# Stability Analysis for Delayed Neural Networks: Reciprocally Convex Approach

Authors: Hongjun Yu; Xiaozhan Yang; Chunfeng Wu; Qingshuang Zeng
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with global stability analysis for a class of continuous neural networks with time-varying delay. The lower and upper bounds of the delay and the upper bound of its first derivative are assumed to be known. By introducing a novel Lyapunov-Krasovskii functional, some delay-dependent stability criteria are derived in terms of linear matrix inequality, which guarantee the considered neural networks to be globally stable. When estimating the derivative of the LKF, instead of applying Jensen’s inequality directly, a substep is taken, and a slack variable is introduced by reciprocally convex combination approach, and as a result, conservatism reduction is proved to be more obvious than the available literature. Numerical examples are given to demonstrate the effectiveness and merits of the proposed method.


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## Body

## 1. Introduction

In recent years, important application in the fields of pattern recognition, signal processing, optimization and associative memories, and so forth has made neural networks (NNs) the eye of attention. Stability analysis of NNs with time-varying delay, as a result, has received much attention ever since, because time delay is frequently encountered in NNs, owing to finite switching speed of amplifiers in the communication and response of neurons, and could cause instability and oscillations in the system. Both delay-independent and delay-dependent stability criteria have been brought up. While delay-independent criteria tend to be more conservative, more attention is given to delay-dependent criteria, because they could make use of the length of the delay.Global stability of various recurrent neural networks has been investigated in [1–4]. In stability analysis of neural networks, the qualitative properties primarily concerned are uniqueness, global asymptotic stability, and global exponential stability of their equilibria. When the system with time delay is described by the following dynamic equation:
(1)x˙(t)=-Cx(t)+Af(x(t))+Bf(x(t-h))+u,
exponential and asymptotic stability analysis has been done by [5–10]. The stability criteria in [5, 9, 10] are delay independent, while those in [6, 8, 11] are delay dependent, and [12] includes both. Also, [8–10] adopted the delay-partitioning approach. After some changes are made on system description, such as setting A=0, adding a new term D∫t-dt‍f(x(s))ds, and introducing a new activation function g(x(t-h)) to substitute for f(x(t-h)), asymptotic stability criteria are derived in [13–17], and uniqueness analysis is done in [5, 18].Various methods have been proposed to reduce conservatism when deriving stability criteria. For example, the free-weighting matrix approach noticed in [6, 7, 19–22] is proved to be very effective since bounding techniques on some cross-product terms are avoided. Stability analyses of NNs with multiple and single time-varying delays are done by [17, 23], respectively. Moreover, a new free-weighting matrix approach is brought up in [24] to estimate the derivative of Lyapunov functional without missing any negative quadratic terms, and thus, improved delay-dependent stability criteria are established. Along with free-weighting matrix approach, [14, 25, 26] adopted the delay partitioning idea to solve delay-dependent stability problem, and the proposed methods have significantly reduced conservatism.When deriving stability criteria for delayed systems, two kinds of approaches are usually used, namely, Lyapunov function approaches [27–29] and Lyapunov-Krasovskii functional (LKF) approaches [17, 23, 24, 30–33]. The former makes no restriction on the derivative of time delay and usually gives a simpler stability criterion or delay-independent criterion while the latter, expressed in the form of LMI, takes the derivative of time delay into account, gives a delay-dependent criterion, and thus can be less conservative since LKF makes use of more information about the system. Discretized LKF method developed in [34] is another method for stability analysis, and [13] made some necessary adjustments to make the method compatible with robust stability problem for NNs with uncertain delays.In addition, the range of time-varying delay for NNs is mostly considered to have a lower bound of zero, as seen in [7, 11, 23, 24], while in practice it may not be restricted to 0, and setting h1 to zero would result in increased conservatism. It is the same with the derivative of time delay. In many papers, time delay of NNs is either constant or unknown, as noticed in [14, 25, 26], while an upper or lower bound could be assumed.In this paper, the stability problem for continuous NNs with time-varying delay is taken into consideration. A novel LKF is brought up, and changes are made to deal with different cases concerning time delay