conservatism of the stability criteria, such as model transformation method, free-weighting matrix method, the method of constructing novel Lyapunov-Krasovskii functionals, delay decomposition technique, and weighting-matrix decomposition method. In [36], the authors derived some less conservative stability criteria by considering some useful terms and using free-weighting matrix technique. By considering the relationship between the time-varying delay and its lower and upper bound, the results obtained in [36] were improved in [37]. By constructing a new Lyapunov-Krasovskii functional and using free-weighting matrix method, some more less conservative criteria than those obtained in [37] were proposed in [38]. Further, the problems of stability analysis of neutral type neural networks with discrete and distributed delays have been investigated in [39]. By using a delay-partitioning approach, a new type of Lyapunov-Krasovskii functionals was constructed to obtain some less conservative stability criteria. However, time delay in [39] is not only constant delay, but also the delay-partitioning approach is equational; hence, this method has some limitations.Motivated by this technique, it is the first attempt to investigate the integral nonuniform partitioning method to be extended for neutral type neural networks with mixed time-varying delays. In the paper, the reduced conservatism of Theorem6 benefits from the construction of the new Lyapunov-Krasovskii functionals in (17), which contain some integral nonuniform partitioning method and triple-integral terms, which play an important role in the improvement of less conservative results. Secondly, a novel handling method is given to establish the relationship among ∫t-ht‍x˙(s)S5x˙(s)ds, ∫t-ht‍xT(s)ds and x(t-h), which play an important role in reducing the conservatism of stability criteria further. Furthermore, compared with previous results by using the first-order convex combination property, our derivation makes full use of the idea of second-order convex combination and the property of quadratic convex function given in the form of a lemma without employing Jensen's inequality. Finally, four numerical examples are given to illustrate the effectiveness and the advantage of the proposed main results.Notation 1.
Notations used in this paper are fairly standard:Rn denotes the n-dimensional Euclidean space, Rn×m is the set of all n×m dimensional matrices; I denotes the identity matrix of appropriate dimensions, T stands for matrix transposition, the natation X>0 (resp., X≥0), for X∈Rn×n means that the matrix is real symmetric positive definite (resp., positive semidefinite); diag{r1,r2,…,rn} denotes block diagonal matrix with diagonal elements ri, i=1,2,…,n, the symbol * represents the elements below the main diagonal of a symmetric matrix, and 〈M〉s is defined as 〈M〉s=(1/2)(M+MT).


## 2. Preliminaries

Consider the following neural networks of neutral type with mixed time-varying delays:(1)z˙(t)=-W0z(t)+W1g(z(t))+W2g(z(t-h(t)))+W3∫t-rt‍gT(z(s))ds+W4y˙(t-h(t))+J,
where z(t)=[z1(t),…,zn(t)]T∈Rn is the neural state vector, g(z(t))=[g(z1(t)),…,g(zn(t))]T is the neuron activation function, J=[J1,…,Jn]T∈Rn is an external constant input vector, and W0=diag{w01,…,w0n}>0, W1, W2, W3, and W4 are the constant matrices of appropriate dimensions.Assumption A.
The time-varying delayh(t) is continuous and differential function that satisfies
(2)0≤h(t)≤h,h˙(t)≤hD<1.Assumption B.
For the constantsδi+, δi-, the nonlinear function gi(·) in (1) satisfies the following condition:
(3)δi-≤gi(α)-gi(β)α-β≤δi+,∀α,β∈R,α≠β,i=1,2,…,n.
Here, we denote Σ¯=diag{δ1+,…,δn+}, Σ_=diag{δ1-,…,δn-}, Σ=diag{max{|δ1+|,|δ1-|},…,max{|δn+|,|δn-|}}=diag{δ1,…,δn}, Σ1=diag{δ1+δ1-,…,δn+δn-}, Σ2=diag{(δ1++δ1-)/2,…,(δn++δn-)/2}.Assumption C.
For given positive scalarsρi satisfies:
(4)0=ρ0<ρ1<⋯<ρl=1(i=1,…,l).
It is clear that under Assumption B, the system (1) has one equilibrium point denoted as z*=[z1*,…,zn*]T. For convenience, we firstly shift the equilibrium point z* to the origin by letting x(t)=z(t)-z*, f(x(t))=g(z(t))-g(z*); then the system (1) can be transformed into
(5)x˙(t)=-W0x(t)+W1f(x(t))+W2f(x(t-h(t)))+W3∫t-rt‍fT(x(s))ds+W4x˙(t-h(t)),
where x(t)=[x1(t),…,xn(t)]∈Rn is the state vector of transformed system, f(x(t))=[f1(x1),…,fn(xn)]T∈Rn. It is easy to check that the transformed neuron activation function fi(·) satisfies
(6)δi-≤fi(α)α≤δi+,fi(0)=0,∀α∈R,α≠0,i=1,2,…,n.
The following lemmas are introduced, which will be used in the proof of the main results.Fact 1 (Boyd et al. [40], (Schur complement)).
For a given symmetric matrixX=XT=[X11X12*X22], where S11∈Rn×n, the following conditions are equivalent: (1)
X<0;(2)
X11<0, X22-X12TX11-1X12<0;(3)
X22<0, X11-X12X22-1X12T<0.Lemma 1 (see [41]).
For symmetric matricesY0, Y1, Y2, and a vector ξt, Let f(α)=α2ξtTY2ξt+αξtTY1ξt+ξtTY0ξt with Y2≤0. Then we have f(α1)<0 and f(α2)<0⇒f(α)<0,  ∀α∈[α1,α2].Lemma 2 (see [42]).
LetW>0, and let ω(s) be an appropriate dimensional vector. Then, we have the following facts for any scalar function β(s)≥0  ∀s∈[a,b]:(1)
-∫ab‍ωT(s)Wω(s)ds≤(b-a)ξtTF1W-1F1ξt+2ξtTF1∫ab‍ω(s)ds;(2)
-∫ab‍β(s)ωT(s)Wω(s)ds≤∫ab‍β(s)dsξtTF2W-1F2ξt+2ξtTF2∫ab‍β(s)ω(s)ds;(3)
-∫ab‍β2(s)ωT(s)Wω(s)ds≤(b-a)ξtTF3W-1F3ξt+2ξtTF3∫ab‍β(s)ω(s)ds,where matrices Fi  (i=1,2,3) and a vector ξt independent of the integral variable are appropriate dimensional arbitrary ones.Lemma 3 (see [43]).
For any constant matrix0<R=RT∈Rn×n, a scalar r>0 and a vector function x:[0,r]→Rn such that the integrations concerned are well defined; then
(7)-∫t-r0‍xT(s)Rx(s)ds≤-1r(∫t-rt‍x(s)ds)TR(∫t-rt‍x(s)ds),-∫-r0‍∫t+θt‍xT(s)Rx(s)dsdθ≤-2r2(∫-r0‍∫t+θt‍x(s)dsdθ)TR(∫-r0‍∫t+θt‍x(s)dsdθ).Lemma 4 (see [44]).
