# Passivity Analysis of Markovian Jumping Neural Networks with Leakage Time-Varying Delays

Authors: N. Mala; A. R. Sudamani Ramaswamy
Journal: Journal of Computational Methods in Physics
 (2013)

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## Abstract

This paper is concerned with the passivity analysis of Markovian
jumping neural networks with leakage time-varying delays. Based on a Lyapunov
functional that accounts for the mixed time delays, a leakage delay-dependent
passivity conditions are derived in terms of linear matrix inequalities (LMIs). The
mixed delays includes leakage time-varying delays, discrete time-varying delays,
and distributed time-varying delays. By employing a novel Lyapunov-Krasovskii
functional having triple-integral terms, new passivity leakage delay-dependent
criteria are established to guarantee the passivity performance. This performance
not only depends on the upper bound of the time-varying leakage delayσ(t) but also depends on the upper bound of the derivative of the time-varying leakage
delay σμ. While estimating the upper bound of derivative of the Lyapunov-Krasovskii functional, the discrete and distributed delays should be treated so as
to appropriately develop less conservative results. Two numerical examples are
given to show the validity and potential of the developed criteria.


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## Body

## 1. Introduction

In the past few decades, neural networks (NNs) have been a hot research topic because of their emerged application in static image processing, pattern recognition, fixed-point computation, associative memory, combinatorial optimization [1–5]. Because the interactions between neurons are generally asynchronous in biological and artificial neural networks, time delays are usually encountered. Since the existence of time delays is frequently one of the main sources of instability for neural networks, the stability analysis for delayed neural networks had been extensively studied and many papers have been published on various types of neural networks with time delays based on the LMI approach [6–14].On the other hand, the main idea of passivity theory is that the passive properties of a system can keep the system internally stable. In addition, passivity theory is frequently used in control systems to prove the stability of systems. The problem of passivity performance analysis has also been extensively applied in many areas such as signal processing, fuzzy control, sliding mode control [15], and networked control [16]. The passivity idea is a promising approach to the analysis of the stability of NNs, because it can lead to more general stability results. It is important to investigate the passivity analysis for neural networks with time delays. More recently, dissipativity or passivity performances of NNs have received increasing attention and many research results have been reported in the literature, for example, [17–21].In practice, the RNNs often exhibit the behavior of finite state representations (also called clusters, patterns, or modes) which are referred to as the information latching problems [22]. In this case, the network states may switch (or jump) between different RNN modes according to a Markovian chain, and this gives rise to the so-called Markovian jumping recurrent neural networks. It has been shown that the information latching phenomenon is recognized to exist universally in neural networks [23, 24], which can be dealt with extracting finite state representation from a trained network, that is, a neural network sometimes has finite modes that switch from one to another at different times. The results related to all kinds of Markovian jump neural networks with time delay can also be found in [25–27] and the references therein. It should be pointed out that all the above mentioned references assume that the considered transition probabilities in the Markov process or Markov chain are time invariant, that is, the considered Markov process or Markov chain is assumed to be homogeneous. It is noted that such kind of assumption is required in most existing results on Markovian jump systems [28, 29]. The detailed discussion about piecewise homogeneous and nonhomogeneous Markovian jumping parameters has been given in [30] and references therein.On the other hand, a typical time delay called as leakage (or “forgetting”) delay may exist in the negative feedback terms of the neural network and it has a great impact on the dynamic behaviors of delayed neural networks and more details are given in [31–36]. In [34] the authors introduced leakage time-varying delay for dynamical systems with nonlinear perturbations and derived leakage delay-dependent stability conditions via constructing a new type of Lyapunov-Krasovskii functional and LMI approach. Recently, the passivity analysis for neural networks of neutral type with Markovian jumping parameters and time delay in the leakage term have been addressed in [37]. With reference to the results above, it has been studied that many results get to be found out for passivity analysis of Markovian jumping neural networks with leakage time-varying delays. Thus, the main purpose of this paper is to shorten such a gap by making the first attempt to deal with the passivity analysis problem for a type of continuous-time neural networks with time-varying transition probabilities and mixed time delays.In this paper, the problem of passivity analysis of Markovian jump neural networks with leakage time-varying delay and discrete and distributed time-varying delays is considered. The Markov process in the under lying neural networks is assumed to be finite piecewise homogeneous, which is a special nonhomogeneous (time-varying) Markov chain. Motivated by [30] a novel Lyapunov-Krasovskii functional is constructed in which the positive definite matrices are dependent on the system mode and a triple-integral term is introduced for deriving the delay-dependent stability conditions. By employing a novel Lyapunov-Krasovskii functional having triple integral terms, new passivity leakage delay-dependent criteria are established to guarantee the passivity performance of the given systems. This performance not only depends on the upper bound of the time-varying leakage delay σ(t) but also depends on the upper bound of the derivative of the time-varying leakage delay σμ. When estimating an upper bound of the derivative of the Lyapunov-Krasovskii functional, we handle the terms related to the discrete and distributed delays appropriately so as to develop less conservative results. Two numerical examples are given to show the validity and potential of the development of the proposed passivity criteria.Notations.  Let ℝn denote the n-dimensional Euclidean space and the superscript “T” denotes the transpose of a matrix or vector. I denote the identity matrix with compatible dimensions. For square matrices M1 and M2, the notation M1>(≥,<,≤)  M2 denotes positive-definite (positive-semidefinite, negative, negative semidefinite) matrix. Let (Ω,𝔉,P) be a complete probability space with a natural filtration {𝔉t}t≥0 and E[·] stand for the correspondent expectation operator with respect to the given probability measure P. Also, let τ>0 and C([-τ,0];ℝn) denote the family of continuously differentiable function ϕ from [-τ,0] to ℝn with the uniform norm ∥ϕ∥τ=max{max-τ≤θ≤0|ϕ(θ)|,max-τ≤θ≤0|ϕ′(θ)|}.


## 2. Problem Description and Preliminaries

Fix a probability space(Ω,ℱ,𝒫), Ω is the sample space, ℱ is the σ-algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ, and consider the following Markov jump neural networks with mixed time-delays:
(1)x˙(t)=-C(r(t))x(t-σ(t))+A(r(t))g(x(t))+B(r(t))g(x(t-τ(t)))+D(r(t))∫t-d(t)t‍g(x(s))ds+u(t),y(t)=g(x(t)),
where x(t-σ(t))=[x1(t-σ(t))x2(t-σ(t))⋯xn(t-σ(t))]T and g(x(t))=[g1(x1(t))g2(x2(t))⋯gn(xn(t))]T,xi(t-σ(t)) are the state of the ith neuron at time t with leakage time varying delay and gi(xi(t)) denotes the neuron activation function; C(r(t))=diag{C1(r1(t))C2(r2(t))⋯Cn(rn(t))} is a diagonal matrix with positive entries; A(r(t))=(aij(r(t)))n×n,B(r(t))=(bij(r(t)))n×n and D(r(t))=(dij(r(t)))n×n, are, respectively, the connection weight matrix, the discretely delayed connection weight matrix, and the distributively delayed connection weight matrix; y(t) is the output of the neural network, and u(t)∈𝕃2[0,∞) is the output; τ(t) and d(t) denote the discrete delay and distributed delay, respectively, and the time varying delay τ(t) satisfies
(2)0≤τ(t)≤τ,0≤τ1≤τ(t)≤τ2,τ˙(t)≤τμ,0≤σ(t)≤σ,σ˙(t)≤σμ,0≤d(t)≤d,
where τ1,τ2,τμ,σμ,σ, and d are some real constants. By the simple transformation, model (1) has an equivalent form as follows:
(3)ddt[x(t)-C(r(t))∫t-σ(t)tx(s)ds]00=-C(r(t))x(t)-C(r(t))x(t-σ(t))σ˙(t)0000+A(r(t))g(x(t))+B(r(t))g(x(t-τ(t)))0000+D(r(t))∫t-d(t)tg(x(s))ds+u(t),y(t)=g(x(t)).
Here, {rt,t≥0} is a right continuous markov chain on the probability space taking values in a finite state space 𝒮={1,2,…,N} with transition rate matrix Π(ηt+h)≜{πij(ηt+h)} given by
(4)pr{rt+h=jrt=i}={πij(ηt+h)h+o(h),j≠i,1+πii(ηt+h)h+o(h),j=i,
in which h≥0, limh→0o(h)/h=0, and πij(ηt+h)≥0 for j≠i is the transition rate from mode i at time t to mode j at time t+h and πii(ηt+h)=-∑j=1,j≠iN‍πij(ηt+h).Similarly, the parameter{ηt,t≥0} is also a right continuous markov chain on the probability space taking values in a finite state space ℳ={1,2,…,T} with transition rate matrix Λ≜{pmn} given by
(5)pr{ηt+h=nηt=m}={pmnh+o(h),n≠m,1+pmnh+o(h),n=m,
in which h≥0, limh→0o(h)/h=0, and pmn≥0 for n≠m, are the transition rate from mode m at time t to mode n at time t+h and pmm=-∑n=m,n≠mT‍pmn.In this paper, we make the following assumption, definition, and lemmas for deriving the main result.Assumption 1.
Each activation functionfi(·) in (1) is continuous and bounded and satisfies
(6)Fi-≤gi(α1)-gi(α2)α1-α2≤Fi+,i=1,2,…,n,
where gi(0)=0,α1,α2∈ℛ,α1≠α2, and Fi- and Fi+ are known real scalars. It follows from (6) that the neural activation function satisfies
(7)Fi-≤gi(α)α≤Fi+,i=1,2,…,n.Lemma 2 (Jensen Inequality).
For any matrixM≥0, any scalars a and b with a≤b and a vector function x(t):[a,b]→ℛn such that the integrals concerned are well defined, the following inequality holds:
(8)(b-a)[∫ab‍x(s)TMx(s)ds]00≥[∫ab‍x(s)ds]TM[∫ab‍x(s)ds].Lemma 3.
For any constant matrixZ=ZT>0 and scalars σ>0,τ1>0,τ2>0 such that the following inequalities hold
(9)-∫-τ20∫t+θtx(s)TZx(s)dsdθ00≤-2τ22(∫-τ20∫t+θtx(s)dsdθ)TZ(∫-τ20∫t+θtx(s)dsdθ)-∫-τ2-τ1∫t+θtx(s)TZx(s)dsdθ00≤-2(τ22-τ12)(∫-τ2-τ1∫t+θtx(s)dsdθ)T0000×Z(∫-τ2-τ1∫t+θtx(s)dsdθ)-∫-σ0∫t+θtx(s)TZx(s)dsdθ00≤-2σ2(∫-σ0∫t+θtx(s)dsdθ)TZ(∫-σ0∫t+θtx(s)dsdθ).The main purpose of this paper is to establish a delay-dependent sufficient condition to ensure that neural networks (1) are passive.Definition 4.
The system (1) is said to be passive, if there exists a scalar ν≥0 such that for all tp≥0 and for all the solutions of (1), the following inequality
(10)2∫0tp‍E{y(s)Tu(s)}ds≥-γ∫0tp‍E{u(s)Tu(s)}ds
holds under zero initial conditions.