and its derivative. When estimating the derivative of LKF, instead of applying Jensen’s inequality directly, a substep is taken, and a slack variable is introduced, and consequently, conservatism reduction is proved to be more obvious than existing results. Numerical examples are given, and analysis is made to demonstrate the effectiveness and merits of the proposed method.In Section1, a brief introduction is presented, and some notations are defined. In Section 2, the stability problem is formulated, and some preliminaries are given. In Section 3, new criteria in the form of one theorem and three corollaries for NNs with time-varying delay are presented. In Section 4, numerical examples are presented, along with results from the other literature. The paper is concluded in Section 5.Notations. The notations used throughout the paper are standard. The superscript “T” stands for matrix transposition; Rn denotes the n-dimensional Euclidean space; the notation P>0 means that P is a real positive definite; I and 0 represent the identity matrix and a zero matrix, respectively; diag(⋯) stands for a block-diagonal matrix; λmin(P)(λmax(P)) denotes the minimum (maximum) eigenvalue of symmetric matrix P; ∥·∥ denotes the Euclidean norm of a vector and its induced norm of a matrix. In symmetric block matrices, we use an asterisk (*) to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Model Descriptions and Preliminaries

The dynamic behavior of a continuous-time neural networks with time delay can be described by the following state equation:(2)dy(t)dt=-Cy(t)+Ag(y(t))+Bg(y(t-h(t)))+u,
where y(t) is the state vector of the neural network; C is a positive matrix; A and B are the connection weight and the delayed connection weight matrices, respectively. g(y(t)) represents the activation function vector of neurons, and u is a constant external bias vector. h(t) denotes axonal signal transmission delay, which is nonnegative, bounded, and has 0<h1<h<h2, 0≤h˙≤γ, and will be written as h for short throughout the paper. The initial conditions associated with system (2) are of the form
(3)yi(θ)=φi(θ),θ∈[-h2,0],i=1,2,…,n,
where φi(θ)∈C([-h2,0],Rn) is the Banach space of continuous functions mapping interval [-h2,0] into Rn.The following assumptions are made on system (2) throughout this paper.(H1)
The activation functionsg(y(t)) are bounded and monotonically nondecreasing on R.(H2)
The activation functionsg(y(t)) satisfy
(4)lm,i≤gi(ω1)-gi(ω2)ω1-ω2≤lp,i,ω1≠ω2.It is known that bounded activation functions always guarantee the existence of an equilibrium point for model (2). For convenience of exposition, in the following, we will shift the equilibrium point y* of model (2) to the origin. The transformation x(t)=y(t)-y* puts system (2) into the following form:
(5)dx(t)dt=-Cx(t)+Af(x(t))+Bf(x(t-h)),
where x(t) is the state vector of the transformed system, and fi(xi)=gi(xi+yi*)-gi(yi*) with fi(0)=0, xi(θ)=ϕi(θ), θ∈[-h2,0], (i=1,…,n). Obviously, the equilibrium point y* of system (2) is globally stable if and only if the origin of system (5) is globally stable. Assume that fi(0)=0, then the functions fi(·) satisfy
(6)fi(ωi)≤lp,iωi,fi(ωi)≥lm,iωi,∀ωi∈R.
Rewriting (6), we can get
(7)Lpω-f(ω)≥0,f(ω)-Lmω≥0,[Lpω-f(ω)]TS[f(ω)-Lmω]≥0,S≥0,
where Lp≜diag(lp,1,lp,2,…,lp,n) and Lm≜diag(lm,1,lm,2,…,lm,n).Lemma 1 (see [35]).
For positive definiteM∈Rn×n, scalars r1 and r2, and a vector function w∈Rn, the following inequality holds:
(8)(∫r1r2‍wT(s)ds)M(∫r1r2‍w(s)ds)≤(r2-r1)∫r1r2‍wT(s)Mw(s)ds.


## 3. Main Results

In this section, some delay-dependent sufficient conditions of the global stability for the neural networks with time-varying delay in (5) are derived. First, we consider the case where the upper bound of h˙ is known, and correspondingly, the global stability condition is given as follows.Theorem 2.
Suppose that in reference system (5), the time delay h satisfies 0<h1<h<h2, 0≤h˙≤γ. Under the condition given in (6), if there exist matrices P>0, Q1>0, Q2>0, R1>0, R2>0, K1>0, K2>0, Si>0,i=1,…,4, S1≜diag(s1,1,s1,2,…,s1,n), S2≜diag(s2,1,s2,2,…,s2,n), and T such that
(9)Θ≜Υ1+Υ2+Υ3+Υ4<0,Ψ≜[R2T*R2]≥0,
where Υ1, Υ2, Υ3, and Υ4 are defined as
(10)Υ1≜DxTPDx˙+Dx˙TPDx+DxTQ1Dx-Dxh1TQ1Dxh1+DxTQ2Dx-Dxh2TQ2Dxh2,Υ2≜DxTS3Dx-(1-γ)DxhTS3Dxh+DfTS4Df-(1-γ)DfhTS4Dfh+2DfTS1Dx˙-2DxTLmTS1Dx˙-2DfTS2Dx˙+2DxTLpTS2Dx˙,Υ3≜DfhTK2LpDxh-DfhTK2Dfh-DxhTLmTK2LpDxh+DxhTLmTK2Dfh+DfTK1LpDx-DfTK1Df-DxTLmTK1LpDx+DxTLmTK1Df,Υ4≜h22Dx˙TR1Dx˙-(Dx-Dxh2)TR1(Dx-Dxh2)+(h2-h1)2Dx˙TR2Dx˙-(Dxh-Dxh1)TR2(Dxh-Dxh1)-(Dxh2-Dxh)TR2(Dxh2-Dxh)-(Dxh-Dxh2)TTT(Dxh1-Dxh)-(Dxh1-Dxh)TT(Dxh-Dxh2),
where Dx˙, Dx, Dxh, Dxh1, Dxh2, Df, and Dfh are defined as
(11)Dfh≜[00000I],Dx≜[I00000],Dxh≜[0I0000],Dxh1≜[00I000],Dxh2≜[000I00],Df≜[0000I0],Dx˙≜[-C000AB],
then system (5) is globally stable. Moreover,
(12)∥x(t)∥≤Δλmin(P)∥ϕ∥,
where Δ is defined as
(13)Δ≜2(λmax(S2Lp)-λmin(S2Lm))+3[h23λmax(R1)+(h2-h1)3λmax(R2)]×(λmax(CTC)+λmax(ATA)λmax(Lp2)a+λmax(BTB)λmax(Lp2))+λmax(P)+h2λmax(S3)+h2λmax(S4)λmax(Lp2)+h1λmax(Q1)+h2λmax(Q2)+2(λmax(S1Lp)-λmin(S1Lm)).Proof.