For any constant matrixR∈Rn×n, R=RT>0, a scalar function h:=h(t)>0, and a vector-valued function x˙:[-h,0]→Rn such that the following integrations are well defined:
(8)-h∫t-ht‍x˙(s)Rx˙(s)ds≤[x(t)x(t-h)]T[-RRR-R][x(t)x(t-h)],-h22∫-h0‍∫t+θt‍x˙T(s)Rx˙(s)dsdθ≤[hx(t)∫t-ht‍x(s)ds]T[-RRR-R][hx(t)∫t-ht‍x(s)ds].Lemma 5.
Letx(t)∈Rn has continuous derived function x˙(t) on interval [0,h]. Then for any matrix Zn×n>0, scalar h>0, the following inequality holds:
(9)-h∫t-ht‍x˙(s)Rx˙(s)ds≤-2h[1h∫t-ht‍x(s)dsx(t-h)]T[R-R-RR][1h∫t-ht‍x(s)dsx(t-h)].Proof.
From Lemma3, we can get
(10)1h(∫t-ht‍∫t-hθ‍x˙(s)dsdθ)TR(∫t-ht‍∫t-hθ‍x˙(s)dsdθ)≤∫t-ht‍(∫t-hθ‍x˙(s)ds)TR(∫t-hθ‍x˙(s)ds)dθ≤∫t-ht‍∫t-hθ‍(θ-(t-h))x˙T(s)Rx˙(s)dsdθ≤∫t-ht‍∫st‍(θ-(t-h))x˙T(s)Rx˙(s)dθds=∫t-ht‍(h22-(s-t+h)22)x˙T(s)Rx˙(s)ds≤h22∫t-ht‍x˙T(s)Rx˙(s)ds.
Notice that
(11)(∫t-ht‍∫t-hθ‍x˙(s)dsdθ)TR(∫t-ht‍∫t-hθ‍x˙(s)dsdθ)=[∫t-ht‍x(s)dshx(t-h)]T[R-R-RR][∫t-ht‍x(s)dshx(t-h)].
Therefore, we get
(12)-∫t-ht‍x˙(s)Rx˙(s)ds≤-2h[1h∫t-ht‍x(s)dsx(t-h)]T[R-R-RR][1h∫t-ht‍x(s)dsx(t-h)].
This completes the proof.


## 3. Main Results

In this section we will give sufficient conditions under which the system (5) is asymptotically stable.Theorem 6.
For given scalarsh>0 and hD<1, the system (5) with the neuron activation function f(x(t)) satisfying the condition (6) is asymptotically stable if there exists P>0,   RiT=Ri>0(i=1,2,3),  QiT=Qi>0(i=1,2),  SiT=Si>0(i=1,…,8),  ZiT=Zi>0(i=1,…,l), diagonal matrices Gi=diag{gi1,g2,…,gin}>0,  Ki=diag{ki1,k2,…,kin}>0,  Li=diag{li1,l2,…,lin}>0,  (i=1,2,3) and M=diag{mi1,mi2,…,min}>0,  (i=1,2,3,4),  Ti(i=1,…,5) and Fi(i=1,…,6) with appropriate dimensions such that the following symmetric linear matrix inequality holds:
(13)Ξ1=[Π0hF1ThF2ThF3T*-hR300**-Q10***-h3Q2]<0,(14)Ξ2=[Π0+hΠ1hF4ThF5ThF6T*-hR300**-Q10***-h3Q2]<0,
where
(15)Π0=〈2[e1e7+e8]P[e2e1-e4]T+2[e20]R1[0e7+e8]T+2[e20]R2[he1e8]T〉s+[e1e1]×(R1+R2)[e1e1]T-(1-hD)[e1e3]R1[e1e3]T-[e1e4]R2[e1e4]T+h[e1e2]R3×[e1e2]T+e2(h2Q1+h3Q2)e2T+〈2F1T[e7e3-e4]T+4F2T[he3-e7]T+6F3T[he3-e7]T+2F4T[e8e1-e3]T-4F5Te8T-6F6Te8T〉s+e9(S1+S2)e9T-(1-hD)e10S1e10T-e11S2e11T-e6S4e6T+e2(S3+S4)e2T-(1-hD)e5S3e5T+e2(hS5+h22S6)e2T-2e1S6e1T+4he1S6e7T+4he1S6e8T+4h2e4S5e7T+4h2e4S5e8T-2he4S5e4T-e7(2h3S5+2h2S6)e7T-e7(4h3S5+4h2S6)e8T-e8(2h3S5+2h2S6)e8T+e9(rS7+r22S8+∑i=1l‍(ρi-ρi-1)rZi)e9T-∑i=1l‍1(ρi-ρi-1)re11+iZie11+iT-1r(e12+⋯+e11+l)S7(e12+⋯+e11+l)T-2r2e12+lS8e12+lT+e1(Σ¯K1-Σ_G1+L1Σ)e2T+2e2(G1-K1+L1)e9T+2e3(Σ¯K2-Σ_G2+L2Σ)e5T+2e5(G2-K2+L2)e10T+2e4(Σ¯K3-Σ_G3+L3Σ)e6T+2e6(G3-K3+L3)e11T+e1(ΣM1Σ-M2Σ1)e1T+2e1M2Σ2e9T-e3M3Σ1e3T+2e3M3Σ2e10T-e9(M1+M2)e9T-e10M3e10T-e4M4Σ1e4T+2e4M4Σ2e11T+e2(-T1-T1T)e2T-e11M4e11T-2e1W0TT1Te2T+2e2T1W1e9T+2e2T1W2e10T+2e2T1W3(e12+⋯+e11+l)T+2e2T1W4e5T-2e1T2e2T+e1(-T2W0-W0TT2T)e1T+2e1T2W1e9T+2e1T2W2e10T+2e1T2W3(e12+⋯+e11+l)T+2e2T2W4e5T-2e2T3Te9T-2e1W0TT3Te9T+2e9T3W2e10T+e9(T3W1+W1TT3T)e9T+2e9T3W3(e12+⋯+e11+l)T+2e5W4TT3Te9T-2e2T4Te10T+e9W1TT4Te10T+e10(T4W2+W2TT4T)e10T+2e10T4W3(e12+⋯+e11+l)T+2e5W4TT4Te10T+2e5W4TT3Te9T-2e2T5Te5T-2e1W0TT5Te5T+e5T5W1e9T+e5T5W2e10T+2e5(T5W4+W4TT4T)e5T+2e5T5W3(e12+⋯+e11+l)T-2e1W0TT4Te10T,Π1=〈2(e20)R1(e1e0)T-4F2Te3T-6F3Te3T+4F5Te1T+6F6Te1T(e1e0)T〉s,
with
(16)〈·〉s=12[(·)+(·)T],ei=[0n×(i-1)In×n0n×(12+l-i)]T(i=1,…,12+l).Proof.