## 3. Main Results

In this section, we derive a new delay-dependent criterion for passivity of the delayed Markovian jumping neural networks (1) using the Lyapunov-Krasovskii functional method combining with LMI approach. For presentation convenience, in the following, we denote
(11)F1=diag{F1-F1+,F2-F2+,…,Fn-Fn+},F2=diag{F1-+F1+2,F2-+F2+2,…,Fn-+Fn+2}.
Now, we establish the following passivity condition for the system (1).Theorem 5.
The given Markovian jumping neural networks (1) is passive if there exist
(12)Pi,m>0,Q1i,m=[Q1i,m1Q1i,m2Q1i,m2TQ1i,m3]>0,Q2i,m=[Q2i,m1Q2i,m2Q2i,m2TQ2i,m3]>0,Q3i,m=[Q3i,m1Q3i,m2Q3i,m2TQ3i,m3]>0,Q4>0,U=[U1U2U2TU3]>0,
positive symmetric matrices S1=S1T>0,  S2=S2T>0,  S3=S3T>0,  T1=T1T>0,  T2=T2T>0,T3=T3T>0; the positive definite matrices W1>0,W2>0 the diagonal matrices Λi,m1>0,Λi,m2>0,Λi,m3>0,  Λi,m4>0,Λi,m5>0,  and a scalar γ>0 such that for any (i,m)∈(𝒮,ℳ) the following LMI holds:
(13)Ξ=(Ξi,j)17×17<0,∑n∈ℳ‍pmnQ1i,n+∑j∈𝒮‍πijmQ1j,m+∑n∈ℳ‍pmnQ3i,n0000+∑j∈𝒮‍πijmQ3j,m<U,∑n∈ℳ‍pmnQ2i,n+∑j∈𝒮‍πijmQ2j,m<U,∑n∈ℳ‍pmnQ3i,n+∑j∈𝒮‍πijmQ3j,m<U,
where
(14)Ξ1,1=-Pi,mCi-CiTpi,m+Q1i,m1+Q2i,m1+τ1U1+Q31+τ2U1+Q4-(τ2-τ1)S1-S3-4τ22T1-4(τ2-τ1)2T2-4σ2T3-F1Λi,m1+[∑n∈ℳ‍pmnPi,n+∑j∈𝒮‍πijmPj,m]+σ2W2,Ξ1,2=Pi,mAi+Q1i,m2+Q2i,m2+τ1U2+Q3i,m2+τ2U2+F2Λi,m1,Ξ1,3=0,Ξ1,4=pi,mBi,Ξ1,5=0,Ξ1,6=0,Ξ1,7=(τ2-τ1)S1,Ξ1,8=0,Ξ1,9=-Pi,mCiσμ,Ξ1,10=0,Ξ1,11=CiTPi,mCi-[∑n∈ℳ‍pmnPi,n+∑j∈𝒮‍πijmPj,m]Ci,Ξ1,12=S3,Ξ1,13=4σT3,Ξ1,14=Pi,mDi,Ξ1,15=4τ2T1,Ξ1,16=4(τ2-τ1)T2,Ξ1,17=Pi,m,Ξ2,2=Q1i,m3+Q2i,m3+τ1U3+Q3i,m3+τ2U3+AiTRAi+d2W1+AiTGAi-Λi,m1,Ξ2,3=0,Ξ2,4=AiTRBi+AiTGBi,Ξ2,5=Ξ2,6=Ξ2,7=Ξ2,8=0,Ξ2,9=-AiTRCi-AiTGCi,Ξ2,10=0,Ξ2,11=-AiTPi,mCi,Ξ2,12=0,Ξ2,13=0,Ξ2,14=AiTRDi+AiTGDi,Ξ2,15=0,Ξ2,16=0,Ξ2,17=AiTR+AiTG-I,Ξ3,3=-(1-τμ)Q1i,m1-F1Λi,m2,Ξ3,4=-(1-τμ)Q1i,m2+F2Λi,m2,Ξ3,5=Ξ3,6=Ξ3,7=Ξ3,8=Ξ3,9=Ξ3,10=Ξ3,11=Ξ3,12=Ξ3,13=Ξ3,14=Ξ3,15=0,Ξ3,16=Ξ3,17=0,Ξ4,4=-(1-τμ)Q1i,m3+BiTRBi+BiTGBi-Λi,m2,Ξ4,5=0,Ξ4,6=Ξ4,7=Ξ4,8=0,Ξ4,9=-BiTRCi-BiTGCi,Ξ4,10=0,Ξ4,11=-BiTPi,mCi,Ξ4,12=Ξ4,13=0,Ξ4,14=BiTRDi+BiTGDi,Ξ4,15=Ξ4,16=0,Ξ4,17=BiTR+BiTG,Ξ5,5=-Q2i,m1-S2-F1Λi,m3,Ξ5,6=-Q2i,m2+F2Λi,m3,Ξ5,7=S2,Ξ6,6=-Q2i,m3-Λi,m3,Ξ7,7=-Q3i,m1-(τ2-τ1)S1-S2-F1Λi,m4,Ξ7,8=-Q3i,m2+F2Λi,m4,Ξ8,8=-Q3i,m3-Λi,m4,Ξ9,9=-(1-σμ)Q4+CiTRCi+CiTGCi-F1Λi,m5,Ξ9,10=F2Λi,m5,Ξ9,11=CiTσμPi,mCi,Ξ9,14=-CiTRDi-CiTGDi,Ξ9,17=-CiTR-CiTG,Ξ10,10=-Λi,m5,Ξ11,11=CiT[∑n∈ℳ‍pmnPi,n+∑j∈𝒮‍πijmPj,m]Ci-W2,Ξ11,14=-CiTPi,mDi,Ξ11,17=-CiTPi,m,Ξ12,12=-S3,Ξ13,13=-4T3,Ξ14,14=DiTRDi+DiTGDi-W1,Ξ14,17=DiTR+DiTG,Ξ15,15=-4T1,Ξ16,16=-4T2,Ξ17,17=R+G-γI,R=τ22(τ2-τ1)S1+(τ2-τ1)2S2+σ2S3,G=τ24T1+(τ22-τ12)2T2+σ4T3,
and the remaining coefficients are all zero.Proof.
Denoteζ=[x(t)Tg(x(t))T]T and consider the following Lyapunov-Krasovskii functional for neural network (1):
(15)V(xt,rt,ηt)=V1(xt,rt,ηt)+V2(xt,rt,ηt)+V3(xt,rt,ηt)+V4(xt,rt,ηt)+V5(xt,rt,ηt),
where
(16)V1(xt,rt,ηt)=[x(t)-Ci∫t-σ(t)tx(s)ds]T00×Pr(t),η(t)[x(t)-Ci∫t-σ(t)tx(s)ds],V2(xt,rt,ηt)=∫t-τ(t)t‍ζ(s)TQ1r(t),η(t)ζ(s)ds+∫t-τ1t‍ζ(s)TQ2r(t),η(t)ζ(s)ds+∫t-τ2t‍ζ(s)TQ3r(t),η(t)ζ(s)ds+∫t-σ(t)t‍x(s)TQ4x(s)ds+∫-τ10‍∫t+θt‍ζ(s)TUζ(s)dsdθ+∫-τ20‍∫t+θt‍ζ(s)TUζ(s)dsdθ,V3(xt,rt,ηt)=τ2(τ2-τ1)+∫-τ20‍∫t+θt‍x˙(s)TS1x˙(s)dsdθ+(τ2-τ1)∫-τ2-τ1‍∫t+θt‍x˙(s)TS2x˙(s)dsdθ+σ∫-σ0‍∫t+θt‍x˙(s)TS3x˙(s)dsdθ,V4(xt,rt,ηt)=∫-d0‍∫t+θt‍g(x(s))TW1g(x(s))dsdθ+σ∫t-σt‍∫θt‍x(s)TW2x(s)dsdθ,V5(xt,rt,ηt)=2τ22∫-τ20‍∫θ0‍∫t+λt‍x˙(s)TT1x˙(s)dsdλdθ+2(τ22-τ12)∫-τ1-τ2‍∫θ0‍∫t+λt‍x˙(s)T000000000000000000000×T2x˙(s)dsdλdθ+2σ2∫-σ0‍∫θ0‍∫t+λt‍x˙(s)TT3x˙(s)dsdλdθ.
Define infinitesimal generator (denoted by 𝕃) of the markov process acting on V(xt,rt,ηt) as follows:
(17)𝕃V(xt,rt,ηt)=limh→01h{E{V(xt+h,rt+h,ηt+h)xt,rt=i,ηt=m}000000000000-V(xt,rt=i,ηt=m){V(xt+h,rt+h,ηt+h)xt,rt=i,ηt=m}}.
It can be calculated that
(18)𝕃V(xt,rt,ηt)00=limh→01h{∑n∈ℳ,n≠m‍pmnh[∑j∈𝒮,j≠i‍πijnhV(xt+h,j,n)00000000000000000000000+(1+πiinh)V(xt+h,i,n)∑j∈𝒮,j≠i‍]00000000000+(1+pmmh)[∑j∈𝒮,j≠i‍πijmhV(xt+h,j,m)00000000000000000000000000+(1+πiimh)V(xt+h,i,m)∑j∈𝒮,j≠i‍]00000000000-V(xt,i,m)+(1+πiimh)V(xt+h,i,m)∑j∈𝒮,j≠i‍]}00=limh→0{+(1+πiimh)V(xt+h,i,m)∑j∈𝒮,j≠i‍]∑n∈ℳ,n≠m‍pmnV(xt+h,i,n)+pmmV(xt+h,i,m)0000000000+1h[∑j∈𝒮,j≠i‍πijmhV(xt+h,j,m)000000000000000+(1+πiimh)V(xt+h,i,m)∑j∈𝒮,j≠i‍]0000000000-1hV(xt,i,m)+(1+πiimh)V(xt+h,i,m)∑j∈𝒮,j≠i‍]}00=limh→0{∑n∈ℳ‍pmnV(xt+h,i,n)+∑j∈𝒮‍πijmV(xt+h,j,m)000000000+1h[V(xt+h,i,m)-V(xt,i,m)]∑n∈ℳ‍}00=∑n∈ℳ‍pmnV(xt,i,n)+∑j∈𝒮‍πijmV(xt,j,m)+V˙(xt,i,m).
From (15), it can be seen that
(19)𝕃V(xt,rt,ηt)=𝕃V1(xt,rt,ηt)+𝕃V2(xt,rt,ηt)+𝕃V3(xt,rt,ηt)+𝕃V4(xt,rt,ηt)+𝕃V5(xt,rt,ηt).
Based on the above equation, along the solution of the neural network (3), we obtain that for each (i,m)∈𝒮×ℳ(20)𝕃V1(xt,rt,ηt)00=[x(t)-Ci∫t-σ(t)t‍x(s)ds]T0000×Pi,mddt[x(t)-Ci∫t-σ(t)t‍x(s)ds]0000+ddt[x(t)-Ci∫t-σ(t)t‍x(s)ds]T0000×Pi,m[x(t)-Ci∫t-σ(t)t‍x(s)ds]0000+[x(t)-Ci∫t-σ(t)t‍x(s)ds]T0000×[∑n∈ℳ‍pmnPi,n+∑j∈𝒮‍πijmPj,m]0000×[x(t)-Ci∫t-σ(t)t‍x(s)ds],𝕃V2(xt,rt,ηt)00≤ζT(t)Q1i,mζ(t)-ζT(t-τ(t))Q1i,mζ(t-τ(t))(1-τμ)0000+∫t-τ(t)t‍ζT(s)[∑n∈ℳ‍pmnQ1i,m+∑j∈𝒮‍πijmQ1j,n]ζ(s)ds0000+ζT(t)Q2i,mζ(t)-ζT(t-τ1)Q2i,mζ(t-τ1)0000+∫t-τ1t‍ζT(s)[∑n∈ℳ‍pmnQ2i,m+∑j∈𝒮‍πijmQ2j,n]ζ(s)ds0000+ζT(t)Q3i,mζ(t)-ζT(t-τ2)Q3i,mζ(t-τ2)0000+∫t-τ2t-τ(t)‍ζT(s)[∑n∈ℳ‍pmnQ3i,m+∑j∈𝒮‍πijmQ3j,n]ζ(s)ds0000+∫t-τ(t)t‍ζT(s)[∑n∈ℳ‍pmnQ3i,m+∑j∈𝒮‍πijmQ3j,n]ζ(s)ds0000+xT(t)Q4x(t)-xT(t-σ(t))Q4x(t-σ(t))(1-σμ)0000+τ1ζT(t)Uζ(t)-∫t-τ1t‍ζT(s)Uζ(s)ds+τ2ζT(t)Uζ(t)0000-∫t-τ2t-τ(t)‍ζT(s)Uζ(s)ds-∫t-τ(t)t‍ζT(s)Uζ(s)ds,𝕃V3(xt,rt,ηt)00=τ22(τ2-τ1)x˙(t)TS1x˙(t)0000-τ2(τ2-τ1)∫-τ20‍x˙(t+θ)TS1x˙(t+θ)dθ0000+(τ2-τ1)2x˙(t)TS2x˙(t)0000-(τ2-τ1)∫-τ2-τ1‍x˙(t+θ)TS2x˙(t+θ)dθ0000+σ2x˙(t)TS3x˙(t)-σ∫-σ0‍x˙(t+θ)TS3x˙(t+θ)dθ00=x˙(t)T[τ22(τ2-τ1)S1+(τ2-τ1)2S2+σ2S3]x˙(t)0000-τ2(τ2-τ1)∫t-τ2t‍x˙(s)TS1x˙(s)ds0000-(τ2-τ1)∫t-τ2t-τ1‍x˙(s)TS2x˙(s)ds0000-σ∫t-σt‍x˙(s)TS3x˙(s)ds,𝕃V4(xt,rt,ηt)00=d2g(x(t))TW1g(x(t))0000-d(t)∫t-d(t)t‍g(x(s))TW1g(x(s))ds0000+σ2x(t)TW2x(t)-σ(t)∫t-σ(t)t‍x(s)TW1x(s)ds,𝕃V5(xt,rt,ηt)00=x˙(t)T[τ24T1+(τ22-τ12)2T2+σ4T3]x˙(t)0000-2τ22∫-τ20‍∫t+θt‍x˙(s)TT1x˙(s)dsdθ0000-(τ22-τ12)∫-τ2-τ1‍∫t+θt‍x˙(s)TT2x˙(s)dsdθ0000-2σ2∫-σ0‍∫t+θt‍x˙(s)TT3x˙(s)dsdθ.
Moreover, based on Lemma 2, we can get the following inequalities:
(21)-(τ2-τ1)∫t-τ2t-τ1‍x˙(s)TS2x˙(s)ds00≤-[∫t-τ2t-τ1‍x˙(s)ds]TS2[∫t-τ2t-τ1‍x˙(s)ds],(22)-d∫t-dt‍g(x(s))TW1g(x(s))ds00≤-[∫t-d(t)t‍g(x(s))ds]TW1[∫t-d(t)t‍g(x(s))ds].
By using Lemma 3, we can also get that
(23)-2τ22∫-τ20‍∫t+θt‍x˙(s)TZx˙(s)dsdθ00≤-4(∫-τ20‍∫t+θt‍x˙(s)dsdθ)TZ(∫-τ20‍∫t+θt‍x˙(s)dsdθ).
Similarly, we can use Lemmas 2 and 3 for other integrals. On the other hand, we have from (6) that for any λ=1,2,…,n,
(24)(gλ(xλ(t))-Fλ-xλ(t))(gλ(xλ(t))-Fλ+xλ(t))≤0,
which is equivalent to
(25)ζT(t)[Fλ+Fλ-e^λe^λT-Fλ++Fλ-2e^λe^λT-Fλ++Fλ-2e^λe^λTe^λe^λT]ζ(t)≤0,
where e^λ denotes the unit column vector having 1 element on its λth row and zeros elsewhere. Thus, for any appropriately dimensioned diagonal matrix Λi,m1>0, the following inequality holds:
(26)0≤ζT(t)[-F1Λi,m1F2Λi,m1*-Λi,m1]ζ(t).
Similarly, for any appropriately dimensioned diagonal matrices Λi,m2>0,Λi,m3>0,Λi,m4>0, and Λi,m5>0, the following inequalities also hold:
(27)0≤ζT(t-τ(t))[-F1Λi,m2F2Λi,m2*-Λi,m2]ζ(t-τ(t)),0≤ζT(t-τ1)[-F1Λi,m3F2Λi,m3*-Λi,m3]ζ(t-τ1),0≤ζT(t-τ2)[-F1Λi,m4F2Λi,m4*-Λi,m4]ζ(t-τ2),0≤ζT(t-σ(t))[-F1Λi,m5F2Λi,m5*-Λi,m5]ζ(t-σ(t)).
Using inequalities (20)–(23) in (19) and adding (26)–(27) in (19), we get
(28)𝕃V(xt,rt,ηt)-2yT(t)u(t)-γuT(t)u(t)≤ρT(t)Ξρ(t),
where ρ(t)=[ρ1T(t)ρ2T(t)ρ3T(t)] with
(29)ρ1(t)=[x(t)g(x(t))x(t-τ(t))g(x(t-τ(t)))x(t-τ1)g(x(t-τ1))],ρ2(t)=[∫t-σ(t)t‍x(t-τ2)g(x(t-τ2))x(t-σ(t))g(x(t-σ(t)))∫t-σ(t)t‍x(s)dsx(t-σ)],ρ3(t)=[∫t-σt‍x(s)ds∫t-d(t)t‍g(x(s))ds∫t-τ2t‍x(s)ds∫t-τ2t-τ1‍x(s)dsu(t)].
Hence we can obtain from (10) that,
(30)𝕃V(xt,rt,ηt)-2y(t)Tu(t)-γu(t)Tu(t)≤0.
Now, to show the passivity of the delayed neural networks in (1), we set
(31)J(tp)=𝔼{∫0tp‍[-γu(t)Tu(t)-2y(t)Tu(t)]dt},
where tp≥0.
Using Dynkin’s formula, we have(32)𝔼[∫0tp‍𝕃V(xt,rt,ηt)dt]=𝔼[V(xtp,rtp,ηtp)]-𝔼[V(x0,r0,η0)].
Now, we can deduce that
(33)J(tp)00=𝔼{∫0tp‍[-γu(t)Tu(t)-2y(t)Tu(t)+𝕃V(xt,rt,ηt)]dt}0000-𝔼[∫0tp‍𝕃V(xt,rt,ηt)dt]00=𝔼{∫0tp‍[-γu(t)Tu(t)-2y(t)Tu(t)+𝕃V(xt,rt,ηt)]dt}0000-𝔼[V(xt,rt,ηt)]+𝔼[V(x0,r0,η0)].
Thus, if (33) holds, then since 𝔼[V(xtp,rtp,ηtp)]≥0 and V(x0,r0,η0)=0 holds under zero initial condition, from (31) it follows that J(tp)≤0 for any tp≥0, which implies that (13) is satisfied and therefore the delayed neural networks (1) are locally passive. Next we shall prove that 𝔼[∥x(t)∥2]→0 as t→∞. Taking expectation on both sides of (28) and integrating from 0 to t we have
(34)∫0t‍𝔼[𝕃V(xs,rs,ηs)]ds-2∫0t‍𝔼[yT(s)u(s)]ds0000-γ∫0t‍𝔼[uT(s)u(s)]ds00≤∫0t‍𝔼[ρT(s)Ξρ(s)]ds.
By using Dynkin’s formula, we have
(35)𝔼[𝕃V(xt,rt,ηt)]-𝔼[𝕃V(x0,r0,η0)]0000-2∫0t‍𝔼[yT(s)u(s)]ds-γ∫0t‍𝔼[uT(s)u(s)]ds00≤∫0t‍𝔼[ρT(s)Ξρ(s)]ds.
Hence
(36)𝔼[𝕃V(xt,rt,ηt)]-∫0t‍𝔼[ρT(s)Ξρ(s)]ds00≤𝔼[𝕃V(x0,r0,η0)]+2∫0t‍𝔼[yT(s)u(s)]ds0000+γ∫0t‍𝔼[uT(s)u(s)]ds<∞,00t≥0.
Using Jenson’s inequality and (36), we have
(37)E∥Ci∫t-σ(t)t‍x(s)ds∥200=E[Ci∫t-σ(t)t‍x(s)ds]T[Ci∫t-σ(t)t‍x(s)ds]00≤λmax(Ci2)E[∫t-σ(t)t‍x(s)ds]T[∫t-σ(t)t‍x(s)ds]00≤λmax(Ci2)λmin(Q4)[∫t-σ(t)t‍Ex(s)ds]TQ4[∫t-σ(t)t‍Ex(s)ds]00≤σ(t)λmax(Ci2)λmin(Q4){∫t-σ(t)t‍ExT(s)Q4x(s)ds}00≤σλmax(Ci2)λmin(Q4){∫t-σt‍ExT(s)Q4x(s)ds}00≤σλmax(Ci2)λmin(Q4)EV1(xt,rt,ηt)00≤σλmax(Ci2)λmin(Q4)EV(xt,rt,ηt)00≤σλmax(Ci2)λmin(Q4)EV(x0,r0,η0),00t≥0.
Similarly, it follows from the definition of V1(xt,rt,ηt) that
(38)E∥x(t)-Ci∫t-σ(t)t‍x(s)ds∥200=E[Ci∫t-σ(t)t‍x(s)ds]T[Ci∫t-σ(t)t‍x(s)ds]00≤EV1(xt,rt,ηt)λmin(Pi,η(t))00≤EV(xt,rt,ηt)λmin(Pi,η(t))00≤EV(x0,r0,η0)λmin(Pi,η(t)),00t≥0.
Hence, it can be obtained that
(39)E∥x(t)∥2=E∥x(t)-Ci∫t-σ(t)t‍x(s)ds+Ci∫t-σ(t)t‍x(s)ds∥2≤2E∥Ci∫t-σ(t)t‍x(s)ds∥2+2E∥x(t)-Ci∫t-σ(t)t‍x(s)ds∥2≤2σλmax(Ci2)λmin(Q4)EV(x0,r0,η0)+2EV(x0,r0,η0)λmin(Pi,η(t))<∞,t≥0,
where
(40)EV(x0,r0,η0)00=E[x(0)-C(r(0))∫-σ(0)0‍x(s)ds]T0000×Pr(0),η(0)[x(0)-C(r(0))∫-σ(0)0‍x(s)ds]0000+∫-τ(0)0‍ζ(s)TQ1r(0),η(0)ζ(s)ds0000+∫-τ10‍ζ(s)TQ2r(0),η(0)ζ(s)ds0000+∫-τ20‍ζ(s)TQ3r(0),η(0)ζ(s)ds0000+∫-σ(0)0‍x(s)TQ4x(s)ds0000+∫-τ10‍∫θ0‍ζ(s)TUζ(s)dsdθ0000+∫-τ20‍∫θ0‍ζ(s)TUζ(s)dsdθ0000+τ2(τ2-τ1)∫-τ20‍∫θ0‍x˙(s)TS1x˙(s)dsdθ0000+(τ2-τ1)∫-τ2-τ1‍∫θ0‍x˙(s)TS2x˙(s)dsdθ0000+σ∫-σ0‍∫θ0‍x˙(s)TS3x˙(s)dsdθ0000+∫-d0‍∫θ0‍g(x(s))TW1g(x(s))dsdθ0000+σ∫-σ0‍∫θt‍x(s)TW2x(s)dsdθ0000+2τ22∫-τ20‍∫θ0‍∫λ0‍x˙(s)TT1x˙(s)dsdλdθ0000+2(τ22-τ12)∫-τ1-τ2‍∫θ0‍∫λ0‍x˙(s)TT2x˙(s)dsdλdθ0000+2σ2∫-σ0‍∫θ0‍∫λ0‍x˙(s)TT3x˙(s)dsdλdθ00≤{2λmaxi∈S(Pi,η(t))(1+σ2maxi∈SCi)00000+τλmax(Q1r(t),η(t))00000+τ1λmax(Q2r(t),η(t))+τ2λmax(Q3r(t),η(t))00000+σλmax(Q4)00000+τ12λmax(U)τ12λmax(U)00000+τ23(τ2-τ1)λmax(S1)00000+(τ2-τ1)3λmax(S2)+σ3λmax(S3)00000+d2λmax(W1)00000+σ3λmax(W2)+2τ24λmax(T1)00000+2(τ22-τ12)(τ1-τ2)2λmax(T2)00000+2σ4λmax(T3)(1+σ2maxi∈SCi)}<∞.