We choose an LKF as(14)V(t)≜V1(t)+V2(t)+V3(t),
where
(15)V1(t)≜xT(t)Px(t)+2∑i=1n‍s1,i∫0xi(t)‍(fi(s)-lm,ixi)ds+2∑i=1n‍s2,i∫0xi(t)‍(lp,ixi-fi(s))ds,V2(t)≜∫t-ht‍[xT(w)S3x(w)+fT(x(w))S4f(x(w))]dw+∫t-h1t‍xT(w)Q1x(w)dw+∫t-h2t‍xT(w)Q2x(w)dw,V3(t)≜h2∫-h20‍∫t+θt‍x˙T(w)R1x˙(w)dwdθ+(h2-h1)∫-h2-h1‍∫t+θt‍x˙T(w)R2x˙(w)dwdθ,Z(w)≜[xT(w)xT(w-h)xT(w-h1)xT(w-h2)fT(x(w))fT(x(w-h))]T.
The derivatives of Vi(t), i=1,2,3, are given, respectively, by
(16)V˙1(t)=xT(t)Px˙(t)+x˙T(t)Px(t)+2[S1(f(x)-Lmx(t))]Tx˙(t)+2[S2(Lpx(t)-f(x))]Tx˙(t)=ZT(t)(DxTPDx˙+Dx˙TPDx+2DfTS1Dx˙ZT(t)a-2DxTLmTS1Dx˙+2DxTLpTS2Dx˙ZT(t)a-2DfTS2Dx˙)Z(t),V˙2(t)≤xT(t)S3x(t)+fT(x(t))S4f(x(t))+xT(t)Q1x(t)+xT(t)Q2x(t)-(1-γ)fT(x(t-h))S4f(x(t-h))-(1-γ)xT(t-h)S3x(t-h)-xT(t-h1)Q1x(t-h1)-xT(t-h2)Q2x(t-h2)=ZT(t)[DxTS3Dx-(1-γ)DxhTS3Dxh+DfTS4Df-(1-γ)DfhTS4Dfh+DxTQ1Dx-Dxh1TQ1Dxh1+DxTQ2Dx-Dxh2TQ2Dxh2]Z(t),V˙3(t)=(h2-h1)2x˙T(t)R2x˙(t)-(h2-h1)∫t-h2t-h1‍x˙T(w)R1x˙(w)dw+h22x˙T(t)R1x˙(t)-h2∫t-h2t‍x˙T(w)R1x˙(w)dw.
By Lemma 1, we can get
(17)V˙3(t)≤h22x˙T(t)R1x˙(t)+(h2-h1)2x˙T(t)R2x˙(t)-(x(t)-x(t-h2))TR1(x(t)-x(t-h2))-(h2-h1)(∫t-h2t-h‍x˙T(w)R2x˙(w)dw(h2as-ah1)+∫t-h2t-h‍x˙T(w)R2x˙(w)dw)≤h22x˙T(t)R1x˙(t)+(h2-h1)2x˙T(t)R2x˙(t)-(x(t)-x(t-h2))TR1(x(t)-x(t-h2))-[h2-h1h2-h(x(t-h)-x(t-h2))Tasa×R2(x(t-h)-x(t-h2))asa+h2-h1h-h1(x(t-h1)-x(t-h))Taas×R2(x(t-h1)-x(t-h))h2-h1h2-h]=h22x˙T(t)R1x˙(t)+(h2-h1)2x˙T(t)R2x˙(t)-(x(t)-x(t-h2))TR1(x(t)-x(t-h2))-h-h1h2-h(x(t-h)-x(t-h2))T×R2(x(t-h)-x(t-h2))-(x(t-h)-x(t-h2))T×R2(x(t-h)-x(t-h2))-h2-hh-h1(x(t-h1)-x(t-h))T×R2(x(t-h1)-x(t-h))-(x(t-h1)-x(t-h))T×R2(x(t-h1)-x(t-h))=h22x˙T(t)R1x˙(t)+(h2-h1)2x˙T(t)R2x˙(t)-(x(t)-x(t-h2))TR1(x(t)-x(t-h2))-[h2-hh-h1(x(t-h1)-x(t-h))-h-h1h2-h(x(t-h)-x(t-h2))]T[R2T*R2]×[h2-hh-h1(x(t-h1)-x(t-h))-h-h1h2-h(x(t-h)-x(t-h2))]-[x(t-h1)-x(t-h)x(t-h)-x(t-h2)]T[R2T*R2]×[x(t-h1)-x(t-h)x(t-h)-x(t-h2)]≤h22x˙T(t)R1x˙(t)+(h2-h1)2x˙T(t)R2x˙(t)-(x(t)-x(t-h2))TR1(x(t)-x(t-h2))-[x(t-h1)-x(t-h)x(t-h)-x(t-h2)]T[R2T*R2]×[x(t-h1)-x(t-h)x(t-h)-x(t-h2)]≤ZT(t)[[Dxh1-DxhDxh-Dxh2]h22Dx˙TR1Dx˙+(h2-h1)2Dx˙TR2Dx˙ZTsa(t)-(Dx-Dxh2)TR1(Dx-Dxh2)ZTsa(t)-[Dxh1-DxhDxh-Dxh2]T[R2T*R2]ZT(t)ss×[Dxh1-DxhDxh-Dxh2]]Z(t).