Consider a novel augmented Lyapunov-Krasovskii functional for the system (5) as follows:
(17)V(xt)=V1(xt)+V2(xt)+V3(xt)+V4(xt)+V5(xt)+V6(xt)+V7(xt),
where
(18)V1(xt)=ξ1T(t)Pξ1(t),V2(xt)=∫t-h(t)t‍ξ2T(t,s){[E1E2]R1[E1E2]T}ξ2(t,s)ds+∫t-ht‍ξ2T(t,s){[E1E2]R2[E1E2]T}ξ2(t,s)ds,(19)V3(xt)=∫t-ht‍(h-t+s)ξ2T(t,s){[E2E3]R3[E2E3]T}ξ2(t,s)ds+∫t-ht‍(h-t+s)2ξ2T(t,s){E3Q1E3T}ξ2(t,s)ds+∫t-ht‍(h-t+s)3ξ2T(t,s){E3Q2E3T}ξ2(t,s)ds,(20)V4(xt)=∫t-h(t)t‍fT(x(s))S1f(x(s))ds+∫t-ht‍fT(x(s))S2f(x(s))ds+∫t-h(t)t‍x˙T(s)S3x˙(s)ds+∫t-ht‍x˙T(s)S4x˙(s)ds,(21)V5(xt)=∫t-ht‍(h-t+s)x˙T(s)S5x˙(s)ds+∫-h0‍∫θ0‍∫t+λt‍x˙T(s)S6x˙(s)dsdλdθ,(22)V6(xt)=∫t-rt‍(r-t+s)fT(x(s))S7f(x(s))ds+∫-r0‍∫θ0‍∫t+λt‍fT(x(s))S8f(x(s))dsdλdθ+2∑i=1l‍∫-ρir-ρi-1r‍∫t+θt‍f(x(s))Zif(x(s))dsdθ,(23)V7(xt)=2∑i=1n‍g1i∫0xi(t)‍(fi(s)-δi-s)ds+2∑i=1n‍k1i∫0xi(t)‍(δi+s-fi(s))ds+2∑i=1n‍l1i∫0xi(t)‍(fi(s)+δis)ds+2∑i=1n‍g2i∫0xi(t-h(t))‍(fi(s)-δi-s)ds+2∑i=1n‍k2i∫0xi(t-h(t))‍(δi+s-fi(s))ds+2∑i=1n‍l2i∫0xi(t-h(t))‍(fi(s)+δis)ds+2∑i=1n‍g3i∫0xi(t-h)‍(fi(s)-δi-s)ds+2∑i=1n‍k3i∫0xi(t-h)‍(δi+s-fi(s))ds+2∑i=1n‍l3i∫0xi(t-h)‍(fi(s)+δis)ds,
with
(24)ξ1T(t)=[xT(t)∫t-ht‍xT(s)ds],ξ2T(t,s)=[xT(t)xT(s)x˙T(s)],ξ3T(t)=[xT(t)x˙T(t)xT(t-h(t))xT(t-h)x˙T(t-h(t))x˙T(t-h)],ξ4T(t)=[∫t-ht-h(t)‍xT(s)ds∫t-h(t)t‍xT(s)dsfT(x(t))fT(x(t-h(t)))fT(x(t-h))∫t-ht-h(t)‍],ξ5T(t)=[∫t-ρ1rt‍fT(x(s))ds⋯∫t-rt-ρl-1r‍fT(x(s))ds],ξT(t)=[ξ3T(t)ξ4T(t)ξ5T(t)∫-r0‍∫t+θt‍fT(x(s))dsdθ],Ei=[0n×(i-1)In×n0n×(3-i)]T(i=1,2,3).
The time derivative of  V(xt) along the trajectory of system (5) is given as
(25)V˙(xt)=V˙1(xt)+V˙2(xt)+V˙3(xt)+V˙4(xt)+V˙5(xt)+V˙6(xt)+V˙7(xt),
where
(26)V˙1(xt)=2ξ1T(t)Pξ˙1(t)=2ξT(t)[e1e7+e8]P[e2e1-e4]Tξ(t),V˙2(xt)≤[xT(t)xT(t)]R1[xT(t)xT(t)]T+2[x˙T(t)0]R1∫t-h(t)t‍[xT(t)xT(s)]Td(s)-(1-hD)×[xT(t)xT(t-h(t))]R1×[xT(t)xT(t-h(t))]T+[xT(t)xT(t)]R2×[xT(t)xT(t)]T+2[x˙T(t)0]R2×∫t-ht‍[xT(t)xT(s)]Td(s)-[xT(t)xT(t-h)]×R2[xT(t)xT(t-h)]T=ξT(t){(e1e4)T(e1e1)R1(e1e1)T+2(e20)×R1(h(t)e1e8)T-(1-hD)×(e1e3)R1(e1e3)T+(e1e1)×R2(e1e1)T+2(e20)R2×(he1e7+e8)T-(e1e4)×R2(e1e4)T}ξ(t),V˙3(xt)=hξ2(t,t){[E2E3]R3[E2E3]T}ξ2(t,t)+x˙T(t)(h2Q1+h3Q2)x˙(t)+Va(x(t))=ξT(t){h(e1e2)R3(e1e2)T+e2(h2Q1+h3Q2)e2T(e1e2)T}ξ(t)+Va(x(t)).
Here, Va(x(t)) is the sum of all integral terms expressed as
(27)Va(x(t))=-∫t-ht‍η2(t,s){[E2E3]R3[E2E3]T}η2(t,s)ds-∫t-ht‍2(h-t+s)η2(t,s){E3Q1E3T}η2(t,s)ds-∫t-ht‍3(h-t+s)2η2(t,s){E3Q2E3T}η2(t,s)ds≤-∫t-ht-h(t)‍η2(t,s){[E2E3]R3[E2E3]T}η2(t,s)ds-∫t-ht-h(t)‍2(h-t+s)η2(t,s){E3Q1E3T}η2(t,s)ds-∫t-ht-h(t)‍3(h-t+s)2η2(t,s){E3Q2E3T}η2(t,s)ds-∫t-h(t)t‍η2(t,s){[E2E3]R3[E2E3]T}η2(t,s)ds-∫t-h(t)t‍2(h(t)-t+s)η2(t,s){E3Q1E3T}η2(t,s)ds-∫t-h(t)t‍3(h(t)-t+s)2η2(t,s){E3Q2E3T}×η2(t,s)ds=V^a(x(t)).
Apply Lemma 2 to V^a(x(t)),
(28)V^a(x(t))≤ξT(t){(h-h(t))F1TR3-1F1+2F1T(e7e3-e4)T+(h-h(t))2F2TQ1-1F2+4F2T[(h-h(t))e3-e7]T+3(h-h(t))F3TQ1-1F3+6F3T[(h-h(t))e3-e7]T+h(t)F4TR3-1F4+2F4T(e8e1-e3)T+h2(t)F5TQ1-1F5+4F5T[h(t)e1-e8]T+3h(t)F6TQ2-1F6+6F6T[h(t)e1-e8]T(e7e3-e4)T}×ξ(t),V˙4(xt)≤fT(x(t))S1f(x(t))-(1-hD)fT(x(t-h(t)))S1f(x(t-h(t)))+fT(x(t))S2f(x(t))-fT(x(t-h))S2f(x(t-h))+x˙T(t)S3x˙(t)-(1-hD)x˙T(t-h(t))S3x˙(t-h(t))+x˙T(t)S4x˙(t)-x˙T(t-h)S4x˙(t-h)=ξT(t){e9(S1+S2)e9T-(1-hD)e10S1e10T-e11S2e11T+e2(S3+S4)e2T-(1-hD)e5S3e5T-e6S4e6T}ξ(t),(29)V˙5(xt)=x˙T(t)(hS5+h22S6)x˙(t)-∫t-ht‍x˙T(s)S5x˙(s)ds-∫-h0‍∫t+θt‍x˙T(s)S6x˙(s)dsdθ.
