# Combined Convex Technique on Delay-Distribution-Dependent Stability for Delayed Neural Networks

Authors: Ting Wang; Tao Li; Mingxiang Xue; Shumin Fei
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

Together with the Lyapunov-Krasovskii functional approach and an improved delay-partitioning idea, one novel sufficient condition is derived to guarantee a class of delayed neural networks to be asymptotically stable in the mean-square sense, in which the probabilistic variable delay and both of delay variation limits can be measured. Through combining the reciprocal convex technique and convex technique one, the criterion is presented via LMIs and its solvability heavily depends on the sizes of both time-delay range and its variations, which can become much less conservative than those present ones by thinning the delay intervals. Finally, it can be demonstrated by four numerical examples that our idea reduces the conservatism more effectively than some earlier reported ones.


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## Body

## 1. Introduction

In past decades, neural networks have been applied to various signal processing problems, such as optimization, image processing, associative memory design, and other engineering fields. In those applications, the key feature of the designed neural network is to be globally stable. Meanwhile, since there inevitably exists communication delay which can induce the oscillation and instability in various dynamical systems, great efforts have been made to analyze the dynamics of delayed systems including delayed neural networks (DNNs) and many elegant results have been reported; see [1–35]. In practical applications, though it is difficult to describe the form of time-delay precisely, the bounds of time-delay and its variation rates still can be measured. Since the Lyapunov functional approach imposes no restriction on delay variation and presents some simple stability criteria, the Lyapunov-Krasovskii functional (LKF) has been widely utilized due to that its analysis can fully make use of the information on time-delay of DNNs as much as possible. Thus recently, the delay-dependent stability has become an important topic of primary significance, in which the main purpose is to derive an allowable delay upper bound guaranteeing the global stability for addressed DNNs in [4–9, 16, 17, 21–35]. Furthermore, in recent years, since the delay-partitioning ideas have been proven to be more effective in reducing the conservatism than some previously reported techniques and received much research attention in [21–27], yet these convex ideas above still need some further improvements since they cannot effectively tackle interval variable delay or cannot fully utilize every delay subintervals, which have been fully addressed in [28].Meanwhile, it can be seen from many existing references that only the deterministic time-delay case was concerned, and the stability criteria were derived based on the information of delay and its variation range. Actually, time-delay in some DNNs is often existent in a stochastic fashion. In practice, to propagate and control the stochastic signals through universal learning networks, a probabilistic universal leaning network (PULN) was proposed. In a PULN, the output signal of the node is transferred to another node by multibranches with arbitrary time-delays which are random and its probability often can be measured by the statistical methods. For this case, if some values of time-delay are very large but its probability of the delay taking such large values is very small, it may lead to a more conservative result if only the information of delay variation range is considered. Thus, recently, some researchers have considered the stability for various systems including DNNs with probability-distributed delays [29–39]. In [36–39], the authors have analyzed the stability and its applications for networked control systems, uncertain linear systems, and T-S fuzzy systems, in which probability delay has been fully considered. As for discrete-time DNNs with probabilistic delay, the global stability has been considered and some pretty results have been proposed in [29–33]. Yet, it has come to our attention that, though some works have studied the dynamics of continuous-time DNNs with probabilistic delay [34, 35], the lower limits of delay variation have not been considered and, in fact, such available information could play an important role in extending the results' application area, which has been illustrated in [40] and yet not taking delay distribution probability into consideration. Presently, as for time-variable delay, the reciprocal convex approach in [41] has been proven to be more effective in reducing the conservatism than some earlier convex techniques [28, 42]. Yet to the authors' best knowledge, few authors have used the combination of the reciprocal convex technique and general convex ones to tackle the global stability for DNNs with probabilistic time-varying delay, which constitutes the main focus of this present work.Together with taking both bounds on probabilistic time-delay and its time variations into consideration, we make some great efforts to investigate the mean-squared stability for DNNs, in which an improved delay-partitioning idea is utilized and a novel Lyapunov functional is chosen. Through combining the reciprocal convex technique and the general convex one, one less conservative condition is given in terms of LMIs, which can present the pretty delay dependence and computational efficiency. Finally, we give four numerical examples to illustrate that our derived results can be less conservative than some existent ones.The notations in the paper are really standard. For symmetric matricesX,Y,X>Y (resp., X≥Y) means that X-Y>0(X-Y≥0) is a positive-definite (resp., positive-semidefinite) matrix; and * denotes the symmetric term in a symmetric matrix, that is,   [XYYTZ]=[XY*Z].


## 2. Problem Formulations and Preliminaries

Consider the delayed neural networks as follows:(2.1)z˙(t)=-Cz(t)+Ag(z(t))+Bg(z(t-τ(t)))+I,
where z=[z1,…,zn]T∈Rn is a real n-vector denoting the state variables associated with the neurons, g(z)=[g1(z1),…,gn(zn)]T represents the neuron activation function, I=[I1,…,In]T∈Rn is a constant input vector, C=diag{c1,…,cn}>0, and A,B are the appropriately dimensional constant matrices.The following assumptions on the system (2.1) are made throughout this paper.Assumption 2.1.