By (16) and (17), we have
(18)V˙(t)≤ZT(t)[[Dxh1-DxhDxh-Dxh2]DxTPDx˙+Dx˙TPDx+2DfTS1Dx˙-DxTLmTS1Dx˙+2DxTLpTS2Dx˙-DfTS2Dx˙+DxTS3Dx-(1-γ)DxhTS3Dxh+DfTS4Df-(1-γ)DfhTS4Dfh+DxTQ1Dx-Dxh1TQ1Dxh1+DxTQ2Dx-Dxh2TQ2Dxh2+h22Dx˙TR1Dx˙+(h2-h1)2Dx˙TR2Dx˙-(Dx-Dxh2)TR1(Dx-Dxh2)-[Dxh1-DxhDxh-Dxh2]T[R2T*R2]×[Dxh1-DxhDxh-Dxh2]]Z(t)=ZT(t)ΩZ(t),
where Ω is defined as
(19)Ω≜DxTPDx˙+Dx˙TPDx+2DfTS1Dx˙-DxTLmTS1Dx˙+2DxTLpTS2Dx˙-DfTS2Dx˙+h22Dx˙TR1Dx˙+(h2-h1)2Dx˙TR2Dx˙-(Dx-Dxh2)TR1(Dx-Dxh2)+DxTS3Dx-(1-γ)DxhTS3Dxh+DfTS4Df-(1-γ)DfhTS4Dfh+DxTQ1Dx-Dxh1TQ1Dxh1+DxTQ2Dx-Dxh2TQ2Dxh2-[Dxh1-DxhDxh-Dxh2]T[R2T*R2][Dxh1-DxhDxh-Dxh2].
Then by (7), (18), and (19), we have
(20)V˙(t)≤ZT(t)ΩZ(t)+(ZT(t)(LpDx-Df)T×S1(Df-LmDx)Z(t))+ZT(t)(LpDxh-Dfh)TS2(Dfh-LmDxh)Z(t)=ZT(t)ΘZ(t).
It is clear that if (9) and (20) hold, then, for any Z(t)≠0, we have V˙(t)<0. It follows that
(21)V(t)≤V(0).
From (15), we get
(22)V(0)=xT(0)Px(0)+2∑i=1n‍∫0xi(0)‍[(Lp,ixi-fi(w))s1,i(fi(w)-Lm,ixi)xT(0)Px(0)+s2∑i=1n‍∫0xi(0)‍+s2,i(Lp,ixi-fi(w))]dw+h2∫-h20‍∫θ0‍x˙T(w)R1x˙(w)dwdθ+(h2-h1)∫-h1-h2‍∫θ0‍x˙T(w)R2x˙(w)dwdθ+∫-h10‍[xT(w)Q1x(w)]dw+∫-h20‍[xT(w)Q2x(w)]dw+∫-h0‍[xT(w)S3x(w)+fT(x(w))S4f(x(w))]dw≤[λmax(P)+h2λmax(S3)+h2λmax(S4)λmax(Lp2)+h1λmax(Q1)+h2λmax(Q2)+2(λmax(S1Lp)-λmin(S1Lm)+λmax(S2Lp)-λmin(S2Lm))]∥ϕ∥2+λmax(R2)(h1-h2)∫-h1-h2‍∫θ0‍x˙T(w)x˙(w)dwdθ+λmax(R1)h2∫-h20‍∫θ0‍x˙T(w)x˙(w)dwdθ.
By a similar method in [9], we have
(23)x˙T(w)x˙(w)≤3[λmax(CTC)+λmax(ATA)λmax(Lp2)aa+λmax(BTB)λmax(Lp2)]∥ϕ∥2.
Thus, V(0)≤Δ∥ϕ∥2, where Δ is defined in (13).
Moreover,(24)V(t)≥xT(t)Px(t)≥λmin(P)∥x(t)∥2.
Therefore, we have(25)∥x(t)∥≤Δλmin(P)∥ϕ∥,
which shows that system (5) is globally stable. This completes the proof.Remark 3.
Theorem2 presents a stability criterion for the delayed neural network. When coping with (h2-h1)∫-h2-h1‍∫t+θt‍x˙T(w)R2x˙(w)dwdθ, instead of using Jensen’s inequality directly, we use a substep which can make the method less conservative, which can be noticed as reciprocally convex combination approach in [36]. It follows that
(26)-h2-h1h2-h-h2-h1h2-has×(x(t-h)-x(t-h2)))-h2-h1h-h1((x(t-h1)-x(t-h))TR2-h2-h1h-h1as×(x(t-h1)-x(t-h)))=-h-h1h2-h((x(t-h)-x(t-h2))TR2-h-h1h2-has×(x(t-h)-x(t-h2)))-((x(t-h)-x(t-h2))TR2asa×(x(t-h)-x(t-h2)))-h2-hh-h1((x(t-h1)-x(t-h))TR2-h2-hh-h1as×(x(t-h1)-x(t-h)))-((x(t-h1)-x(t-h))TR2asa×(x(t-h1)-x(t-h)))=-[h2-hh-h1(x(t-h1)-x(t-h))-h-h1h2-h(x(t-h)-x(t-h2))]T[R2T*R2]×[h2-hh-h1(x(t-h1)-x(t-h))-h-h1h2-h(x(t-h)-x(t-h2))]-[x(t-h1)-x(t-h)x(t-h)-x(t-h2)]T[R2T*R2]×[x(t-h1)-x(t-h)x(t-h)-x(t-h2)]≤-[x(t-h1)-x(t-h)x(t-h)-x(t-h2)]T[R2T*R2]×[x(t-h1)-x(t-h)x(t-h)-x(t-h2)].