From Lemmas 4 and 5, we have
(30)-∫t-ht‍x˙T(s)S5x˙(s)ds≤-2h[1h∫t-ht‍x(s)dsx(t-h)]T[S5-S5-S5S5]×[1h∫t-ht‍x(s)dsx(t-h)],-∫-h0‍∫t+θt‍x˙T(s)S6x˙(s)dsdθ≤2h2[hx(t)∫t-ht‍x(s)ds]T[-S6S6S6-S6][hx(t)∫t-ht‍x(s)ds].
Then combining (29) and (30), we can obtain that
(31)V˙5(xt)≤ξT(t){e2(hS5+h22S6)e2T-2e1S6e1T+4he1S6e7T+4he1S6e8T+4h2e4S5e7T+4h2(t)e4S5e8T-2he4S5e4T-e7(2h3S5+2h2S6)e7T-e7(4h3S5+4h2S6)e8T-e8(2h3S5+2h2S6)e8T(hS5+h22S6)}ξ(t),(32)V˙6(xt)=rfT(x(t))S7f(x(t))-∫t-rt‍fT(x(s))S7f(x(s))ds+r22fT(x(t))S8f(x(t))-∫-r0‍∫t+θt‍f(x(s))S8f(x(s))dsdθ+∑i=1l‍(ρi-ρi-1)rfT(x(t))Zif(x(t))-∑i=1l‍∫t-ρirt-ρi-1r‍fT(x(s))Zif(x(s))ds.
From Lemma 3, we get
(33)-∫t-rt‍fT(x(s))S7fT(x(s))ds≤-1r∫t-rt‍fT(x(s))dsS7∫t-rt‍f(x(s))ds,-∫-r0‍∫t+θt‍fT(x(s))S8f(x(s))dsdθ≤-2r2∫-r0‍∫t+θt‍fT(x(s))dsdθS8×∫-r0‍∫t+θt‍f(x(s))dsdθ,-∫t-ρirt-ρi-1r‍fT(x(s))Zif(x(s))ds≤-1(ρi-ρi-1)r∫t-ρirt-ρi-1r‍fT(x(s))dsZi×∫t-ρirt-ρi-1r‍f(x(s))ds.
Then combining (32) and (33) we can have that
(34)V˙6(xt)≤rfT(x(t))S7f(x(t))+r22fT(x(t))S8f(x(t))+∑i=1l‍(ρi-ρi-1)rfT(x(t))Zif(x(t))-1r∑i=1l‍∫t-ρirt-ρi-1r‍fT(x(s))dsS7∑i=1l‍∫t-ρirt-ρi-1r‍f(x(s))ds-2r2∫-r0‍∫t+θt‍fT(x(s))dsdθS8∫-r0‍∫t+θt‍f(x(s))ds-∑i=1l‍1(ρi-ρi-1)r∫t-ρirt-ρi-1r‍fT(x(s))dsZi×∫t-ρirt-ρi-1r‍f(x(s))ds=ξT(t){e9(rS7+r22S8+∑i=1l‍(ρi-ρi-1)rZi)e9T-2r2e12+lS8e12+lT-∑i=1l‍1(ρi-ρi-1)re11+i×Zie11+iT-1r(e12+⋯+e11+l)×S7(e12+⋯+e11+l)T∑i=1l‍}ξ(t),(35)V˙7(xt)≤2[fT(x(t))(G1-K1+L1)+xT(t)(Σ¯K1-Σ_G1+L1Σ)]×x˙(t)+2(1-hD)×[fT(x(t-h(t)))(G2-K2+L2)+xT(t-h(t))(Σ¯K2-Σ_G2+L2Σ)]×x˙(t-h(t))+2[fT(x(t-h))(G3-K3+L3)+xT(t-h)(Σ¯K3-Σ_G3+L3Σ)]×x˙(t-h)=2ξT(t){e1(Σ¯K1-Σ_G1+L1Σ)e2T+2e2(G1-K1+L1)e9T+2e3(Σ¯K2-Σ_G2+L2Σ)e5T+2e5(G2-K2+L2)e10T+2e4(Σ¯K3-Σ_G3+L3Σ)e6T+2e6(G3-K3+L3)e11T}ξ(t).
From (6), for any positive diagonal matrices Mi=diag{mi1,…,min}(i=1,2,3,4) one can easily check
(36)0≤[xT(t)ΣM1Σx(t)-fT(x(t))M1f(x(t))]+[-xT(t)M2Σ1x(t)+2xT(t)M2Σ2f(x(t))-fT(x(t))M2f(x(t))]+[(x(t-h(t)))-xT(t-h(t))M3×Σ1x(t-h(t))+2xT(t-h(t))×M3Σ2f(x(t-h(t)))-fT(x(t-h(t)))×M3f(x(t-h(t)))xT]+[(x(t-h))-xT(t-h)M4Σ1x(t-h)+2xT(t-h)M4Σ2f×(x(t-h))-fT(x(t-h))M4f(x(t-h))xT]=ξT(t){e1(ΣM1Σ-M2Σ1)e1T+2e1M2Σ2e9T-e3M3Σ1e3T+2e3M3Σ2e10T-e9(M1+M2)e9T-e10M3e10T-e4M4Σ1e4T+2e4M4Σ2e11T-e11M4e11T}ξ(t).
Furthermore, for arbitrary matrices T1, T2, T3, T4, T5 with appropriate dimensions, we have
(37)2x˙T(t)T1×[∫t-rt‍-x˙(t)-W0x(t)+W1f(x(t))+W2f(x(t-h(t)))+W3∫t-rt‍fT(x(s))ds+W4(x˙(t-h(t)))]=ξT(t){e2(-T1-T1T)e2T-2e1W0TT1Te2T+2e2T1W1e9T+2e2T1W2e10T+2e2T1W3(e12+⋯+e11+l)T+2e2T1W4e5T}ξ(t)=0,(38)2xT(t)T2×[∫t-rt‍-x˙(t)-W0x(t)+W1f(x(t))+W2f(x(t-h(t)))+W3∫t-rt‍fT(x(s))ds+W4(x˙(t-h(t)))]=ξT(t){-2e1T2e2T+e1(-T2W0-W0TT2T)e1T+2e1T2W1e9T+2e1T2W2e10T+2e1T2W3(e12+⋯+e11+l)T+2e2T2W4e5T}ξ(t)=0,(39)2fT(x(t))T3×[∫t-rt‍-x˙(t)-W0x(t)+W1f(x(t))+W2f(x(t-h(t)))+W3∫t-rt‍fT(x(s))ds+W4(x˙(t-h(t)))∫t-rt‍]=ξT(t){-2e2T3Te9T-2e1W0TT3Te9T+e9(T3W1+W1TT3T)e9T+2e9T3W2e10T+2e9T3W3(e12+⋯+e11+l)T+2e5W4TT3Te9T}ξ(t)=0,(40)2fT(x(t-h(t)))T4×[(x˙(t-h(t)))-x˙(t)-W0x(t)+W1f(x(t))+W2f(x(t-h(t)))+W3∫t-rt‍fT(x(s))ds+W4(x˙(t-h(t)))]=ξT(t){-2e2T4Te10T-2e1W0TT4Te10T+e9W1TT4Te10T+e10(T4W2+W2TT4T)e10T+2e10T4W3(e12+⋯+e11+l)T+2e5W4TT4Te10T}ξ(t)=0,(41)x˙T(t-h(t))T5×[(x˙(t-h(t)))-x˙(t)-W0x(t)+W1f(x(t))+W2f(x(t-h(t)))+W3∫t-rt‍fT(x(s))ds+W4(x˙(t-h(t)))]=ξT(t){-2e2T5Te5T-2e1W0TT5Te5T+e5T5W1e9T+e5T5W2e10T+2e5T5W3(e12+⋯+e11+l)T+2e5(T5W4+W4TT4T)e5T}ξ(t)=0.