The time-varying delayτ(t) satisfies 0≤τ1≤τ(t)≤τ3. Moreover, consider the information of probability distribution of τ(t), two sets and functions are defined as Ω1={t,τ(t)∈[τ1,τ2)}, Ω2={t,τ(t)∈[τ2,τ3]}, and
(2.2)τ1(t)={τ(t),fort∈Ω1,τ-1,fort∈Ω2,τ2(t)={τ(t),fort∈Ω2,τ-2,fort∈Ω1,(2.3)μ1≤τ˙1(t)≤μ2,μ3≤τ˙2(t)≤μ4,
where τ2∈[τ1,τ3], τ-1∈[τ1,τ2), and τ-2∈[τ2,τ3]. It is easy to check that t∈Ω1 means that the event τ(t)∈[τ1,τ2) occurs and t∈Ω2 means that the event τ(t)∈[τ2,τ3] occurs. Therefore, a stochastic variable δ(t) can be defined as
(2.4)δ(t)={1,fort∈Ω1,0,fort∈Ω2.Assumption 2.2.
δ
(
t
) is a Bernoulli distributed sequence with
(2.5)Prob{δ(t)=1}=E{δ(t)}=δ0,Prob{δ(t)=0}=1-E{δ(t)}=1-δ0,
where 0≤δ0≤1 is a constant and E{δ(t)} is the mathematical expectation of δ(t). It is easy to check that E{δ(t)-δ0}=0.Assumption 2.3.
For the constantsρj+,ρj-, the nonlinear function gj(·) in (2.1) satisfies the following condition:
(2.6)ρj-≤gj(α)-gj(β)α-β≤ρj+,∀α,β∈R,α≠β,j=1,2,…,n.
Here, we denote Σ-=diag{ρ1+,…,ρn+}, Σ=diag{ρ1-,…,ρn-}, and
(2.7)Σ1=diag{ρ1+ρ1-,…,ρn+ρn-},Σ2=diag{ρ1++ρ1-2,…,ρn++ρn-2}.
It is clear that under Assumptions 2.1–2.3, system (2.1) has one equilibrium point z*=[z1*,…,zn*]T. For convenience, we shift the equilibrium point z* to the origin by letting x=z-z*, f(x)=g(x+z*)-g(z*), and the system (2.1) can be converted to
(2.8)x˙(t)=-Cx(t)+Af(x(t))+Bf(x(t-τ(t))),
where f(x)=[f1(x1),…,fn(xn)]T. Based on the methods in [37–39], the system above can be equivalently converted to
(2.9)x˙(t)=-Cx(t)+Af(x(t))+δ(t)Bf(x(t-τ1(t)))+(1-δ(t))Bf(x(t-τ2(t))).
It is easy to check that the function fj(·) satisfies fj(0)=0, and
(2.10)ρj-≤fj(α)α≤ρj+,∀α∈R,α≠0,j=1,2,…,n.
Then, the problem to be addressed in the paper can be formulated as developing a condition ensuring that the system (2.9) is asymptotically stable.
In order to obtain the stability criterion for system (2.9), the following lemmas are introduced.Lemma 2.4 (see [27]).
For any constant matrixX∈Rn×n, X=XT≥0, a scalar functional h:=h(t)≥0, and a vector function x˙:[-h,0]→Rn such that the following integration is well defined, then -h∫t-hh‍x˙T(s)Xx˙(s)ds≤[x(t)-x(t-h)]TX[x(t)-x(t-h)].Lemma 2.5 (see [41]).
Let the functionsf1(t),f2(t),…,fN(t):Rm→R have the positive values in an open subset D of Rm and satisfy (1/α1)f1(t)+(1/α2)f2(t)+⋯+(1/αN)fN(t):D→R with αi>0 and ∑i=1N‍αi=1, then the reciprocal convex technique of fi(t) over the set D satisfies
(2.11)∑i1αifi(t)≥∑ifi(t)+∑i≠jgi,j(t),∀gi,j(t):Rm→R,[fi(t)gi,j(t)gi,jT(t)fj(t)]≥0.


## 3. Delay-Distribution-Dependent Stability

Firstly, we can rewrite the system (2.9) as
(3.1)x˙(t)=-Cx(t)+Af(x(t))+δ0Bf(x(t-τ1(t)))+(1-δ0)Bf(x(t-τ2(t)))+(δ(t)-δ0)×B[f(x(t-τ1(t)))-f(x(t-τ2(t)))].
Now letting l,m be positive integers, we, respectively, divide the delay intervals [τ1,τ2] and [τ2,τ3] into l,m segments averagely. Moreover, we introduce the  following denotations:
(3.2)δ1=τ2-τ1l,δ2=τ3-τ2m,ρ1(t)=τ1(t)-τ1l,ρ2(t)=τ2(t)-τ2m.