From (39) and (40), it can be deduced that the trivial solution of system (1) is locally passive. Then the solutions x(t)=x(t,0,ϕ) of system (1) is bounded on [0,∞). considering (1), we know that x˙(t) is bounded on [0,∞), which leads to the uniform continuity of the solution x(t) on [0,∞). From (36), we note that the following inequality holds:
(41)λmin(Ξ)∫0t‍E[xT(s)x(s)]ds00≤𝔼[𝕃V(xt,rt,ηt)]0000-∫0t‍𝔼[ρT(s)Ξρ(s)]ds00≤𝔼[𝕃V(x0,r0,η0)]+2∫0t‍𝔼[yT(s)u(s)]ds0000+γ∫0t‍𝔼[uT(s)u(s)]ds<∞,t≥0.
By Barbalats’ lemma [38], it holds that 𝔼[∥x(t)∥2]→0 as t→∞ and this completes the proof of the global passivity of the system (1).Remark 6.
Whenσ(t)=σ, the system (1) becomes
(42)x˙(t)=-C(r(t))x(t-σ(t))+A(r(t))g(x(t))+B(r(t))g(x(t-τ(t)))+D(r(t))∫t-d(t)t‍g(x(s))ds+u(t),y(t)=g(x(t)).
The system (42) can be written in its equivalent form as follows:
(43)ddt[x(t)-C(r(t))∫t-σt‍x(s)ds]00=-C(r(t))x(t)+A(r(t))g(x(t))0000+B(r(t))g(x(t-τ(t)))+D(r(t))0000×∫t-d(t)t‍g(x(s))ds+u(t),y(t)=g(x(t)).
The time varying delay τ(t) satisfies
(44)0≤τ1≤τ(t)≤τ2,00τ˙(t)≤τμ,000≤d(t)≤d,
where τ1,τ2,τμ,d are some constants and the leakage delay σ≥0 is a constant.
Now, the passivity condition for the neural networks (43) is given in the following corollary and the result follows from Theorem 5.Corollary 7.
Neural networks (43) are passive if there exist
(45)Pi,m>0,0000Q1i,m=[Q1i,m1Q1i,m2Q1i,m2TQ1i,m3]>0,Q2i,m=[Q2i,m1Q2i,m2Q2i,m2TQ2i,m3]>0,0000Q3i,m=[Q3i,m1Q3i,m2Q3i,m2TQ3i,m3]>0,Q4>0,0000U=[U1U2U2TU3]>0,
positive symmetric matrices S1=S1T>0,S2=S2T>0,S3=S3T>0,T1=T1T>0,T2=T2T>0,T3=T3T>0; the positive definite matrices W1>0,W2>0; the diagonal matrices Λi,m1>0,Λi,m2>0,Λi,m3>0,Λi,m4>0,Λi,m5>0;  and a scalar γ>0 such that for any (i,m)∈(𝒮,ℳ) the following LMI holds:
(46)Ξ=(Ξi,j)15×15<0,∑n∈ℳ‍pmnQ1i,n+∑j∈𝒮‍πijmQ1j,m+∑n∈ℳ‍pmnQ3i,n+∑j∈𝒮‍πijmQ3j,m<U,∑n∈ℳ‍pmnQ2i,n+∑j∈𝒮‍πijmQ2j,m<U,∑n∈ℳ‍pmnQ3i,n+∑j∈𝒮‍πijmQ3j,m<U,
where
(47)Ξ1,1=-Pi,mCi-CiTPi,m+Q1i,m1+Q2i,m1+τ1U1+Q3i,m1+τ2U1+Q4-(τ2-τ1)S1-S3-4τ22T1-4(τ2-τ1)2T2-4σ2T3-F1Λi,m1+[∑n∈ℳ‍pmnPi,n+∑j∈𝒮‍πijmPj,m]+σ2W2,Ξ1,2=Pi,mAi+Q1i,m2+Q2i,m2+τ1U2+Q3i,m2+τ2U2+F2Λi,m1,Ξ1,4=Pi,mBi,0000Ξ1,7=(τ2-τ1)S1,0000Ξ1,9=S3,Ξ1,11=CiTPi,mCi-[∑n∈ℳ‍pmnPi,n+∑j∈𝒮‍πijmPj,m]Ci+4σT3,Ξ1,12=Pi,mDi,0000Ξ1,13=4τ2T1,Ξ1,14=4(τ2-τ1)T2,0000Ξ1,15=Pi,m,Ξ2,2=Q1i,m3+Q2i,m3+τ1U3+Q3i,m3+τ2U3+AiTRAi+d2W1+AiTGAi-Λi,m1,Ξ2,4=AiTRBi+AiTGBi,0000Ξ2,9=-AiTRCi-AiTGCi,Ξ2,11=-AiTPi,mCi,0000Ξ2,12=AiTRDi+AiTGDi,Ξ2,15=AiTR+AiTG-I,Ξ3,3=-(1-τμ)Q1i,m1-F1Λi,m2,Ξ3,4=-(1-τμ)Q1i,m2+F2Λi,m2,Ξ4,4=-(1-τμ)Q1i,m3+BiTRBi+BiTGBi-Λi,m2,Ξ4,9=-BiTRCi-BiTGCi,Ξ4,11=-BiTPi,mCi,Ξ4,12=BiTRDi+BiTGDi,Ξ4,15=BiTR+BiTG,Ξ5,5=-Q2i,m1-S2-F1Λi,m3,Ξ5,6=-Q2i,m2+F2Λi,m3,Ξ5,7=S2,Ξ6,6=-Q2i,m3-Λi,m3,Ξ7,7=-Q3i,m1-(τ2-τ1)S1-S2-F1Λi,m4,Ξ7,8=-Q3i,m2+F2Λi,m4,Ξ8,8=-Q3i,m3-Λi,m4,Ξ9,9=-Q4+CiTRCi+CiTGCi-S3-F1Λi,m5,Ξ9,10=F2Λi,m5,0000Ξ9,12=-CiTRDi-CiTGDi,Ξ9,15=-CiTR-CiTG,0000Ξ10,10=-Λi,m5,Ξ11,11=CiT[∑n∈ℳ‍pmnPi,n+∑j∈𝒮‍πijmPj,m]Ci-W2-4T3,Ξ11,12=-CiTpi,mDi,0000Ξ11,15=-CiTpi,m,Ξ12,12=-DiTRDi-DiTGDi-W1,Ξ12,15=DiTR+DiTG,Ξ13,13=-4T1,Ξ14,14=-4T1,Ξ15,15=R+G-γI,R=τ22(τ2-τ1)S1+(τ2-τ1)2S2+σ2S3,G=τ24T1+(τ22-τ12)2T2+σ4T3,
and the remaining coefficients are all zero.Proof.
We can define the Lyapunov functional for the above neural networks as in Theorem5 by replacing σ(t) by σ. The proof is the same as that of Theorem 5, and hence it is omitted.