The combination of (26)–(37) gives that
(42)V(xt)≤ξT(t){Π0+h(t)Π1+Πh}ξ(t),
where Π0, Π1 are defined in (13) and (14), respectively, and
(43)Πh=(h-h(t))F1TR3-1F1+(h-h(t))2F2TQ1-1F2+3(h-h(t))F3TQ1-1F3+h(t)F4TR3-1F4+h2(t)F5TQ1-1F5+3h(t)F6TQ2-1F6.
Note that the scalar valued function ξT(t){Π0+h(t)Π1+Πh}ξ(t) is quadratic function on the scalar h(t) and the coefficient of second order is ξT(t)[F2TQ1-1F2+F5TQ1-1F5]ξ(t)≥0 since Q1>0. This means that the function ξT(t){Π0+h(t)Π1+Πh}ξ(t) is a convex quadratic function for h(t). Finally, apply Fact 1 and Lemma 1 in order, then we get
(44)Ξ1<0in(22),Ξ2<0in(23)(45)⟺[Π0+h(t)Π1+Πh]h(t)=0<0,[Π0+h(t)Π1+Πh]h(t)=h<0,(46)⟺Π0+h(t)Π1+Πh<0,∀h(t)∈[0,h],
which means the asymptotic stability of the system (19). This completes the proof.Remark 7.
In Theorem6, the augmented vector ξ(t) has integrating terms of activation function f(x(t)) which are ∫t-ρ1rt‍fT(x(s))ds,…,∫t-rt-ρl-1r‍fT(x(s))ds and ∫-r0‍∫t+θt‍fT(x(s))dsdθ. By these terms, more past history of f(x(t)) can be used, which lead to less conservative results.Remark 8.
Compared with those in previous articles, Ours constructed a new type of Lyapunov-Krasovskii functional which has three differences:(1) an independent augmented variable ∫t-ht‍xT(s)ds; (2) the cross terms between entries in ξ1(t), ξ2(t,s), respectively; (3) quadratic terms multiplied by first, second, and third degrees of a scalar function h-t+s by 1  means the number increase of the integral by 1.Remark 9.
Compared with traditional approach to deal with term like∫t-ht‍x˙T(s)S5x˙(s)ds, Lemma 5 provides a new handling method. This new handling method can establish the relationship among ∫t-h(t)t‍xT(s)ds, ∫t-ht-h(t)‍xT(s)ds  and x(t-h), which may significantly reduce the conservatism of stability criteria.Remark 10.
WhenW3=0, the system (5) reduces to
(47)x˙(t)=-W0x(t)+W1f(x(t))+W2f(x(t-h(t)))+W4x˙(t-h(t)).
Similarly, based on Theorem 6, we can obtain the asymptotical stability for system (47) as follows.Theorem 11.
For given scalarsh>0 and hD<1, the system (47) with the neuron activation function f(x(t)) satisfying the condition (6) is asymptotically stable if there exists P>0, RiT=Ri>0  (i=1,2,3),  QiT=Qi>0(i=1,2), SiT=Si>0(i=1,…,6),  diagonal matrices Gi=diag{gi1,gi2,…,gin}>0, Ki=diag{ki1,ki2,…,kin}>0, Li=diag{li1,li2,…,lin}>0, (i=1,2,3), and M=diag{mi1,mi2,…,min}>0,(i=1,2,3,4), Ti(i=1,…,5) and Fi(i=1,…,6) with appropriate dimensions such that the following symmetric linear matrix inequality holds:
(48)Ξ^1=[Π^0hF1ThF2ThF3T*-hR300**-Q10***-h3Q2]<0,(49)Ξ^2=[Π^0+hΠ^1hF4ThF5ThF6T*-hR300**-Q10***-h3Q2]<0,
where
(50)Π^0=〈2[e^1e^7+e^8]P[e2e^1-e4]T+2[e^20]R1[0e^8]T+2[e^20]R2[he^1e^7+e^8]T〉s+[e^1e^1]×(R1+R2)[e^1e^1]T-(1-hD)[e^1e^3]×R1[e^1e^3]T-[e^1e^4]R2[e^1e^4]T+h[e^1e^2]R3×[e^1e^2]T+e^2(h2Q1+h3Q2)e2T+〈2F1T[e^7e^3-e^4]T+4F2T[he^3-e^7]T+6F3T[he^3-e^7]T+2F4T[e^8e^1-e^3]T-4F5Te^8T-6F6Te^8TF1T[e^7e^3-e^4]T〉s+e^9(S1+S2)e^9T-(1-hD)e^10S1e^10T-e^11S2e^11T-e^6S4e^6T+e^2(S3+S4)e^2T-(1-hD)e^5S3e5T+e^2(hS5+h22S6)e^2T-2e^1S6e^1T+4he^1S6e^7T+4h2e^4S5e^7T+4h2e^4S5e^8T-e^7(2h3S5+2h2S6)e^7T-e^7(4h3S5+4h2S6)e^8T-e^8(2h3S5+2h2S6)e^8T+e^1(Σ¯K1-Σ_G1+L1Σ)e^2T+2e^2(G1-K1+L1)e^9T+2e^3(Σ¯K2-Σ_G2+L2Σ)×e^5T+2e^5(G2-K2+L2)e^10T+2e^4(Σ¯K3-Σ_G3+L3Σ)e^6T+2e^6(G3-K3+L3)×e^11T+e1(ΣM1Σ-M2Σ1)e^1T+2e^1M2Σ2e^9T-e^3M3Σ1e^3T+2e^3M3Σ2e^10T-e^9(M1+M2)e^9T-e^10M3e^10T-e^4M4Σ1e^4T+2e^4M4Σ2e^11T+e^2(-T1-T1T)e^2T-e^11M4e^11T-2e^1W0TT1Te^2T+2e^2T1W1e^9T+2e^2T1W2e^10T-2e1T2e^2T+e^1(-T2W0-W0TT2T)e^1T+2e^1T2W1e^9T+2e^1T2W2e^10T+2e^2T2W4e^5T-2e^1W0TT3Te^9T+2e^9T3W2e^10T+e^9(T3W1+W1TT3T)e^9T+2e^5W4TT3Te^9T-2e^2T4Te^10T-2e^1W0TT4Te^10T+e^9W1TT4Te^10T+e^10(T4W2+W2TT4T)e^10T+2e^5W4TT3Te^9T-2e^2T5Te^5T-2e^1W0TT5Te^5T+e^5T5W1e^9T+e^5T5W2e^10T-2he^4S5e^4T+4he^1S6e^8T+2e^5(T5W4+W4TT4T)e^5T+2e^5W4TT4Te^10T+2e^2T1W4e^5T-2e^2T3Te^9T,Π^1=〈2(e^20)R1(e^10)T-4F2Te^3T-6F3Te^3T+4F5Te^1T+6F6Te^1T(e^10)T〉s,
with
(51)〈·〉s=12[(·)+(·)T],e^i=[0n×(i-1)In×n0n×(11)]T(i=1,…,11).Remark 12.