Then, based on (2.10) and (3.2), we can construct the following Lyapunov-Krasovskii functional candidate:
(3.3)V(xt)=V1(xt)+V2(xt)+V3(xt),
where
(3.4)V1(xt)=xT(t)Px(t)+∫-τ10‍∫t+θt‍τ1x˙T(s)Qx˙(s)dsdθ+2∑i=1n‍ki∫0xi‍[fi(s)-ρi-s]ds+2∑i=1n‍li∫0xi‍[ρi+s-fi(s)]ds,V2(xt)=∫t-τ1t‍[x(s)f(x(s))]T[P1H1*Q1][x(s)f(x(s))]ds+∑i=12∫t-τi-δit-τi‍[σi(s)hi(σi(s))]T[P~iH~i*Q~i][σi(s)hi(σi(s))]ds+∑i=1l∫t-τ1-(i-1)δ1-ρ1(t)t-τ1-(i-1)δ1‍[x(s)f(x(s))]T[X1iY1i*Z1i][x(s)f(x(s))]ds+∑i=1l∫t-τ1-iδ1t-τ1-(i-1)δ1-ρ1(t)‍[x(s)f(x(s))]T[X2iY2i*Z2i][x(s)f(x(s))]ds+∑i=1m∫t-τ2-(i-1)δ2-ρ2(t)t-τ2-(i-1)δ2‍[x(s)f(x(s))]T[X3iY3i*Z3i][x(s)f(x(s))]ds+∑i=1m∫t-τ2-iδ2t-τ2-(i-1)δ2-ρ2(t)‍[x(s)f(x(s))]T[X4iY4i*Z4i][x(s)f(x(s))]ds,V3(xt)=∑i=1l∫-τ1-iδ1-τ1-(i-1)δ1‍∫t+θt‍δ1x˙T(s)W1ix˙(s)dsdθ+∑i=1m∫-τ2-iδ2-τ2-(i-1)δ2‍∫t+θt‍δ2x˙T(s)W2ix˙(s)dsdθ,
with K=diag{k1,…,kn},L=diag{l1,…,ln}, n×n constant matrices P,Q,P1,H1,Q1,Xji,Yji,Zji(j=1,2,3,4), Wji(j=1,2), ln×ln constant matrices P~1,Q~1,H~1, mn×mn constant matrices P~2,Q~2,H~2, and
(3.5)σ1T(s)=[xT(s)xT(s-δ1)⋯xT(s-(l-1)δ1)],h1T(σ1(s))=[fT(x(s))fT(x(s-δ1))⋯fT(x(s-(l-1)δ1))],σ2T(s)=[xT(s)xT(s-δ2)⋯xT(s-(m-1)δ2)],h2T(σ2(s))=[fT(x(s))fT(x(s-δ2))⋯fT(x(s-(m-1)δ2))].
Denoting a parameter set Φ={P,Q,K,L,W1i,W2h,[P1H1*Q1],[P~jH~j*Q~j],[XjiYji*Zji],[XkhYkh*Zkh], j=1,2,i=1,…,l;k=3,4,h=1,…,m}, then we give one proposition which is essential in the following deduction.Proposition 3.1.
If the parameter setΦ satisfies the following condition:
(3.6)P>0,Q>0,K>0,L>0,W1i>0,W2h>0,[P1H1*Q1]>0,[P~jH~j*Q~j]>0,[XjiYji*Zji]>0,[XkhYkh*Zkh]>0,j=1,2,k=3,4;i=1,…,l;h=1,…,m,
then the Lyapunov-Krasovskii functional (3.3) is definitely positive.Moreover, in order to simplify the subsequent proof, we also give some notations in the following:(3.7)μ-1=μ2-μ1,μ-3=μ4-μ3,Σ~i=diag{Σi,…,Σi}ln×ln,Σ-i=diag{Σi,…,Σi}mn×mn,X~i=diag{Xi1,…,Xil},Y~i=diag{Yi1,…,Yil},Z~i=diag{Zi1,…,Zil},i=1,2,S~1=diag{S11,…,S1l},W~1=diag{W11,…,W1l},V~1=diag{V11,…,V1l},R~1=diag{R11,…,R1l},(3.8)X~j=diag{Xj1,…,Xjm},Y~j=diag{Yj1,…,Yjm},Z~j=diag{Zj1,…,Zjm},j=3,4,S~2=diag{S21,…,S2m},W~2=diag{W21,…,W2m},V~2=diag{V21,…,V2m},R~2=diag{R21,…,R2m}.Theorem 3.2.
For two given positive integersl,m, and time-delay satisfying (2.2), the delayed neural network (3.1) is globally asymptotically stable in the mean square sense, if there exists one parameter set Φ satisfying Proposition 3.1, n×n matrices Ei(i=1,2), and n×n diagonal matrices Ui>0(i=1,2,3,4,5),V1i>0,R1i>0,S1i(i=1,…,l) making [W1iS1i*W1i]≥0, V2j>0,R2j>0,S2i(j=1,…,m) making [W2iS2i*W2i]≥0 such that the following LMIs in (3.9) hold:
(3.9)ϒ1Θϒ1T+ϒ2Ξϒ2T+μ-1l∑i=1lI1iT[XeiYei*Zei]I1i+μ-3m∑i=1m‍I2iT[XfiYfi*Zfi]I2i<0,e=1,2;f=3,4,
where