## 4. Problem without Switching

### 4.1. Description and Preliminaries

In this section, we derive passivity criterion for the delayed neural networks using the Lyapunov-Krasovskii functional without Markovian jumping parameters.Consider the following neural networks with mixed time-delays:(48)x˙(t)=-Cx(t-σ(t))+Ag(x(t))+Bg(x(t-τ(t)))+D∫t-d(t)t‍g(x(s))ds+u(t),y(t)=g(x(t)).
Or, it has an equivalent form as follows:
(49)ddt[x(t)-C∫t-σ(t)t‍x(s)ds]00=-Cx(t)-Cx(t-σ(t))σ˙(t)+Ag(x(t))0000+Bg(x(t-τ(t)))0000+D∫t-d(t)t‍g(x(s))ds+u(t),y(t)=g(x(t)).
Now, we establish the following passivity condition for the system (49).Theorem 8.
Neural network (49) is passive if there exist
(50)P>0,Q1=[Q11Q12Q12TQ13]>0,Q2=[Q21Q22Q22TQ23]>0,Q3=[Q31Q32Q32TQ33]>0,Q4>0,U=[U1U2U2TU3]>0,
positive symmetric matrices S1=S1T>0,S2=S2T>0,S3=S3T>0, T1=T1T>0,T2=T2T>0, T3=T3T>0; the positive definite matrices W1>0,W2>0; the diagonal matrices Λ1>0,Λ2>0,Λ3>0,Λ4>0,Λ5>0;  and a scalar γ>0 such that the following LMI holds:
(51)Ξ=(Ξi,j)20×20<0,
where
(52)Ξ1,1=-PC-CTp+Q11+Q21+τ1U1+Q31+τ2U1+Q4-(τ2-τ1)S1-S3-4τ22T1-4(τ2-τ1)2T2-4σ2T3-F1Λ1+σ2W2,Ξ1,2=PA+Q12+Q22+τ1U2+Q32+τ2U2+F2Λ1,Ξ1,4=pB,Ξ1,7=(τ2-τ1)S1,Ξ1,9=-PCσμ,Ξ1,11=CTPC,Ξ1,12=S3,Ξ1,13=4σT3,Ξ1,14=PDi,Ξ1,17=4τ2T1,Ξ1,19=4(τ2-τ1)T2,Ξ1,20=P,Ξ2,2=Q13+Q23+τ1U3+Q33+τ2U3+ATRA+d2W1+ATGA-Λ1,Ξ2,4=ATRB+ATGB,Ξ2,9=-ATRC-ATGC,Ξ2,11=-ATPC,Ξ2,14=ATRD+ATGD,Ξ2,20=ATR+ATG-I,Ξ3,3=-(1-τμ)Q11-F1Λ2,Ξ3,4=-(1-τμ)Q12+F2Λ2,Ξ4,4=-(1-τμ)Q13+BTRB+BTGB-Λ2,Ξ4,9=-BTRC-BTGC,Ξ4,11=-BTPC,Ξ4,14=BTRD+BTGD,Ξ4,20=BTR+BTG,Ξ5,5=-Q21-S2-F1Λ3,Ξ5,6=-Q22+F2Λ3,Ξ5,7=S2,Ξ6,6=-Q23-Λ3,Ξ7,7=-Q31-(τ2-τ1)S1-S2-F1Λ4,Ξ7,8=-Q32+F2Λ4,Ξ8,8=-Q33-Λ4,Ξ9,9=-(1-σμ)Q4+CTRC+CTGC-F1Λ5,Ξ9,10=F2Λ5,Ξ9,11=CTσμPC,Ξ9,14=-CTRD-CTGD,Ξ9,20=-CTR-CTG,Ξ10,10=-Λ5,Ξ11,11=-W2,Ξ11,14=-CTPD,Ξ11,20=-CTP,Ξ12,12=-S3,Ξ13,13=-4T3,Ξ14,14=DTRD+DTGD-W1,Ξ14,20=DTR+DTG,Ξ15,15=-1τ1U1,Ξ15,16=-1τ1U2,Ξ16,16=-1τ1U3,Ξ17,17=-1τ2U1-4T1,Ξ17,18=-1τ2U2,Ξ18,18=-1τ2U3,Ξ19,19=-4T2,Ξ20,20=R+G-γI,R=τ22(τ2-τ1)S1+(τ2-τ1)2S2+σ2S3,G=τ24T1+(τ22-τ12)2T2+σ4T3,
and the remaining coefficients are all zero.Proof.
Denoteζ=[x(t)Tg(x(t))T]T and consider the following Lyapunov-Krasovskii functional for neural network (49):
(53)V(xt)=V1(xt)+V2(xt)+V3(xt)+V4(xt)+V5(xt),
where
(54)V1(xt)=[x(t)-Ci∫t-σ(t)t‍x(s)ds]T×P[x(t)-Ci∫t-σ(t)t‍x(s)ds],V2(xt)=∫t-τ(t)t‍ζ(s)TQ1ζ(s)ds+∫t-τ1t‍ζ(s)TQ2ζ(s)ds+∫t-τ2t‍ζ(s)TQ3ζ(s)ds+∫t-σ(t)t‍x(s)TQ4x(s)ds+∫-τ10‍∫t+θt‍ζ(s)TUζ(s)dsdθ+∫-τ20‍∫t+θt‍ζ(s)TUζ(s)dsdθ,V3(xt)=τ2(τ2-τ1)+∫-τ20‍∫t+θt‍x˙(s)TS1x˙(s)dsdθ+(τ2-τ1)∫-τ2-τ1‍∫t+θt‍x˙(s)TS2x˙(s)dsdθ+σ∫-σ0‍∫t+θt‍x˙(s)TS3x˙(s)dsdθ,V4(xt)=∫-d0‍∫t+θt‍g(x(s))TW1g(x(s))dsdθ+σ∫t-σt‍∫θt‍x(s)TW2x(s)dsdθ,V5(xt)=2τ22∫-τ20‍∫θ0‍∫t+λt‍x˙(s)TT1x˙(s)dsdλdθ+2(τ22-τ12)∫-τ1-τ2‍∫θ0‍∫t+λt‍x˙(s)TT2x˙(s)dsdλdθ+2σ2∫-σ0‍∫θ0‍∫t+λt‍x˙(s)TT3x˙(s)dsdλdθ.
Taking time derivative acting on V(xt) along the neural networks (49) is defined as follows:
(55)V˙1(xt)=[x(t)-C∫t-σ(t)t‍x(s)ds]T×Pddt[x(t)-C∫t-σ(t)t‍x(s)ds]+ddt[x(t)-C∫t-σ(t)t‍x(s)ds]T×P[x(t)-C∫t-σ(t)t‍x(s)ds],V˙2(xt)≤ζT(t)Q1ζ(t)-ζT(t-τ(t))Q1ζ(t-τ(t))×(1-τμ)+ζT(t)Q2ζ(t)-ζT(t-τ1)Q2ζ(t-τ1)+ζT(t)Q3ζ(t)-ζT(t-τ2)Q3ζ(t-τ2)+xT(t)Q4x(t)-xT(t-σ(t))Q4x(t-σ(t))(1-σμ)+∫-τ10‍ζT(t)Uζ(t)dθ-∫t-τ1t‍ζT(s)Uζ(s)ds+∫-τ20‍ζT(t)Uζ(t)dθ-∫t-τ2t‍ζT(s)Uζ(s)ds,V˙3(xt)=x˙(t)T[τ22(τ2-τ1)S1+(τ2-τ1)2S2+σ2S3]x˙(t)-τ2(τ2-τ1)∫t-τ2t‍x˙(s)TS1x˙(s)ds-(τ2-τ1)∫t-τ2t-τ1‍x˙(s)TS2x˙(s)ds-σ∫t-σt‍x˙(s)TS3x˙(s)ds,V˙4(xt)=d2g(x(t))TW1g(x(t))-d(t)∫t-d(t)t‍g(x(s))TW1g(x(s))ds+σ2x(t)TW2x(t)-σ(t)∫t-σ(t)t‍x(s)TW1x(s)ds,V˙5(xt)=x˙(t)T[τ24T1+(τ22-τ12)2T2+σ4T3]x˙(t)-2τ22∫-τ20‍∫t+θt‍x˙(s)TT1x˙(s)dsdθ-(τ22-τ12)∫-τ2-τ1‍∫t+θt‍x˙(s)TT2x˙(s)dsdθ-2σ2∫-σ0‍∫t+θt‍x˙(s)TT3x˙(s)dsdθ.
Similarly like Theorem 5 we can use Lemmas 2 and 3 for the integrals. On the other hand, we have from (5) that for any λ=1,2,…,n,
(56)(gλ(xλ(t))-Fλ-xλ(t))(gλ(xλ(t))-Fλ+xλ(t))≤0,
which is equivalent to
(57)ζT(t)[Fλ+Fλ-eλeλT-Fλ++Fλ-2eλeλT-Fλ++Fλ-2eλeλTeλeλT]ζ(t)≤0,
where eλ denotes the unit column vector having 1 element on its λth row and zeros elsewhere. Thus, for any appropriately dimensioned diagonal matrix Λ1>0, the following inequality holds:
(58)0≤ζT(t)[-F1Λ1F2Λ1*-Λ1]ζ(t).
Similarly, for any appropriately dimensioned diagonal matrices Λ2>0,Λ3>0,Λ4>0, and Λ5>0, the following inequalities also hold:
(59)0≤ζT(t-τ(t))[-F1Λ2F2Λ2*-Λ2]ζ(t-τ(t)),0≤ζT(t-τ1)[-F1Λ3F2Λ3*-Λ3]ζ(t-τ1),0≤ζT(t-τ2)[-F1Λ4F2Λ4*-Λ4]ζ(t-τ2),0≤ζT(t-σ(t))[-F1Λ5F2Λ5*-Λ5]ζ(t-σ(t)).
Using inequalities (55) and adding (58)-(59) to V˙(xt), we get
(60)V˙(xt)-2yT(t)u(t)-γuT(t)u(t)≤ρT(t)Ξρ(t),
where ρ(t)=[ρ1T(t)ρ2T(t)ρ3T(t)ρ4T(t)] with
(61)ρ1(t)=[x(t)g(x(t))x(t-τ(t))g(x(t-τ(t)))x(t-τ1)g(x(t-τ1))(x(t-τ(t)))],ρ2(t)=[∫t-σ(t)t‍x(t-τ2)g(x(t-τ2))x(t-σ(t))g(x(t-σ(t)))∫t-σ(t)t‍x(s)ds],ρ3(t)=[∫t-τ1t‍x(t-σ)∫t-σt‍x(s)ds∫t-d(t)t‍g(x(s))ds∫t-τ1t‍x(s)ds∫t-τ1t‍g(x(s))ds],ρ4(t)=[∫t-τ2t‍x(s)ds∫t-τ2t‍g(x(s))ds∫t-τ2t-τ1‍x(s)dsu(t)∫t-τ2t‍].
Hence we can obtain from (51) that
(62)V˙(xt)-2y(t)Tu(t)-γu(t)Tu(t)≤0.
The remaining part of the proof is the same as Theorem 5.Remark 9.
In this paper, Theorem5 provides passivity criteria for the Markovian jumping neural networks with leakage time varying delays. Such stability criterion is derived based on the assumption that the leakage time varying delays are differentiable and the values of σμ are known. A new set of triple integral terms have been introduced in the Lyapunov-Krasovskii functional to derive the leakage delay-dependent passivity conditions via LMI approach. New type of Lyapunov-Krasovskii functional is constructed in which the positive definite matrices Q1i,m,Q2i,m,Q3i,m are dependent on the system mode and a triple-integral term is introduced for deriving the delay-dependent passivity conditions.