WhenW4=0, the system (5) reduces to
(52)x˙(t)=-W0x(t)+W1f(x(t))+W2f(x(t-h(t)))+W3∫t-rt‍fT(x(s))ds.
Similarly, based on Theorem 6, we can obtain the asymptotical stability for system (52) as follows.Theorem 13.
For given scalarsh>0 and hD<1, the system (52) with the neuron activation function f(x(t)) satisfying the condition (6) is asymptotically stable if there exists P>0, RiT=Ri>0  (i=1,2,3),  QiT=Qi>0(i=1,2),  SiT=Si>0(i=1,2,5…,8),  ZiT=Zi>0(i=1,…,l), diagonal matrices G1=diag{g1,g2,…,gn}>0,  K1=diag{k1,k2,…,kn}>0,  L1=diag{l1,l2,…,ln}>0, and M=diag{mi1,mi2,…,min}>0,(i=1,2,3,4), Ti(i=1,…,4) and Fi(i=1,…,6) with appropriate dimensions such that the following symmetric linear matrix inequality holds:
(53)Ξ~1=[Π~0hF1ThF2ThF3T*-hR300**-Q10***-h3Q2]<0,(54)Ξ~2=[Π~0+hΠ~1hF4ThF5ThF6T*-hR300**-Q10***-h3Q2]<0,
where
(55)Π~0=〈2[e~1e~5+e~6]P[e~2e~1-e~4]T+2[e~20]R1[0e~6]T+2[e~20]×R2[he~1e~5+e~6]T〉s+[e~1e~1]×(R1+R2)[e~1e~1]T-(1-hD)[e~1e3]×R1[e~1e~3]T-[e~1e4]R2[e~1e~4]T+h[e~1e~2]R3×[e~1e~2]T+e~2(h2Q1+h3Q2)×e~2T+〈2F1T[e~5e~3-e~4]T+4F2T[he~3-e~5]T+6F3T[he~3-e~5]T+2F4T[e~6e~1-e~3]T-4F5Te~6T-6F6Te~6T[e~5e~3-e~4]T〉s+e~7(S1+S2)e~7T-(1-hD)e~8S1e~8T-e~9S2e~9T+e~2(hS5+h22S6)×e~2T-2e~1S6e~1T+4he~1S6e~5T+e~7W1TT4Te~8T+e~8(T4W2+W2TT4T)e~8T+4he~1×S6e~6T+4h2e~4S5e~5T+4h2e~4S5e~6T-2he~4S5e4T-e~5(2h3S5+2h2S6)e~5T-e~5(4h3S5+4h2S6)e~6T-e~6(2h3S5+2h2S6)e~6T+e~7(rS7+r22S8+∑i=1l‍(ρi-ρi-1)rZi)e~7T-∑i=1l‍1(ρi-ρi-1)re~9+iZie~9+iT-1r(e~10+⋯+e~9+l)×S7(e~10+⋯+e9+l)T-2r2e~10+lS8e~10+lT+e~1(Σ¯K1-Σ_G1+L1Σ)e~2T+2e~2(G1-K1+L1)e7T+e~1(ΣM1Σ-M2Σ1)e1T+2e1M2Σ2e7T-e3M3Σ1e3T+2e7T3W2e8T+2e~3M3Σ2e~8T-e~7(M1+M2)e~7T-e~8M3e~8T-e~4M4Σ1e~4T+2e~4M4Σ2e~9T+e~2(-T1-T1T)e~2T-e~9M4e~9T-2e~1W0TT1Te~2T+2e2T1W1e~7T+2e~2T1W2e~8T+2e~2T1W3×(e~10+⋯+e~9+l)T-2e~1×W0TT4Te~8T-2e~1T2e2T+e~1(-T2W0-W0TT2T)e~1T+2e~1T2W1e~7T+2e~1T2W2e8T-2e~2T4Te~8T+2e~1T2W3×(e~10+⋯+e~9+l)T-2e~2T3Te~7T-2e1W0TT3Te~7T+e~7(T3W1+W1TT3T)e~7T+2e~7T3W3(e~10+⋯+e9+l)T+2e~8T4W3(e~10+⋯+e~9+l)T,Π~1=〈2(e~20)R1(e~1e~0)T-4F2Te~3T-6F3Te~3T+4F5Te~1T+6F6Te~1T(e~1e~0)T〉s,
with
(56)〈·〉s=12[(·)+(·)T],e~i=[0n×(i-1)In×n0n×(10+l-i)]T(i=1,…,10+l).Remark 14.
WhenW3=0 and W4=0, the system (5) reduces to
(57)x˙(t)=-W0x(t)+W1f(x(t))+W2f(x(t-h(t))).
Similarly, based on Theorem 6, we can obtain the asymptotical stability for system (57) as follows.Theorem 15.