(3.10)I1i=[0n·(m+l+1+i)n-In0n·(2l+2m+1)n-In*0n·(m+l+1+i)nIn0n·(2l+2m+1)nIn*],I2i=[0n·(m+2l+1+i)n-In0n·(2l+2m+1)n-In*0n·(m+2l+1+i)nIn0n·(2l+2m+1)nIn*],Θ=[Θ11Q00Θ15000Θ1900Θ1,12Θ1,13*Θ22000-H10000000**Θ33000U2Σ2000000***Θ44000U3Σ200000****Θ55000Θ590000*****-Q10000000******-U2000000*******-U30000********Θ9900Θ9,12Θ9,13*********-U4Σ10U4Σ20**********-U5Σ10U5Σ2***********-U40************-U5],Ξ=[Ξ-1Ξ-2*Ξ-3],Ξ-1=[Ξ11000S~10*Ξ2200S~10**Ξ3300S~2***Ξ440S~2****Ξ550*****Ξ66],Ξ-2=diag{Ξ17,Ξ28,Ξ39,Ξ4,10,Ξ5,11,Ξ6,12},Ξ-3=diag{Ξ77,Ξ88,Ξ99,Ξ10,10,Ξ11,11,Ξ12,12},ϒ1=[In*0n·nIn*0n·(l+1)nIn*0n·(m+l+1)nIn*0n·2(m+l+1)nIn*0n·(2m+2l+3)nIn*0n·(2m+3l+3)nIn*0n·3(m+l+1)nIn*0n·4(m+l+1)nIn*0n·(4m+4l+5)nIn*0n·(4m+4l+6)nIn*0n·(4m+4l+7)nIn*0n·(4m+4l+8)nIn],ϒ2=[0ln·nIln*0ln·2nIln*0mn·(l+1)nImn*0mn·(l+2)nImn*0ln·(m+l+2)nIln*0mn·(m+2l+2)nImn*0ln·(2m+2l+3)nIln*0ln·(2m+2l+4)nIln*0mn·(2m+3l+3)nImn*0mn·(2m+3l+4)nImn*0ln·(3m+3l+4)nIln*0mn·(3m+4l+4)nImn*],
with the notations * in Iji(j=1,…,6) and ϒg(g=1,2) denoting the appropriately dimensional zero matrices making them (4l+4m+9)n columns:
(3.11)Θ11=-E1TC-CTE1+P1-Q-U1Σ1,Θ15=E1TA+H1+U1Σ2,Θ19=P-ΣK+Σ-L-E1T-CTE2,Θ1,12=δ0E1TB,Θ1,13=(1-δ0)E1TB,Θ22=-P1-Q,Θ33=-U2Σ1,Θ44=-U3Σ1,Θ55=-U1+Q1,Θ59=K-L+ATE2,Θ99=-E2T-E2+τ12Q+∑i=1lδ12W1i+∑i=1mδ22W2i,Θ9,12=δ0E2TB,Θ9,13=(1-δ0)E2TB,Ξ11=P~1+X~1-W~1-V~1Σ~1,Ξ22=-P~1-X~2-W~1,Ξ33=P~2+X~3-W~2-V~2Σ-1,Ξ44=-P~2-X~4-W~2,Ξ55=(1-μ2l)X~2-(1-μ1l)X~1-2W~1-R~1Σ~1,Ξ66=(1-μ4m)X~4-(1-μ3m)X~3-2W~2-R~2Σ-1,Ξ77=Q~1+Z~1-V~1,Ξ88=-Q~1-Z~2,Ξ99=-V~2+Q~2+Z~3,Ξ10,10=-Q~2-Z~4,Ξ11,11=(1-μ2l)Z~2-(1-μ1l)Z~1-R~1,Ξ12,12=(1-μ4m)Z~4-(1-μ3m)Z~3-R~2,Ξ17=H~1+Y~1+V~1Σ~2,Ξ28=-H~1-Y~2,Ξ39=H~2+Y~3+V~2Σ-2,Ξ4,10=-H~2-Y~4,Ξ5,11=(1-μ2l)Y~2-(1-μ1l)Y~1+R~1Σ~1,Ξ6,12=(1-μ4l)Y~4-(1-μ3l)Y~3+R~2Σ-2.Proof.
Firstly, we show the uniqueness of the equilibrium point by the method of contradiction. Here we can denote the equilibrium point asz* of DNNs (2.9), then we have
(3.12)0=-Cz*+Af(z*)+δ(t)Bf(z*)+(1-δ(t))Bf(z*)=-Cz*+(A+B)f(z*).
Now we suppose that the other equilibrium point z*≠0 exists, then it follows that
(3.13)2(z*)TE1T[-Cz*+(A+B)f(z*)]=0,(3.14)-(z*)TU1Σ1z*+2(z*)TU1Σ2f(z*)-fT(z*)U1f(z*)≥0.
Then combining (3.13) and (3.14) yields that
(3.15)[z*f(z*)]T[Π11Π12Π12TΠ22][z*f(z*)]≥0
with Π11=-E1TC-CTE1-U1Σ1, Π12=E1T(A+B)+U1Σ2, and Π22=-U1. Meanwhile, it is noted that
(3.16)[z*f(z*)]T[P1H1*Q1][z*f(z*)]≥0.
Yet on the other hand, let
(3.17)Ψ=[InIn0n×(4l+4m+7)n0n×(2l+2m+2)nIn0n×(2l+2m+4)nInIn].
Then multiplying the term Υ1ΘΥ1T+Υ2ΞΥ2T in (3.9) by Ψ and ΨT on its left hand and right one, respectively, and using (3.15) and (3.16), we can deduce that
(3.18)[Π11+P1Π12+H1Π12T+H1TΠ22+Q1]<0⇒Π=[Π11Π12Π12TΠ22]<-[P1H1H1TQ1].
Thus we can derive Π<0, which contradicts with (3.15) and implies z*=0. This is to say, the origin of the DNNs (3.1) is the unique equilibrium point.