## 4.1. Description and Preliminaries

In this section, we derive passivity criterion for the delayed neural networks using the Lyapunov-Krasovskii functional without Markovian jumping parameters.Consider the following neural networks with mixed time-delays:(48)x˙(t)=-Cx(t-σ(t))+Ag(x(t))+Bg(x(t-τ(t)))+D∫t-d(t)t‍g(x(s))ds+u(t),y(t)=g(x(t)).
Or, it has an equivalent form as follows:
(49)ddt[x(t)-C∫t-σ(t)t‍x(s)ds]00=-Cx(t)-Cx(t-σ(t))σ˙(t)+Ag(x(t))0000+Bg(x(t-τ(t)))0000+D∫t-d(t)t‍g(x(s))ds+u(t),y(t)=g(x(t)).
Now, we establish the following passivity condition for the system (49).Theorem 8.
Neural network (49) is passive if there exist
(50)P>0,Q1=[Q11Q12Q12TQ13]>0,Q2=[Q21Q22Q22TQ23]>0,Q3=[Q31Q32Q32TQ33]>0,Q4>0,U=[U1U2U2TU3]>0,
positive symmetric matrices S1=S1T>0,S2=S2T>0,S3=S3T>0, T1=T1T>0,T2=T2T>0, T3=T3T>0; the positive definite matrices W1>0,W2>0; the diagonal matrices Λ1>0,Λ2>0,Λ3>0,Λ4>0,Λ5>0;  and a scalar γ>0 such that the following LMI holds:
(51)Ξ=(Ξi,j)20×20<0,
where
(52)Ξ1,1=-PC-CTp+Q11+Q21+τ1U1+Q31+τ2U1+Q4-(τ2-τ1)S1-S3-4τ22T1-4(τ2-τ1)2T2-4σ2T3-F1Λ1+σ2W2,Ξ1,2=PA+Q12+Q22+τ1U2+Q32+τ2U2+F2Λ1,Ξ1,4=pB,Ξ1,7=(τ2-τ1)S1,Ξ1,9=-PCσμ,Ξ1,11=CTPC,Ξ1,12=S3,Ξ1,13=4σT3,Ξ1,14=PDi,Ξ1,17=4τ2T1,Ξ1,19=4(τ2-τ1)T2,Ξ1,20=P,Ξ2,2=Q13+Q23+τ1U3+Q33+τ2U3+ATRA+d2W1+ATGA-Λ1,Ξ2,4=ATRB+ATGB,Ξ2,9=-ATRC-ATGC,Ξ2,11=-ATPC,Ξ2,14=ATRD+ATGD,Ξ2,20=ATR+ATG-I,Ξ3,3=-(1-τμ)Q11-F1Λ2,Ξ3,4=-(1-τμ)Q12+F2Λ2,Ξ4,4=-(1-τμ)Q13+BTRB+BTGB-Λ2,Ξ4,9=-BTRC-BTGC,Ξ4,11=-BTPC,Ξ4,14=BTRD+BTGD,Ξ4,20=BTR+BTG,Ξ5,5=-Q21-S2-F1Λ3,Ξ5,6=-Q22+F2Λ3,Ξ5,7=S2,Ξ6,6=-Q23-Λ3,Ξ7,7=-Q31-(τ2-τ1)S1-S2-F1Λ4,Ξ7,8=-Q32+F2Λ4,Ξ8,8=-Q33-Λ4,Ξ9,9=-(1-σμ)Q4+CTRC+CTGC-F1Λ5,Ξ9,10=F2Λ5,Ξ9,11=CTσμPC,Ξ9,14=-CTRD-CTGD,Ξ9,20=-CTR-CTG,Ξ10,10=-Λ5,Ξ11,11=-W2,Ξ11,14=-CTPD,Ξ11,20=-CTP,Ξ12,12=-S3,Ξ13,13=-4T3,Ξ14,14=DTRD+DTGD-W1,Ξ14,20=DTR+DTG,Ξ15,15=-1τ1U1,Ξ15,16=-1τ1U2,Ξ16,16=-1τ1U3,Ξ17,17=-1τ2U1-4T1,Ξ17,18=-1τ2U2,Ξ18,18=-1τ2U3,Ξ19,19=-4T2,Ξ20,20=R+G-γI,R=τ22(τ2-τ1)S1+(τ2-τ1)2S2+σ2S3,G=τ24T1+(τ22-τ12)2T2+σ4T3,
and the remaining coefficients are all zero.Proof.
Denoteζ=[x(t)Tg(x(t))T]T and consider the following Lyapunov-Krasovskii functional for neural network (49):
(53)V(xt)=V1(xt)+V2(xt)+V3(xt)+V4(xt)+V5(xt),
where
(54)V1(xt)=[x(t)-Ci∫t-σ(t)t‍x(s)ds]T×P[x(t)-Ci∫t-σ(t)t‍x(s)ds],V2(xt)=∫t-τ(t)t‍ζ(s)TQ1ζ(s)ds+∫t-τ1t‍ζ(s)TQ2ζ(s)ds+∫t-τ2t‍ζ(s)TQ3ζ(s)ds+∫t-σ(t)t‍x(s)TQ4x(s)ds+∫-τ10‍∫t+θt‍ζ(s)TUζ(s)dsdθ+∫-τ20‍∫t+θt‍ζ(s)TUζ(s)dsdθ,V3(xt)=τ2(τ2-τ1)+∫-τ20‍∫t+θt‍x˙(s)TS1x˙(s)dsdθ+(τ2-τ1)∫-τ2-τ1‍∫t+θt‍x˙(s)TS2x˙(s)dsdθ+σ∫-σ0‍∫t+θt‍x˙(s)TS3x˙(s)dsdθ,V4(xt)=∫-d0‍∫t+θt‍g(x(s))TW1g(x(s))dsdθ+σ∫t-σt‍∫θt‍x(s)TW2x(s)dsdθ,V5(xt)=2τ22∫-τ20‍∫θ0‍∫t+λt‍x˙(s)TT1x˙(s)dsdλdθ+2(τ22-τ12)∫-τ1-τ2‍∫θ0‍∫t+λt‍x˙(s)TT2x˙(s)dsdλdθ+2σ2∫-σ0‍∫θ0‍∫t+λt‍x˙(s)TT3x˙(s)dsdλdθ.
Taking time derivative acting on V(xt) along the neural networks (49) is defined as follows:
(55)V˙1(xt)=[x(t)-C∫t-σ(t)t‍x(s)ds]T×Pddt[x(t)-C∫t-σ(t)t‍x(s)ds]+ddt[x(t)-C∫t-σ(t)t‍x(s)ds]T×P[x(t)-C∫t-σ(t)t‍x(s)ds],V˙2(xt)≤ζT(t)Q1ζ(t)-ζT(t-τ(t))Q1ζ(t-τ(t))×(1-τμ)+ζT(t)Q2ζ(t)-ζT(t-τ1)Q2ζ(t-τ1)+ζT(t)Q3ζ(t)-ζT(t-τ2)Q3ζ(t-τ2)+xT(t)Q4x(t)-xT(t-σ(t))Q4x(t-σ(t))(1-σμ)+∫-τ10‍ζT(t)Uζ(t)dθ-∫t-τ1t‍ζT(s)Uζ(s)ds+∫-τ20‍ζT(t)Uζ(t)dθ-∫t-τ2t‍ζT(s)Uζ(s)ds,V˙3(xt)=x˙(t)T[τ22(τ2-τ1)S1+(τ2-τ1)2S2+σ2S3]x˙(t)-τ2(τ2-τ1)∫t-τ2t‍x˙(s)TS1x˙(s)ds-(τ2-τ1)∫t-τ2t-τ1‍x˙(s)TS2x˙(s)ds-σ∫t-σt‍x˙(s)TS3x˙(s)ds,V˙4(xt)=d2g(x(t))TW1g(x(t))-d(t)∫t-d(t)t‍g(x(s))TW1g(x(s))ds+σ2x(t)TW2x(t)-σ(t)∫t-σ(t)t‍x(s)TW1x(s)ds,V˙5(xt)=x˙(t)T[τ24T1+(τ22-τ12)2T2+σ4T3]x˙(t)-2τ22∫-τ20‍∫t+θt‍x˙(s)TT1x˙(s)dsdθ-(τ22-τ12)∫-τ2-τ1‍∫t+θt‍x˙(s)TT2x˙(s)dsdθ-2σ2∫-σ0‍∫t+θt‍x˙(s)TT3x˙(s)dsdθ.
Similarly like Theorem 5 we can use Lemmas 2 and 3 for the integrals. On the other hand, we have from (5) that for any λ=1,2,…,n,
(56)(gλ(xλ(t))-Fλ-xλ(t))(gλ(xλ(t))-Fλ+xλ(t))≤0,
which is equivalent to
(57)ζT(t)[Fλ+Fλ-eλeλT-Fλ++Fλ-2eλeλT-Fλ++Fλ-2eλeλTeλeλT]ζ(t)≤0,
where eλ denotes the unit column vector having 1 element on its λth row and zeros elsewhere. Thus, for any appropriately dimensioned diagonal matrix Λ1>0, the following inequality holds:
(58)0≤ζT(t)[-F1Λ1F2Λ1*-Λ1]ζ(t).
Similarly, for any appropriately dimensioned diagonal matrices Λ2>0,Λ3>0,Λ4>0, and Λ5>0, the following inequalities also hold:
(59)0≤ζT(t-τ(t))[-F1Λ2F2Λ2*-Λ2]ζ(t-τ(t)),0≤ζT(t-τ1)[-F1Λ3F2Λ3*-Λ3]ζ(t-τ1),0≤ζT(t-τ2)[-F1Λ4F2Λ4*-Λ4]ζ(t-τ2),0≤ζT(t-σ(t))[-F1Λ5F2Λ5*-Λ5]ζ(t-σ(t)).
Using inequalities (55) and adding (58)-(59) to V˙(xt), we get
(60)V˙(xt)-2yT(t)u(t)-γuT(t)u(t)≤ρT(t)Ξρ(t),
where ρ(t)=[ρ1T(t)ρ2T(t)ρ3T(t)ρ4T(t)] with
(61)ρ1(t)=[x(t)g(x(t))x(t-τ(t))g(x(t-τ(t)))x(t-τ1)g(x(t-τ1))(x(t-τ(t)))],ρ2(t)=[∫t-σ(t)t‍x(t-τ2)g(x(t-τ2))x(t-σ(t))g(x(t-σ(t)))∫t-σ(t)t‍x(s)ds],ρ3(t)=[∫t-τ1t‍x(t-σ)∫t-σt‍x(s)ds∫t-d(t)t‍g(x(s))ds∫t-τ1t‍x(s)ds∫t-τ1t‍g(x(s))ds],ρ4(t)=[∫t-τ2t‍x(s)ds∫t-τ2t‍g(x(s))ds∫t-τ2t-τ1‍x(s)dsu(t)∫t-τ2t‍].
Hence we can obtain from (51) that
(62)V˙(xt)-2y(t)Tu(t)-γu(t)Tu(t)≤0.
The remaining part of the proof is the same as Theorem 5.Remark 9.
In this paper, Theorem5 provides passivity criteria for the Markovian jumping neural networks with leakage time varying delays. Such stability criterion is derived based on the assumption that the leakage time varying delays are differentiable and the values of σμ are known. A new set of triple integral terms have been introduced in the Lyapunov-Krasovskii functional to derive the leakage delay-dependent passivity conditions via LMI approach. New type of Lyapunov-Krasovskii functional is constructed in which the positive definite matrices Q1i,m,Q2i,m,Q3i,m are dependent on the system mode and a triple-integral term is introduced for deriving the delay-dependent passivity conditions.