For given scalarsh>0 and hD<1, the system (57) with the neuron activation function f(x(t)) satisfying the condition (6) is asymptotically stable if there exists P>0, RiT=Ri>0  (i=1,2,3), QiT=Qi>0(i=1,2), SiT=Si>0(i=1,2,5,6), diagonal matrices G1=diag{gi1,g2,…,gin}>0,  K1=diag{k1,k2,…,kn}>0,  L1=diag{l1,l2,…,ln}>0,  and M=diag{mi1,mi2,…,min}>0,  (i=1,2,3,4), Ti(i=1,…,4) and Fi(i=1,…,6) with appropriate dimensions such that the following symmetric linear matrix inequality holds:
(58)Ξ˘1=[Π˘0hF1ThF2ThF3T*-hR300**-Q10***-h3Q2]<0,(59)Ξ˘2=[Π˘0+hΠ˘1hF4ThF5ThF6T*-hR300**-Q10***-h3Q2]<0,
where
(60)Π˘0=〈2[e˘1e˘5+e˘6]P[e2e˘1-e4]T+2[e˘20]R1[0e6]T+2[e˘20]R2[he˘1e˘5+e˘6]T〉s+[e˘1e^1]×(R1+R2)[e˘1e˘1]T-(1-hD)[e˘1e˘3]R1[e˘1e˘3]T-[e˘1e˘4]R2[e˘1e˘4]T+h[e˘1e˘2]R3×[e˘1e˘2]T+e˘2(h2Q1+h3Q2)e2T+〈2F1T[e˘5e˘3-e˘4]T+4F2T[he˘3-e˘5]T+6F3T[he˘3-e˘5]T+2F4T[e˘6e˘1-e˘3]T-4F5Te˘6T-6F6Te˘6T[e˘5e˘3-e˘4]T〉s+e˘7(S1+S2)e˘7T-(1-hD)e˘8S1e˘8T-e˘9S2e˘9T+e˘2(hS5+h22S6)e˘2T-2e˘1S6e˘1T+2he˘1S6e˘5T+2he˘1S6e˘6T+2h2e˘4S5e˘5T+2h2e˘4S5e˘6T-2he˘4S5e˘4T-e˘5(2h3S5+2h2S6)e˘5T-e˘5(2h3S5+2h2S6)e˘6T-e˘6(2h3S5+2h2S6)e˘6T+2e˘2(G1-K1+L1)e˘7T+e˘1(Σ¯K1-Σ_G1+L1Σ)e˘2T+e˘1(ΣM1Σ-M2Σ1)e˘1T+2e˘1M2Σ2e˘7T-e˘3M3Σ1e˘3T+2e˘3M3Σ2e^8T-e˘7(M1+M2)e˘7T-e˘8M3e˘8T-e˘4M4Σ1e˘4T+2e˘4M4Σ2e^9T+e˘2(-T1-T1T)e˘2T-e˘9M4e˘9T-2e˘1W0TT1Te˘2T+2e˘2T1W1e˘7T+2e˘2T1W2e˘8T-2e1T2e˘2T+e˘1(-T2W0-W0TT2T)e˘1T+2e˘1T2W2e˘8T-2e˘2T3Te˘7T-2e˘1W0TT3Te˘7T+2e˘7T3W2e˘8T+e˘7(T3W1+W1TT3T)e˘7T-2e˘2T4Te˘8T-2e˘1W0TT4Te˘8T+e˘7W1TT4Te˘8T+e˘8(T4W2+W2TT4T)e˘8T+2e˘1T2W1e˘7T,Π^1=〈2(e^20)R1(e^10)T-4F2Te^3T-6F3Te^3T+4F5Te^1T+6F6Te^1T(e^10)T〉s,
with
(61)〈·〉s=12[(·)+(·)T],e^i=[0n×(i-1)In×n0n×(9i)]T(i=1,…,9).


## 4. Numerical Examples

In this section, four numerical examples are given to show the effectiveness and improvement of the main results proposed in the paper.Example 1.
Consider the following neural networks of neutral type with discrete and distributed delays:(62)x˙(t)=-W0x(t)+W1f(x(t))+W2f(x(t-h(t)))+W3∫t-rt‍fT(x(s))ds+W4x˙(t-h(t)),
where
(63)W1=[-2.5573-1.38131.9574-1.1398-1.0226-0.88450.5045-0.21111.03781.55320.66451.1902-0.38960.7079-0.3398-2.2543],W2=[0.2853-0.07930.46940.5354-0.59551.3352-0.90360.5529-0.1497-0.6065-0.1641-0.2037-0.4348-1.3474-0.6275-2.2543],W3=[0.02650.11570.0578-0.09300.3186-0.1363-0.08590.07420.2037-0.20490.01120.1457-0.3161-0.2469-0.0736-2.2543],W4=[-0.30540.36820.1761-0.0235-0.0546-0.2089-0.07540.26680.45630.00230.14400.6928-0.0115-0.23490.20040.1574],W0=diag{1.6305,1.9221,2.5973,1.3775},Σ¯=diag{1.0275,0.9960,0.3223,0.2113},Σ_=diag{0,0,0,0}.From Table1, it can be seen that the stability criterion proposed in the paper is less conservative than those obtained by Feng et al. in [32] and Lakshmanan et al. in [39]. Besides, from Tables 1 and 2, the maximum allowable time-delay bounds h will become larger with the values of l becoming larger. Moreover, Figures 1 and 2 show that the state vector x(t) stabilizes to zero asymptotically with different hD and initial values. By using the Matlab LMI toolbox, we solve LMIs (13) and (14) for the case hD=0 and h=4.6531 and obtain
(64)P11=[0.0048-0.00090.0017-0.0003-0.00090.0072-0.00100.00040.0017-0.00100.0045-0.0009-0.00030.0004-0.00090.0042],R211=[0.0022-0.00120.0013-0.0004-0.00120.0043-0.00130.00060.0013-0.00130.0021-0.0007-0.00040.0006-0.00070.0015],R222=[0.0063-0.00060.0011-0.0007-0.00060.0085-0.00150.00060.0011-0.00150.0050-0.0010-0.00070.0006-0.00100.0034],R311=[0.0030-0.00040.0006-0.0005-0.00040.0041-0.00090.00040.0006-0.00090.0026-0.0006-0.00050.0004-0.00060.0018],Z1=[0.0307-0.0024-0.0035-0.0050-0.00240.01810.0044-0.0137-0.00350.00440.0136-0.0027-0.0050-0.0137-0.00270.0302],Z2=[0.0022-0.0000-0.00030.0005-0.00000.0017-0.0000-0.0001-0.0003-0.00000.00260.00100.0005-0.00010.00100.0045],S1=[0.00650.00330.0006-0.00230.00330.00580.0017-0.00310.00060.00170.0049-0.0010-0.0023-0.0031-0.00100.0073],S2=[0.0066-0.0003-0.00160.0010-0.00030.00490.0006-0.0003-0.00160.00060.00520.00110.0010-0.00030.00110.0063],T1=[0.0087-0.00390.0032-0.0014-0.00370.0144-0.00170.00210.0030-0.00160.0061-0.0009-0.00070.0012-0.00170.0092],T2=[0.0131-0.00080.0044-0.0002-0.00430.0237-0.00740.0003-0.00120.00290.0099-0.0003-0.00160.0014-0.00250.0086].Table 1
Maximum allowable time-delay boundsh=r in Example 1.
h
D
=
0
[32]
(m=l=2) [39]
(m=l=3) [39]
(l=2) Theorem 6
(l=3) Theorem 6
h
=
r
1.8630
2.7442
3.0942
4.6531
4.8652Table 2
Maximum allowable time-delay boundsh=r for different values hD in Example 1.
h
D
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Theorem6 (l=2)
4.3682
4.1238
3.9862
3.6782
3.5132
3.1230
2.8623
2.5219
1.6725
Theorem6 (l=3)
4.5132
4.2685
4.1032
3.8658
3.7216
3.3275
2.9273
2.6312
1.7321Figure 1
The simulation of Example1 for h=r=4.8652, where the initial value is [-3,-2,2,3]T, hD=0, l=3.Figure 2
The simulation of Example1 for h=r=4.5312, where the initial value is [-0.8,-0.6,0.4,0.8]T, hD=0.1, l=3.Example 2.