Next, through directly calculating and using the denotations in (3.7)-(3.8), the stochastic differential of Vi(xt)(i=1,2) in (3.3) along the trajectories of system (3.1) yields
(3.19)ℒV1(xt)=2xT(t)Px˙(t)+x˙T(t)τ12Qx˙(t)-∫t-τ1tτ1x˙T(s)Qx˙(s)ds+2[fT(x(t))(K-L)+xT(t)(Σ-L-ΣK)]x˙(t)≤2xT(t)Px˙(t)+x˙T(t)τ12Qx˙(t)-[x(t)-x(t-τ1)]TQ[x(t)-x(t-τ1)]+2[fT(x(t))(K-L)+xT(t)(Σ-L-ΣK)]x˙(t),ℒV2(xt)=[xT(t)P1x(t)+2xT(t)H1f(x(t))+fT(x(t))Q1f(x(t))]-[xT(t-τ1)P1x(t-τ1)+2xT(t-τ1)H1f(x(t-τ1))+fT(x(t-τ1))Q1f(x(t-τ1))]+[σ1T(t-τ1)P~1σ1(t-τ1)+2σ1T(t-τ1)H~1h1(σ1(t-τ1))+h1T(σ1(t-τ1))Q~1h1(σ1(t-τ1))]-[σ1T(t-τ1-δ1)P~1×σ(t-τ1-δ1)+2σ1T(t-τ1-δ1)H~1h1(σ(t-τ1-δ1))+h1T(σ(t-τ1-δ1))Q~1×h1(σ(t-τ1-δ1))]+[σ2T(t-τ2)P~2σ2(t-τ2)+2σ2T(t-τ2)H~2h2(σ2(t-τ2))+h2T(σ2(t-τ2))Q~2h2(σ2(t-τ2))]-[σ2T(t-τ2-δ2)P~2σ(t-τ2-δ2)+2σ2T(t-τ2-δ2)H~2h2(σ(t-τ2-δ2))+h2T(σ(t-τ2-δ2))Q~2h2(σ(t-τ2-δ2))]+[σ1T(t-τ1)X~1σ1(t-τ1)+σ1T(t-τ1-ρ1(t))(1-τ˙1(t)l)(X~2-X~1)×σ1(t-τ1-ρ1(t))-σ1T(t-τ1-δ1)X~2σ1(t-τ1-δ1){(1-τ˙1(t)l)}]+[2σ1T(t-τ1)Y~1h1(σ1(t-τ1))+2σ1T(t-τ1-ρ1(t))×(1-τ˙1(t)l)(Y~2-Y~1)×h1(σ1(t-τ1-ρ1(t)))-2σ1T(t-τ1-δ1)Y~2h1(σ1(t-τ1-δ1)){(1-τ˙1(t)l)}]+[h1T(σ1(t-τ1))Z~1h1(σ1(t-τ1))+h1T(σ1(t-τ1-ρ1(t)))(1-τ˙1(t)l)×(Z~2-Z~1)h1(σ1(t-τ1-ρ1(t)))-h1T(σ1(t-τ1-δ1))Z~2h1T(σ1(t-τ1-δ1)){{(1-τ˙1(t)l)}}]+[σ2T(t-τ2)X~3σ2(t-τ2)+σ2T(t-τ2-ρ2(t))(1-τ˙2(t)m)(X~4-X~3)×σ2(t-τ2-ρ2(t))-σ2T(t-τ2-δ2)X~4σ2(t-τ2-δ2){{(1-τ˙1(t)l)}}]+[2σ2T(t-τ2)Y~3h2(σ2(t-τ2))+2σ2T(t-τ2-ρ2(t))×(1-τ˙2(t)m)×(Y~4-Y~3)h2(σ2(t-τ2-ρ2(t)))-2σ2T(t-τ2-δ2)Y~4h2(σ2(t-τ2-δ2)){(1-τ˙2(t)m)}]+[h2T(σ2(t-τ2))Z~3h2(σ2(t-τ2))+h2T(σ2(t-τ2-ρ2(t)))(1-τ˙2(t)m)×(Z~4-Z~3)h2(σ2(t-τ2-ρ2(t)))-h2T(σ2(t-τ2-δ2))Z~4hT(σ2(t-τ2-δ2)){(1-τ˙2(t)m)}].
Moreover, we can compute out ℒV3(xt) as follows:
(3.20)ℒV3(xt)=∑i=1l‍x˙T(t)(δ12W1i)x˙(t)-∑i=1l‍∫t-τ1-iδ1t-τ1-(i-1)δ1‍δ1x˙T(s)W1ix˙(s)ds+∑i=1m‍x˙T(t)(δ22W2i)x˙(t)-∑i=1m‍∫t-τ2-iδ2t-τ2-(i-1)δ2‍x˙T(s)(δ2W2i)x˙(s)ds.
Then, by resorting to Lemmas 2.4 and 2.5  [W1iS1i*W1i]≥0,[W2iS2i*W2i]≥0, and using the donations W~i,S~i(i=1,2), the following inequalities can be derived:
(3.21)-∑i=1l∫t-τ1-iδ1t-τ1-(i-1)δ1‍δ1x˙T(s)W1ix˙(s)ds≤∑i=1l[-[x(t-τ1-(i-1)δ1-ρ1(t))-x(t-τ1-iδ1)]TW1i×[x(t-τ1-(i-1)δ1-ρ1(t))-x(t-τ1-iδ1)]-2[x(t-τ1-(i-1)δ1-ρ1(t))-x(t-τ1-iδ1)]TS1i×[x(t-τ1-(i-1)δ1)-x(t-τ1-(i-1)δ1-ρ1(t))]-[x(t-τ1-(i-1)δ1)-x(t-τ1-(i-1)δ1-ρ1(t))]TW1i×[x(t-τ1-(i-1)δ1)-x(t-τ1-(i-1)δ1-ρ1(t))]{[x(t-τ1-(i-1)δ1-ρ1(t))-x(t-τ1-iδ1)]TW1i}]=-[σ1T(t-τ1-ρ1(t))W~1σ(t-τ1-ρ1(t))-2σ1T(t-τ1-ρ1(t))×S~1σ1(t-τ1-δ1)+σ1T(t-τ1-δ1)W~1σ1(t-τ1-δ1)]-[σ1T(t-τ1-ρ1(t))W~1σ1(t-τ1-ρ1(t))-2σ2T(t-τ2-ρ2(t))S~1σ2(t-τ2)+σ2T(t-τ2)W~2σ2(t-τ2)],-∑i=1m∫t-τ2-iδ2t-τ2-(i-1)δ2‍δ2x˙T(s)W2ix˙(s)ds≤-[σ2T(t-τ2-ρ2(t))W~2σ(t-τ2-ρ2(t))-2σ2T(t-τ2-ρ2(t))×S~2σ2(t-τ2-δ2)+σ2T(t-τ2-δ2)W~2σ2(t-τ2-δ2)]-[σ2T(t-τ2-ρ2(t))W~2σ2(t-τ2-ρ2(t))-2σ2T(t-τ2-ρ2(t))S~2σ2(t-τ2)+σ2T(t-τ2)W~2σ2(t-τ2)].