If no substep is taken, it will follow that
(27)-(h2-h1)∫t-h2t-h1‍x˙T(w)R2x˙(w)dw≤-[x(t-h1)-x(t-h)+x(t-h)asa-x(t-h2)]TR2a×[x(t-h1)-x(t-h)+x(t-h)as-x(t-h2)]=-[x(t-h1)-x(t-h)x(t-h)-x(t-h2)]T[R2R2R2R2]×[x(t-h1)-x(t-h)x(t-h)-x(t-h2)].We can see that compared to (27), (26) is relatively free since T could be more than nonnegative, and consequently, its LMI could suffer bigger delay. For the same reason, if the middle term could be found between
(28)-[h2-hh-h1(x(t-h1)-x(t-h))-h-h1h2-h(x(t-h)-x(t-h2))]T[R2T*R2]asas×[h2-hh-h1(x(t-h1)-x(t-h))-h-h1h2-h(x(t-h)-x(t-h2))]
and 0, conservatism would be further reduced.Remark 4.
Based on Theorem2, we can determine the maximum admissible delay h1 and h2 at a known upper bound of h˙. Moreover, the relationship between f(x) and x could be specified using Lm and Lp. As to the case when h˙ is unknown, we can refer to Corollary 5, where some changes are made on the LKF.Corollary 5.
Suppose that the time delayh in reference system (5) satisfies 0<h1<h<h2. Under the condition given by (6) and (7), if there exist matrices P>0, Q1>0, Q2>0, R1>0, R2>0, K1>0, K2>0, S1≜diag(s1,1,s1,2,…,s1,n)>0, S2≜diag(s2,1,s2,2,…,s2,n)>0, and T such that the following matrix inequalities hold:
(29)Θ~≜Υ~1+Υ~2+Υ~3+Υ~4<0,Ψ~≜[R2T*R2]≥0,
where Υ~1, Υ~2, Υ~3, and Υ~4 are defined as
(30)Υ~1≜DxTPDx˙+Dx˙TPDx+DxTQ1Dxasas-Dxh1TQ1Dxh1+DxTQ2Dx-Dxh2TQ2Dxh2,Υ~2≜2DfTS1Dx˙-2DxTLmTS1Dx˙asas-2DfTS2Dx˙+2DxTLpTS2Dx˙,Υ~3≜DfhTK2LpDxh-DfhTK2Dfh-DxhTLmTK2LpDxhasas+DxhTLmTK2Dfh+DfTK1LpDx-DfTK1Dfasas-DxTLmTK1LpDx+DxTLmTK1Df,Υ~4≜h22Dx˙TR1Dx˙-(Dx-Dxh2)TR1(Dx-Dxh2)asas+(h2-h1)2Dx˙TR2Dx˙asas-(Dxh-Dxh1)TR2(Dxh-Dxh1)asas-(Dxh2-Dxh)TR2(Dxh2-Dxh)asas-(Dxh-Dxh2)TTT(Dxh1-Dxh)asas-(Dxh1-Dxh)TT(Dxh-Dxh2),
where Dx˙, Dx, Dxh, Dxh1, Dxh2, Df, and Dfh are defined in (11), then system (5) is globally stable. Moreover,
(31)∥x(t)∥≤Δ~λmin(P)∥ϕ∥,
where Δ~ is defined as
(32)Δ~≜λmax(P)+h1λmax(Q1)+h2λmax(Q2)+2(λmax(S1Lp)-λmin(S1Lm)AS+λmax(S2Lp)-λmin(S2Lm))+3[h23λmax(R1)+(h2-h1)3λmax(R2)]×[λmax(CTC)+λmax(ATA)λmax(Lp2)AS+λmax(BTB)λmax(Lp2)].Proof.
We choose an LKF as(33)V~(t)≜V~1(t)+V~2(t)+V~3(t),
where
(34)V~1(t)≜xT(t)Px(t)+2∑i=1n‍s1,i∫0xi(t)‍(fi(s)-lm,ixi)dsasasas+2∑i=1n‍s2,i∫0xi(t)‍(lp,ixi-fi(s))ds,V~2(t)≜∫t-h1t‍xT(w)Q1x(w)dwasasas+∫t-h2t‍xT(w)Q2x(w)dw,V~3(t)≜h2∫-h20‍∫t+θt‍x˙T(w)R1x˙(w)dwdθasasas+(h2-h1)∫-h2-h1‍∫t+θt‍x˙T(w)R2x˙(w)dwdθ.
Since the result can be obtained directly from Theorem2, the rest of the proof for Corollary 5 is omitted.Remark 6.