Using this example, we will show the improvement of our results for (52) by two cases (A) and (B). Consider the following parameters as in [32, 39]:Case A
(65)
W
0
=
[
6
0
0
0
5
0
0
0
7
]
,
W
1
=
[
1.2
-
0.8
0.6
0.5
-
1.5
0.7
-
0.8
-
1.2
-
1.4
]
,
W
2
=
[
-
1.4
0.9
0.5
-
0.6
1.2
0.8
0.5
-
0.7
1.1
]
,
W
3
=
[
-
1.8
0.7
-
0.8
0.6
1.4
1
-
0.4
-
0.6
1.2
]
,
Σ
¯
=
[
0.5
0
0
0
0.5
0
0
0
0.5
]
,
Σ
_
=
[
0
0
0
0
0
0
0
0
0
]
.Case B
(66)
W
0
=
[
0.9
0
0
0.8
]
,
W
1
=
[
1
-
1.7
-
1.6
1
]
,
W
2
=
[
1
0.6
0.5
0.8
]
,
W
3
=
[
0.4
0.3
0.1
0.2
]
,
Σ
¯
=
[
0.25
0
0
0.25
]
,
Σ
_
=
[
0
0
0
0
]
.For the above two Cases A and B of numerical examples, the maximum allowable upper bounds for guaranteeing the asymptotical stability of the corresponding systems obtained from Theorem13 are listed in Table 3. Table 3 clearly shows that our results have larger improvement over those results of previous literature. In Case A, when hD=0, h=r, the simulation result is given as Figure 3, which implies that under the given conditions, the state vector x(t) stabilizes to zero asymptotically.Table 3
Maximum allowable time-delay upper boundsh=r for hD=0 in Example 2.
h
D
=
0
[9]
[45]
[46]
[12]
[21]
[39]
Theorem13
Case A
4.3163
4.4697
4.3324
4.8374
4.4879
4.3763
4.9532
Case B
2.8266
2.9137
2.8317
2.7953
6.8279
7.2368
7.8698Figure 3
The simulation of Example2 of Case A for h=r=4.9532, where the initial value is [-0.2,0.5,-0.4]T, hD=0.Example 3.
Consider the following delayed neural network given in (47) with parameters:
(67)W0=[2002],W1=[1.60.30.30.5],W2=[0.20.10.10.2],W4=[0.15000.15],Σ¯=[1001],Σ_=[0000].The corresponding results for maximum allowable upper bounds of the time-varying delayh(t) are given in Table 4. From Table 4, it can be clearly seen that the maximum allowable time-delay bounds h will become small with the values of hD becoming large. Furthermore, when hD=0.1, the state trajectories of the system (47) are shown in Figure 4.Table 4
Maximum allowable time delay boundsh for different values hD in Example 3.
h
D
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Theorem11
9.0123
8.6789
8.4245
8.1237
7.9028
7.4532
7.1279
6.8325
6.0142Figure 4
The simulation of Example3 for h=r=9.0123, where the initial value is [-3,3]T, hD=0.1.Example 4.
In this Example, we give four Cases (A)–(C) that show the potential benefits and effectiveness of the developed method for delayed neural networks. Consider the delayed system (57) with the following parameters:Case A
(68)
W
1
=
[
-
0.0373
0.4852
-
0.3351
0.2336
-
1.6033
0.5988
-
0.3224
1.2352
0.3394
-
0.0860
-
0.3824
-
0.5785
-
0.1311
0.3253
-
0.9534
-
0.5015
]
,
W
2
=
[
0.8674
-
1.2405
-
0.5325
0.0220
0.0474
-
0.9164
0.0360
0.9816
1.8495
2.6117
-
0.3788
0.8428
-
2.0413
0.5179
1.1734
-
0.2775
]
,
W
0
=
diag
{
1.2769,0.6231,0.9230,0.4480
}
,
Σ
¯
=
diag
{
0.1137,0.1279,0.7994,0.2368
}
,
Σ
_
=
diag
{
0,0
,
0,0
}
.Case B
(69)
W
1
=
[
0.0530
0.0454
0.0987
0.2750
]
,
W
2
=
[
0.2381
0.9320
0.0388
0.5062
]
,
W
0
=
diag
{
1.5,0.7
}
,
Σ
¯
=
diag
{
0.3,0.8
}
,
Σ
_
=
diag
{
0,0
}
.Case C
(70)
W
1
=
[
1
1
-
1
-
1
]
,
W
2
=
[
0.88
1
1
1
]
,
W
0
=
{
2,2
}
,
Σ
¯
=
diag
{
0.4,0.8
}
,
Σ
_
=
diag
{
0,0
}
.From Tables5, 6, and 7, it is clearly shown that Theorem 15 is less conservative than those in the mentioned literature. In Case A, when hD=0.1, the simulation result is given as Figure 5, which implies that under the given conditions, the state vector x(t) stabilizes to zero asymptotically. Hence, this example indicates fully that the method proposed in the paper plays a major role in reducing conservatism.Table 5
Maximum allowable upper boundsh for different hD in Case A.
h
D
0.1
0.5
0.9
[36]
3.27
2.15
1.31
[37]
3.27
2.22
1.58
[38]
3.30
2.53
2.08
[47]
3.35
2.59
2.13
[48]
3.75
2.73
2.27
[49]
3.70
3.12
2.59
[50]
3.91
2.79
2.33
[42]
4.21
3.15
2.91
Theorem15
5.19
3.92
3.36Table 6
Maximum allowable time-delay upper boundsh in Case B.
h
D
=
0
0.4
0.45
0.5
0.55
[36]
3.99
3.27
3.05
2.98
[48]
4.38
3.60
3.33
3.23
[49] (m=2)
4.39
3.67
3.46
3.41
[19]
4.4801
4.0626
3.8083
3.7064
[18] (m=2)
5.2420
4.4301
4.1055
3.9231
Theorem 1 in [51]
5.0588
4.2603
4.0604
4.0185
Theorem 2 in [51]
5.3079
4.5267
4.2924
4.1903
Corollary 1 in [51]
9.7094
7.7523
6.8570
6.2977
Theorem15
10.2358
8.8378
7.9531
7.4532Table 7
Maximum allowable time-delay boundsh for different values hD in Case C.
Method
[52]
[48]
[53]
[42]
[54]
Theorem15
h
D
=
0.8
2.3571
2.8854
2.9144
3.1409
4.1626
6.9785
h
D
=
0.9
1.6050
1.9631
1.9095
1.6375
3.9766
6.3457Figure 5
The simulation of Example4 for h=5.19, where the initial value is [-0.2,0.5,-0.4,0.2]T, hD=0.1.Finally, we provide the number of decision variables involved in the LIM in each of the Theorems6–15 in Table 8.Table 8
Number of decision variables involved in the stability criteria.
Method
Number of decision variables
Method
Number of decision variables
Theorem6
[
(
12
+
l
)
2
+
8
(
12
+
l
)
+
6
]
n
2
+
(
16
+
l
)
n
Theorem11
215
n
2
+
15
n
Theorem13
[
(
10
+
l
)
2
+
8
(
10
+
l
)
+
6
]
n
2
+
(
14
+
l
)
n
Theorem15
159
n
2
+
13
n


## 5. Conclusions

This paper is concerned with the stability analysis of neutral type neural networks with mixed time-varying delays. Some improved delay-dependent stability results are established by using a novel approach for the networks. Improved delay-dependent stability criteria in terms of linear matrix inequalities (LMIs) are derived by employing a new type of Lyapunov-Krasovskii functionals with three and four integral terms. Different from previous results by using the first-order convex combination property, our derivation applies the idea of second-order convex combination and the property of quadratic convex function. obtained results are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness and the advantage of the proposed main results.



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*Source: 450175-2013-11-24.xml*