Then, it follows from (3.21) that ℒV3(xt) satisfies
(3.22)ℒV3(xt)≤∑i=1l‍x˙T(t)(δ12W1i)x˙(t)+∑i=1l‍x˙T(t)(δ22W2i)x˙(t)=-[σ1T(t-τ1-ρ1(t))W~1σ(t-τ1-ρ1(t))=-2σ1T(t-τ1-ρ1(t))S~1σ1(t-τ1-δ1)+σ1T(t-τ1-δ1)W~1σ1(t-τ1-δ1)]====-[σ1T(t-τ1-ρ1(t))W~1σ1(t-τ1-ρ1(t))-2σ2T(t-τ2-ρ2(t))S~1σ2(t-τ2)====+σ2T(t-τ2)W~1σ2(t-τ2)]====-[σ2T(t-τ2-ρ2(t))W~2σ(t-τ2-ρ2(t))-2σ2T(t-τ2-ρ2(t))×S~2σ2(t-τ2-δ2)====+σ2T(t-τ2-δ2)W~2σ2(t-τ2-δ2)]==-[σ2T(t-τ2-ρ2(t))×W~2σ2(t-τ2-ρ2(t))====-2σ2T(t-τ2-ρ2(t))S~2σ2(t-τ2)+σ2T(t-τ2)W~2σ2(t-τ2)].
From (2.10), for any n×n diagonal matrices Ui>0(i=1,2,3,4,5), V1i>0, R1i>0(i=1,…,l), V2j>0, R2j>0(j=1,…,m), and setting V~1,R~1 in (3.7), V~2,R~2 in (3.8), the following inequality holds:
(3.23)0≤[-xT(t)U1Σ1x(t)+2xT(t)U1Σ2f(x(t))-fT(x(t))U1f(x(t))]+[-xT(t-τ2)U2Σ1x(t-τ2)+2xT(t-τ2)U2Σ2f(x(t-τ2))-fT(x(t-τ2))U2f(x(t-τ2))]+[-xT(t-τ3)U3Σ1x(t-τ3)+2xT(t-τ3)×U3Σ2f(x(t-τ3))-fT(x(t-τ3))U3f(x(t-τ3))]+[-xT(t-τ1(t))×U4Σ1x(t-τ1(t))+2xT(t-τ1(t))U4Σ2f(x(t-τ1(t)))-fT(x(t-τ1(t)))U4f(x(t-τ1(t)))]+[-xT(t-τ2(t))U5Σ1x(t-τ2(t))+2xT(t-τ2(t))U5Σ2f(x(t-τ2(t)))-fT(x(t-τ2(t)))U5f(x(t-τ2(t)))]+[-σ1T(t-τ1)V~1Σ~1σ1(t-τ1)+2σ1T(t-τ1)V~1Σ~2h1(σ1(t-τ1))-h1T(σ1(t-τ1))V~1h1(σ1(t-τ1))]+[-σ1T(t-τ1-ρ1(t))R~1Σ~1σ1(t-τ1-ρ1(t))+2σ1T(t-τ1-ρ1(t))R~1Σ~2h1(σ1(t-τ1-ρ1(t)))-h1T(σ1(t-τ1-ρ1(t)))R~1h1(σ1(t-τ1-ρ1(t)))]+[-σ2T(t-τ2)V~2Σ-1σ2(t-τ2)+2σ2T(t-τ2)V~2Σ-2h2(σ2(t-τ2))-h2T(σ2(t-τ2))×V~2h2(σ2(t-τ2))]+[-σ2T(t-τ2-ρ2(t))R~2Σ-1σ2(t-τ2-ρ2(t))+2σ2T(t-τ2-ρ2(t))×R~2Σ-2h2(σ2(t-τ2-ρ2(t)))-h2T(σ2(t-τ2-ρ2(t)))R~2h2(σ2(t-τ2-ρ2(t)))].
Moreover, together with (3.1) and any n×n constant matrices E1,E2, one can deduce
(3.24)0=[xT(t)E1T+x˙T(t)E2T]×[-x˙(t)-Cx(t)+Af(x(t))+δ0Bf(x(t-τ1(t)))+(1-δ0)Bf(x(t-τ2(t)))+(δ(t)-δ0)B[f(x(t-τ1(t)))-f(x(t-τ2(t)))]].
Now adding the right terms (3.19) and (3.22)–(3.24) to ℒV(xt) and taking the mathematical expectation on its both sides, we can deduce
(3.25)E{ℒV(xt)}≤ζT(t){ϒ1Θϒ1T+τ˙1(t)-μ1l∑i=1l‍I1iT[X1iY1i*Z1i]I1i+μ2-τ˙1(t)l∑i=1l‍I1iT[X2iY2i*Z2i]I1i+ϒ2Ξϒ2T+τ˙2(t)-μ3m∑i=1m‍I2iT[X3iY3i*Z3i]I2i+μ4-τ˙2(t)m∑i=1m‍I2iT[X4iY4i*Z2i]I2i}ζ(t):=ζT(t)Λ(t)ζ(t),
where Θ,Ξ,ϒl(l=1,2),Iki(k=1,2) are presented in (3.9), and
(3.26)ζT(t)=[xT(t)σ1T(t-τ1)σ2T(t-τ2)xT(t-τ3)σT(t-τ1-ρ1(t))×σT(t-τ2-ρ2(t))fT(x(t))h1T(σ1(t-τ1))h2T(σ2(t-τ2))fT(x(t-τ3))×h1T(σ1(t-τ1-ρ1(t)))h2T(σ2(t-τ2-ρ2(t))){xT(t)σ1T(t-τ1)σ2T(t-τ2)xT(t-τ3)σT(t-τ1-ρ1(t))}×x˙T(t)xT(t-τ1(t))xT(t-τ2(t))fT(x(t-τ1(t)))fT(x(t-τ2(t)))].