## 5. Numerical Examples

In this chapter, we provide two simple examples presented here in order to illustrate the usefulness of our main results. Our aim is to examine the passivity analysis of given delayed neural networks.Example 1.
Consider the delayed neural networks (1) with the following parameters and having Markovian jumping parameters as below:
(63)x˙(t)=-C(r(t))x(t-σ(t))+A(r(t))g(x(t))+B(r(t))g(x(t-τ(t)))+D(r(t))∫t-d(t)t‍g(x(s))ds+u(t),y(t)=g(x(t)),
where
(64)C1=[2.4003.5],C2=[2.6003.2],A1=[0.41.6-0.50.2],A2=[0.51.2-0.50.8],B1=[0.90.50.70.5],B2=[0.30.60.5-1.2],D1=[1.1-1.60.40.9],D2=[0.60.41.2-0.6]
and the activation functions are taken as follows: g1(α)=g2(α)=tanh(α). It is found that F1=[0000]  and   F2=[1001]. Furthermore, the transition probability matrices are
(65)Π1=[-0.90.50.70.6],Π2=[-0.50.50.7-0.8],Π3=[-0.70.90.5-0.8],Λ=[-0.70.20.250.5-1.20.30.30.6-0.5].
We choose the lower and upper bounds of delay values of τ(t), σ(t), and d(t) are τ1=0.2,τ2=1.5,σ=0.3,σμ=0.4,τμ=0.6,d=0.5. By applying MATLAB LMI toolbox, we obtain the feasible solution as follows:
(66)P11=[6.01380.20290.20295.5897],P12=[5.94530.21800.21805.5357],P13=[6.17530.51520.51525.9589],P21=[9.2206-0.4935-0.49355.5949],P22=[8.3958-0.3461-0.34615.2798],P23=[8.4741-0.2735-0.27355.2809],Q1111=[96.50320.89930.8993101.4537],Q1112=[-20.1310-0.5306-0.5306-22.4636],Q1113=[63.6503-0.1830-0.183058.2996],Q1121=[109.97241.78021.7802115.7604],Q1122=[-25.8818-2.1403-2.1403-27.4127],Q1123=[69.92940.23720.237263.8244],Q1211=[121.1754-0.3055-0.3055116.3565],…,U1=[141.10524.95614.9561148.6923],U2=[-60.7043-2.3397-2.3397-67.6228],U3=[44.32040.26130.261346.7241],S1=[0.1771-0.0191-0.0191-0.1934],S2=[0.2957-0.0315-0.03150.3223],S3=[30.0136-0.8922-0.8922-31.0764],W1=[152.5941-29.2867-28.2867162.9678],W2=[172.8581-2.3076-2.3076174.9177],T1=[0.6547-0.0596-0.05960.7113],T2=[0.6766-0.0639-0.06390.7368],T3=[32.2540-0.0557-0.055732.3317],L111=[358.610600408.4828],L112=[365.055000414.4924],L113=[485.646000535.5544],L121=[380.737500419.7776],L122=[374.827900416.9314],L123=[484.063900529.9711],L211=[99.36110090.0987],L212=[96.86890090.9087],L213=[87.23980080.2453],L221=[88.416300110.4032],L222=[93.539500115.5929],…,L521=[111.717700134.5697],L522=[110.661500132.6384],L523=[111.463600133.5637],γ=166.0447.
This shows that the given Markovian jumping neural networks (1) or (3) are globally passive with respect to the passive control.Example 2.
Consider the delayed neural network (49) with the following parameters and without markovian jumping parameters as below:
(67)x˙(t)=-Cx(t-σ(t))+Ag(x(t))+Bg(x(t-τ(t)))+D∫t-d(t)t‍g(x(s))ds+u(t),y(t)=g(x(t)),
where
(68)C=[2.2002.5],A=[1.2-1.5-1.71.2],B=[1.10.50.50.8],D=[0.80.20.20.3].
Further, we have the matrices
(69)F1=[0000],F2=[1001].
Here, the bounds of time delays of τ(t), σ(t), and d(t) are chosen as follows: τ1=0.5,τ2=1,σ=0.1,σμ=0.1,τμ=0.2,d=0.5. By applying MATLAB LMI toolbox, we obtain the feasible solution as follows:
(70)P=[0.28990.03710.03710.2688],Q11=[4.9833-0.0919-0.09194.8960],Q12=[-1.36850.03910.0391-1.3516],Q13=[2.73590.79770.79772.6571],Q21=[4.6114-0.0801-0.08014.4621],Q22=[-1.20200.04330.0433-1.1716],Q23=[2.01590.17570.17572.0771],Q31=[4.5805-0.0773-0.07734.4345],Q32=[-1.19790.04040.0424-1.1685],Q33=[2.01580.17700.17702.0774],Q4=[5.20090.74310.74315.4021],U1=[5.7038-0.0258-0.02585.5568],U2=[-1.41440.06490.0649-1.3884],U3=[2.93860.20220.20223.0092],S1=[0.23600.05150.05150.2458],S2=[0.42590.07980.07980.4379],S3=[2.93800.04950.04952.9462],W1=[4.86730.53580.53584.1789],W2=[5.76160.40260.40265.7553],T1=[0.08070.04600.04600.0859],T2=[0.35530.06320.06320.3633],T3=[1.9454-0.0007-0.00071.9459],L1=[17.36730016.8346],L2=[3.5738003.2436],L3=[2.5698002.5662],L4=[2.5591002.5552],L5=[4.0154004.2815],γ=6.5268.
This shows that the given Markovian jumping neural networks (49) are globally passive with respect to the passive control.


## 6. Conclusion

In this paper, stochastic stability analysis of Markovian jump neural networks with leakage time-varying delay and discrete and distributed time-varying delays is considered. The Markov process in the underlying neural networks is finite piecewise homogeneous. A leakage delay-dependent passivity conditions have been derived in terms of LMIs by constructing novel Lyapunov-Krasovskii functional having triple integral terms. This performance not only depends on the upper bound of the time-varying leakage delayσ(t) but also depends on the upper bound of the derivative of the time-varying leakage delay σμ. Two numerical examples have been provided to demonstrate the effectiveness of the proposed methods for both with and without Markovian jumping parameters.



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*Source: 172906-2013-07-18.xml*