Corollary5 presents the stability criterion when the upper bound of h˙ is unknown. Since γ is unknown and under current conditions, it cannot be estimated or substituted, the first term in V~2(t) of ∫-h0‍[xT(w)S3x(w)+fT(x(w))S4f(x(w))]dw should be changed or eliminated from the LKF. In Corollary 5, the term is eliminated because the other two terms in V~2(t) can serve the same function, and it is unnecessary to keep any extra x(t-h) or f(x(t-h)). The rest of the terms were reserved because they will not generate any h˙-related terms when estimating the derivative of the LKF.Corollary 7.
Suppose that the time delayh in reference system (5) satisfies 0<h<h2 and 0≤h˙≤γ. Under the condition given by (6) and (7), if there exist matrices P>0, Q2>0, R1>0, R2>0, K1>0,K2>0, Si>0,i=1,…,4, S1≜diag(s1,1,s1,2,…,s1,n), S2≜diag(s2,1,s2,2,…,s2,n), and T such that the following matrix inequalities hold:
(35)Θ^≜Υ^1+Υ^2+Υ^3+Υ^4<0,Ψ^≜[R2T*R2]≥0,
where Υ^1, Υ^2, Υ^3, and Υ^4 are defined as
(36)Υ^1≜D^xTPD^x˙+D^x˙TPD^x+D^xTQ2D^x-D^xh2TQ2D^xh2,Υ^2≜D^xTS3D^x-(1-γ)D^xhTS3D^xh+D^fTS4D^fasas-(1-γ)D^fhTS4D^fh+2D^fTS1D^x˙-2D^xTLmTS1D^x˙asas-2D^fTS2D^x˙+2D^xTLpTS2D^x˙,Υ^3≜D^fhTK2LpD^xh-D^fhTK2D^fh-D^xhTLmTK2LpD^xhasas+D^xhTLmTK2D^fh+D^fTK1LpD^x-D^fTK1D^fasas-D^xTLmTK1LpD^x+D^xTLmTK1D^f,Υ^4≜h22D^x˙TR1D^x˙-(D^x-D^xh2)TR1(D^x-D^xh2)asas+h22D^x˙TR2D^x˙-(D^xh-D^x)TR2(D^xh-D^x)asas-(D^xh2-D^xh)TR2(D^xh2-D^xh)asas-(D^xh-D^xh2)TTT(D^x-D^xh)asas-(D^x-D^xh)TT(D^xh-D^xh2),
where D^x˙, D^x, D^xh, D^xh2, D^f, and D^fh are defined as
(37)D^x˙≜[-C00AB],D^x≜[I0000],D^xh≜[0I000],D^xh2≜[00I00],D^f≜[000I0],D^fh≜[0000I],
then system (5) is globally stable. Moreover,
(38)∥x(t)∥≤Δ^λmin(P)∥ϕ∥,
where Δ^ is defined as
(39)Δ^≜λmax(P)+h2λmax(S3)+h2λmax(S4)λmax(Lp2)+h2λmax(Q2)+2(λmax(S1Lp)-λmin(S1Lm)+λmax(S2Lp)-λmin(S2Lm))+3h23λmax(R1+R2)(λmax(CTC)asas3h23λmax(R1+R2)+λmax(ATA)λmax(Lp2)asas3h23λmax(R1+R2)+λmax(BTB)λmax(Lp2)).Proof.
We choose an LKF as(40)V^(t)≜V^1(t)+V^2(t)+V^3(t),
where
(41)V^1(t)≜xT(t)Px(t)+2∑i=1n‍s1,i∫0xi(t)‍(fi(s)-lm,ixi)dsasasas+2∑i=1n‍s2,i∫0xi(t)‍(lp,ixi-fi(s))ds,V^2(t)≜∫t-ht‍[x(w)TS3x(w)+fT(x(w))S4f(x(w))]dwasasas+∫t-h2t‍xT(w)Q2x(w)dw,V^3(t)≜h2∫-h20‍∫t+θt‍x˙T(w)(R1+R2)x˙(w)dwdθ.
Since the result can be obtained directly from Theorem2, the rest of the proof for Corollary 7 is omitted.Remark 8.
Corollary7 presents the stability criterion when the lower bound h1 is zero. If h1 is zero, the second term in V^2(t) of ∫t-h1t‍xT(w)Q1x(w)dw should be changed or eliminated from the LKF. In Corollary 7, the term is eliminated because there is no need to introduce an extra variable Q1, while P and other matrices can serve the same function. Moreover, x˙T(w)R1x˙(w)dwdθ and (h2-h1) ∫-h2-h1‍∫t+θt‍x˙T(w)R2x˙(w)dwdθ in V^3(t) can be merged because when h1 is zero, they have the same form. But R1 and R2 can be reserved, because when estimating the upper bound of the LKF, it is still useful to introduce a T like Theorem 2.Corollary 9.