Then utilizing the general convex technique in [28, 39], the LMIs described by (3.9) can guarantee Λ(t)<0, which indicates that there must exist a positive scalar χ>0 such that E{ℒV(xt)}≤ζT(t)Λ(t)ζ(t)≤-χ∥x(t)∥2<0 for any x(t)≠0. Then, it follows from the Lyapunov-Krasovskii stability theorem that the system (3.1) is asymptotically stable in the mean-square sense, and it completes the proof.Remark 3.3.
Transmitted delays are always existent in various dynamical networks due to the finite switching speed of amplifiers in electronic neural networks or the finite signal propagation time in biological networks. Furthermore, with the development of network delay tomography, the probability of time-delay distribution can be estimated. Thus, when the probability is available, it will be helpful to utilize such information to reduce the conservatism [29–35]. Yet, up till now, few authors have utilized the delay-partitioning idea to investigate the stability of DNNs with probabilistic time-varying delay, and, in this work, we applied one improved idea, which can fully consider the information of delay subinterval. Moreover, though the stability criterion in (3.9) is not presented in the forms of standard LMIs, it is still convenient and straightforward to check the feasibility without tuning any parameters by resorting to LMI in Matlab Toolbox.Remark 3.4.
Presently, in the literatures [21–28], various convex combination techniques have been widely employed and improved to tackle constant or time-varying delays owing to that they can help reduce the conservatism efficiently. In [41], the authors put forward the reciprocal convex approach, which can consider those important terms ignored and be more effective than the ones [21–28]. Yet it has come to our attention that the reciprocal convex one cannot efficiently tackle the case that both the bounds on delay derivatives are available. In this paper, we first combine the reciprocal convex technique and the convex ones to study the stability for DNNs with probabilistic time-varying delay.Remark 3.5.
As forV2(xt) in (3.3), if we denote [X2iY2i*Z2i]=[X4iY4i*Z4i]=0 (resp., [X1iY1i*Z1i]=[X3iY3i*Z3i]=0), our results can be true as only μ2,μ4 (resp., μ1,μ3) are available. If we set [XjiYji*Zji]=0  (j=1,2,3,4) in (3.3) simultaneously, Theorem 3.2 still holds as that μi(i=1,2,3,4) are unknown, or τi(t)(i=1,2) are not differentiable. Moreover, the number of free-weighting matrices in Theorem 3.2 is much smaller than the ones of these present results [34, 37], and the reciprocal convex technique is used, it can induce much more computational simplicity in a mathematical point of view.Remark 3.6.
In view of the delay-partitioning idea employed in this work, with integersl,m increasing, the dimension of the derived LMIs will become higher and it will take more computing time to check them. Yet, if the lower bound of τ1(t) is set and l≥5, the maximum allowable delay upper bound will become unapparently larger and approach to an approximate upper limitation [22–27]. Thus, if we want to employ the idea to real cases, we do not necessarily partition two delay intervals into too many segments.Remark 3.7.
In order to give the more general results, the delay intervals[τ1,τ2] and [τ2,τ3] are, respectively, divided into l,m subintervals in this work, which makes the condition of Theorem 3.2 seem to be very complicated. Yet if we set l=m≤5, our results will avoid the complexity in some degree. Moreover, if we choose the simple Lyapunov-Krasovskii functional in (3.3) with H1=Y1i=Y2i=Y3i=Y4i=0 and H~i=0, the condition of Theorem 3.2 also will become much less complicated.


## 4. Numerical Examples

In this section, four numerical examples will be presented to illustrate the derived results. Firstly, we will utilize a numerical example to illustrate the significance of studying the lower bound of delay derivative.Example 4.1.
As a special case ofδ0=1, we revisit the delayed neural networks considered in [21, 28] with
(4.1)C=[2002],A=[11-1-1],B=[0.88111],Σ=[0000],Σ-=[0.4000.8],
and τ1=0 is set. If we do not consider the existence of μ1, then, by utilizing Theorem 3.2 and Remark 3.5, the corresponding maximum allowable upper bounds (MAUBs) τmax for different μ2 derived by the results in [21] and in the paper can be summarized in Table 1, which demonstrates that Theorem 3.2 of l=1 is somewhat more conservative than the one in [21, 28]. Yet, if we set μ1=0.5, it is easy to verify that our results can yield much less conservative results than the ones in [21, 28], which can be shown in Table 2.
Based on Tables1 and 2, it is indicated that the conservatism of stability criterion can be greatly deduced if we take the available μ1 into consideration. Moreover, though the delay-partitioning idea has been used in [28], the corresponding MAUBs τmax derived by [28] and Theorem 3.2 are summarized in Table 3, which shows that our idea can be more efficient than the one in [18] even for l=1,2.Table 1
Calculated MAUBsτmax for l=1, unavailable μ1 in Example 4.1.