Suppose that the time delayh in reference system (5) satisfies 0<h<h2. Under the condition given by (6) and (7), if there exist matrices P>0, Q2>0, R1>0, R2>0, K1>0, K2>0, Si>0,i=1,2, Si>0,i=1,…,4, S1≜diag(s1,1,s1,2,…,s1,n), S2≜diag(s2,1,s2,2,…,s2,n), and T such that the following matrix inequalities hold:
(42)Θ¯≜Υ¯1+Υ¯2+Υ¯3+Υ¯4<0,Ψ¯≜[R2T*R2]≥0,
where Υ¯1, Υ¯2, Υ¯3, and Υ¯4 are defined as
(43)Υ¯1≜D^xTPD^x˙+D^x˙TPD^x+D^xTQ2D^x-D^xh2TQ2D^xh2,Υ¯2≜2D^fTS1D^x˙-2D^xTLmTS1D^x˙asas-2D^fTS2D^x˙+2D^xTLpTS2D^x˙,Υ¯3≜D^fhTK2LpD^xh-D^fhTK2D^fh-D^xhTLmTK2LpD^xhasas+D^xhTLmTK2D^fh+D^fTK1LpD^x-D^fTK1D^fasas-D^xTLmTK1LpD^x+D^xTLmTK1D^f,Υ¯4≜h22D^x˙TR1D^x˙-(D^x-D^xh2)TR1(D^x-D^xh2)asas+h22D^x˙TR2D^x˙-(D^xh-D^x)TR2(D^xh-D^x)asas-(D^xh2-D^xh)TR2(D^xh2-D^xh)asas-(D^xh-D^xh2)TTT(D^x-D^xh)asas-(D^x-D^xh)TT(D^xh-D^xh2),
where D^x˙, D^x, D^xh, D^xh2, D^f, and D^fh are defined in (16), then system (5) is globally stable. Moreover,
(44)∥x(t)∥≤Δ¯λmin(P)∥ϕ∥,
where Δ¯ is defined as
(45)Δ¯≜2(λmax(S1Lp)-λmin(S1Lm)+λmax(S2Lp)-λmin(S2Lm))+λmax(P)+h2λmax(Q2)+3h23λmax(R1+R2)(λmax(CTC)+3h23λmax(R1+R2)+λmax(ATA)λmax(Lp2)+3h23λmax(R1+R2)+λmax(BTB)λmax(Lp2)).Proof.
We choose an LKF as(46)V¯(t)≜V¯1(t)+V¯2(t)+V¯3(t),
where
(47)V¯1(t)≜xT(t)Px(t)+2∑i=1n‍s1,i∫0xi(t)(fi(s)-lm,ixi)dsasasas+2∑i=1n‍s2,i∫0xi(t)‍(lp,ixi-fi(s))ds,V¯2(t)≜∫t-h2t‍xT(w)Q2x(w)dw,V¯3(t)≜h2∫-h20‍∫t+θt‍x˙T(w)(R1+R2)x˙(w)dwdθ.
Since the result can be obtained directly from Theorem2, the rest of the proof for Corollary 9 is omitted.Remark 10.
Corollary9 presents the stability criterion when h1 is zero and the upper bound of time delay’s derivative, or γ, is unknown. If h1 is zero and γ is unknown, the first and second terms of ∫-h0‍[xT(w)S3x(w)+fT(x(w))S4f(x(w))]dw and ∫t-h1t‍xT(w)Q1x(w)dw in V¯2(t) should be eliminated from the LKF. There is no need to introduce an extra variable Q1 or keep any x(t-h) or f(x(t-h)), and the reason is the same with Corollaries 5 and 7. Moreover, the terms in V¯3(t) can be merged, while R1 and R2 should be reserved for the same reason with Corollary 7.


## 4. Numerical Examples

In this section, examples are provided to demonstrate the advantages of the proposed stability criteria.Example 11.
Consider the delayed neural network in (5) with the following parameters, which has also been investigated by [13, 39]:
(48)C=[2002],A=[11-1-1],B=[0.88111],Lp=[0.4000.8],Lm=[0000].
Our objective is to find the allowable maximum time delay h2 such that the system is stable under different h1 and h˙. The simulation results from the available literature are shown in Table 1, along with results from Theorem 2 and Corollaries 5, 7, and 9. It is clear that the conservatism reduction proves to be more obvious than those in [23, 24, 37, 38].Table 1
Allowable maximum time delayh2 under different h1 and γ in this paper along with results from other papers for comparison.
Methods
h
1
γ
=
0.8
γ
=
0.9
Unknownγ
[37]
h
1
=
0
h
2
=
2.3534
h
2
=
1.605
h
2
=
1.5103
[37]
h
1
=
1
h
2
=
3.2575
h
2
=
2.4769
h
2
=
2.3606
[37]
h
1
=
2
h
2
=
4.2552
h
2
=
3.4769
h
2
=
3.3606
[23]
h
1
=
0
h
2
=
1.2281
h
2
=
0.8636
h
2
=
0.8298
[24]
h
1
=
0
h
2
=
1.6831
h
2
=
0.1493
h
2
=
1.088
[38]
h
1
=
0
h
2
=
2.8854
h
2
=
1.9631
h
2
=
1.781
h
1
=
0
h
2
=
5.088(Corollary 7)
h
2
=
4.881(Corollary 7)
h
2
=
5.023(Corollary 9)
h
1
=
1
h
2
=
4.751(Theorem 2)
h
2
=
4.066(Theorem 2)
h
2
=
5.22(Corollary 5)
h
1
=
2
h
2
=
4.689(Theorem 2)
h
2
=
4.377(Theorem 2)
h
2
=
4.788(Corollary 5)In addition, whenC, A, B, and Lp take the same values as (48), and Lm takes a different value as
(49)Lm=[0.1000.2],
results under the same system with (49) are shown in Table 2. Since Lm is specified in (49), allowable maximum time delay h2 is expected to be different from those of (48). As shown in Table 2, with h1 getting bigger, the difference between h2 and h1 becomes smaller, to ensure stability of (49). Moreover, allowable maximum h2 of (49) is apparently smaller than their counterparts of (48) because Lm in (49) is a positive definite, while its counterpart in (48) is zero, which means that f(x(t)) in (49) is more closely related to x(t) than (48).Table 2
Allowable maximum time delayh2 of (49) under different h1 and γ.
Theorem2
γ
=
0.3
γ
=
0.6
γ
=
0.9
h
1
=
2
h
2
=
3.836
h
2
=
3.664
h
2
=
4.135
h
1
=
2.5
h
2
=
4.152
h
2
=
4.21
h
2
=
3.904
h
1
=
3
h
2
=
3.355
h
2
=
3.687
h
2
=
3.908Example 12.
Consider the delayed system in (5) with
(50)C=[1-1.50-0.6-0.210-11],A=[0.20.10.1-0.10.1000.1-0.5],Lp=[100010001],B=[-0.040.20.038-0.030.10.0240.01-0.1-0.014],Lm=[0.10000.10000.1].
This example is used to demonstrate the effectiveness of Theorem2 in Table 3 and Corollaries 5, 7, and 9 in Table 4. Both Lp and Lm have taken positive definite values, and allowable maximum time delay h2 under different h1 and γ is presented in Tables 3 and 4.Table 3
Allowable maximum time delayh2 of (50) under different h1 and γ.
Theorem2
γ
=
0
γ
=
0.3
γ
=
0.5
γ
=
0.8
γ
=
1
h
1
=
0.1
h
2
=
1.064
h
2
=
1.019
h
2
=
1.044
h
2
=
1.075
h
2
=
1.062
h
1
=
0.2
h
2
=
0.992
h
2
=
1.014
h
2
=
1.021
h
2
=
1.044
h
2
=
1.076
h
1
=
0.3
h
2
=
0.987
h
2
=
0.988
h
2
=
1.008
h
2
=
1.0151
h
2
=
1.0171
h
1
=
0.9
h
2
=
0.995
h
2
=
1.002
h
2
=
1.008
h
2
=
1.015
h
2
=
1.011
Theorem2
γ
=
1.2
γ
=
1.5
γ
=
1.8
γ
=
2
γ
=
5
h
1
=
0.1
h
2
=
1.072
h
2
=
1.0811
h
2
=
1.103
h
2
=
1.101
h
2
=
1.114
h
1
=
0.2
h
2
=
1.041
h
2
=
1.055
h
2
=
1.062
h
2
=
1.054
h
2
=
1.088
h
1
=
0.3
h
2
=
1.013
h
2
=
1.036
h
2
=
1.033
h
2
=
1.036
h
2
=
1.048
h
1
=
0.9
h
2
=
1.013
h
2
=
1.01
h
2
=
1.015
h
2
=
1.017
h
2
=
1.004Table 4
Allowable maximum time delayh2 of (50) under different h1 and γ.
Corollary5
h
1
=
0.1
h
1
=
0.15
h
1
=
0.2
h
1
=
0.25
h
1
=
0.3
Unknownγ
h
2
=
1.022
h
2
=
1.015
h
2
=
0.981
h
2
=
0.972
h
2
=
0.971
Corollary5
h
1
=
0.5
h
1
=
0.9
Unknownγ
h
2
=
0.983
h
2
=
0.995
Corollary7
γ
=
0
γ
=
0.3
γ
=
0.5
γ
=
0.8
γ
=
1
h
1
=
0
h
2
=
1.097
h
2
=
1.107
h
2
=
1.111
h
2
=
1.1191
h
2
=
1.121
Corollary7
γ
=
1.2
γ
=
1.5
γ
=
1.8
γ
=
2
γ
=
5
h
1
=
0
h
2
=
1.133
h
2
=
1.132
h
2
=
1.142
h
2
=
1.1272
h
2
=
1.1471
Corollary9
Unknownγ
h
1
=
0
h
2
=
1.089As seen in Table3, values of h2 change periodically with different γ under the same h1, but the difference between h2 and h1 decreases readily with increasing h1 under the same γ. It can be expected that whatever value γ takes, when h1 is big enough, h1 and h2 would converge to a single point. Moreover, it can be seen in Table 3 that the point is the same with the convergence point under γ=0, which is also the allowable maximum constant time delay.Table4 is used to demonstrate the effectiveness of Corollaries 5, 7, and 9. Allowable maximum time delay is presented under unknown γ for Corollary 5, h1=0 for Corollary 7, and both unknown γ and h1=0 for Corollary 9. It can be seen that when γ is unknown, h2 is apparently smaller than otherwise. Moreover, sums of h1 and h2 are about the same with those under h1=0.1 but smaller than those under h1≥0.2, which signifies that h1 and h2 have a near-linear relationship under 0≤h1≤0.1.


## 5. Conclusion

This paper has investigated the global stability of the neural networks with time-varying delay. By introducing a novel LKF, delay-dependent global stability criteria have been obtained. By reciprocally convex combination approach, a substep is taken, and a slack variable is introduced to estimate the derivative of LKF, and as a result, the proposed method is expected to be less conservative than the available literature. The proposed criteria have been formulated in terms of linear matrix inequalities and, thus, can be readily solved by standard computing software. Numerical examples are given, and analysis is made under different ranges and derivatives of time delay. The conservatism reduction has been proved to be more obvious than existing results.



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