Methods∖μ2
0.6
0.8
0.9
1.2
Theorem3.2 [21]
3.5209
2.8654
1.9508
—
Theorem3.2 [28]
3.4877
2.8456
1.9149
1.1168
Theorem3.2
3.5102
2.8554
1.9447
1.1212Table 2
Calculated MAUBsτmax for l=1 and μ1=0.5 in Example 4.1.
Methods∖μ2
0.6
0.8
0.9
1.2
Theorem3.2 [28]
3.5871
2.8813
1.9652
1.2052
Theorem3.2
3.6198
2.9018
2.0012
1.2853Table 3
Calculated MAUBsτmax for various l and μ1=0.5 in Example 4.1.
Methods∖μ2
0.8
0.9
unknownμ2
Theorem3.2 [28]
l=1
2.8813
1.9655
1.2052
l=2
3.1486
2.1967
1.4073
Theorem3.2
l=1
2.8873
1.9677
1.2100
l=2
3.1678
2.2011
1.4244Example 4.2.
Considering the special case ofδ0=1, we consider the delayed neural networks (2.1) with
(4.2)C=diag{1.2769,0.6231,0.9230,0.4480},Σ=03×3,Σ-=diag{0.1137,0.1279,0.7994,0.2368},A=[-0.03730.4852-0.33510.2336-1.60330.5988-0.32241.23520.3394-0.0860-0.3824-0.5785-0.13110.3253-0.9534-0.5015],B=[0.8674-1.2405-0.53250.02200.0474-0.91640.03600.98161.84952.6117-0.37880.8428-2.04130.51791.1734-0.2775],
which has been addressed extensively; see [26, 27] and the references therein. Together with the delay-partitioning idea and for different μ2, the work [27] has calculated the MAUBs τmax such that the origin of the system is globally asymptotically stable for τ1(t) satisfying 3=τ1≤τ1(t)≤τ2≤τmax. By resorting to Theorem 3.2 and Remark 3.5, the corresponding results can be given Table 4, which indicates that our delay-partitioning idea can be more effective than the relevant ones in [27] even for l=1,2 and μ1=0.Table 4
Calculated MAUBsτmax for various l,μ2 in Example 4.2.
Methods∖μ2
0.1
0.5
0.9
unknownμ2
Theorem3.2 [27]
l=1
3.33
3.16
3.10
3.09
l=2
3.65
3.32
3.26
3.24
Theorem3.2
l=1
3.35
3.22
3.15
3.13
l=2
3.77
3.42
3.34
3.32Example 4.3.
We still consider the DNNs with the following parameters [24, 28] by setting δ0=1:
(4.3)C=[4.18890000.71600001.9985],B=[-0.1052-0.5069-0.1121-0.0257-0.28080.02120.1205-0.21530.1315],Σ-=[0.41290003.89930001.016],A=Σ=03×3, and time-varying delay τ(t) satisfies 0≤τ(t)≤τ2,0≤τ˙(t)≤μ2. Then, the MAUBs for this example with time-varying delay τ(t) are given in Table 5 by using the delay-partitioning idea in [28] and the one in this paper, which can illustrate that our delay-decomposition idea is superior over the one by [28].Table 5
Calculated MAUBsτmax for l=2,μ2 in Example 4.3.
Methods∖μ2
0.1
0.5
0.9
1
Theorem3.2 [28]
2.0011
1.3450
1.0687
1.0685
Theorem3.2
2.0148
1.3485
1.0701
1.0698Example 4.4.
Consider DNN (2.3) of the following parameters, which has been considered in [33]:
(4.4)C=[7006],A=[0.2-40.10.3],B=[0.40.20.10.7],f(x)=[tanh(-0.2x1)tanh(x2)].
If we set τ1=0, τ2=0.4, μ1=μ3=0.1, and μ2=μ4, the relevant MAUBs τmax are computed and listed in Tables 6 and 7 by setting l=m=1, for various μ2, and the delay probability distribution δ0=0.2, which can guarantee the addressed system to be asymptotically stable in the mean-squared sense.
Based on Tables6 and 7, even for m=l=1, our results still can be less conservative than the one in [33]. As for m=1, l=2, one verifies that our results can reduce the conservatism much more evidently.Table 6
Calculated MAUBsτmax for l=m=1, various μ2, and δ0=0.2 in Example 4.4.
Methods∖μ2
0.2
0.6
1
1.5
2.5
Theorem3.2 [33]
1.1281
1.1279
1.1278
1.1277
1.1276
Theorem3.2 [28]
1.3149
1.3086
1.3031
1.3026
1.3025
Theorem3.2
1.4242
1.4189
1.4134
1.4131
1.4130Table 7
Calculated MAUBsτmax for l=m=1, various μ2, and δ0=0.8 in Example 4.4.
Methods∖μ2
0.2
0.6
1
1.5
2.5
Theorem3.2 [33]
1.7177
1.6869
1.5758
1.5757
1.5757
Theorem3.2 [28]
1.9778
1.8863
1.7742
1.7741
1.7740
Theorem3.2
2.0285
1.9678
1.9112
1.8785
1.7876


## 5. Conclusions

This paper has investigated the asymptotical stability for DNNs with probability-distribution delay. Through employing an improved idea of delay partitioning and constructing one novel Lyapunov-Krasovskii functional, one stability criterion with significantly reduced conservatism has been established in terms of LMIs. The proposed stability condition benefits from both bounds of time-varying delay and variation and combined convex technique. Four numerical examples have been given to demonstrate the effectiveness of the derived criteria and the improvements over some existent ones.



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*Source: 426350-2012-10-02.xml*