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Research article

A study on a line congruence as surface in the space of lines

  • In this work, we introduce a line congruence as surface in the space of lines in terms of the E. Study map. This provides the ability to derive some formulae of surfaces theory into line spaces. In addition, the well known equation of the Plucker's conoid has been obtained and its kinematic-geometry are examined in details. At last, an example of application is investigated and explained in detail.

    Citation: Rashad A. Abdel-Baky, Monia F. Naghi. A study on a line congruence as surface in the space of lines[J]. AIMS Mathematics, 2021, 6(10): 11109-11123. doi: 10.3934/math.2021645

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  • In this work, we introduce a line congruence as surface in the space of lines in terms of the E. Study map. This provides the ability to derive some formulae of surfaces theory into line spaces. In addition, the well known equation of the Plucker's conoid has been obtained and its kinematic-geometry are examined in details. At last, an example of application is investigated and explained in detail.



    Numerous studies on recurrent neural networks (RNNs), including those on bidirectional associative memory neural networks, cellular neural networks, and Hopfield neural networks, among others, have been proposed. In the meantime, neural networks (NNs) have been employed as a tool to address issues that have arisen in associative memory, signal processing, image processing, static image treatment, pattern recognition, and optimization. Additionally, it seems that time delays are crucial to networked control systems, chemical reactions, communication systems, etc. The authors of [1], [2] investigated time-delayed artificial neural network electronic implementations. Several studies have looked into different sorts of delays inside NNs in this context; see the papers [38][40] and references therein. The passivity theory is widely applied in many engineering problems. Indeed, it is intimately related to the circuit analysis which is a useful and significant tool to analyze the stability of nonlinear systems, signal processing and chaos control. Thus, it has been employed in various fields of science and engineering [7], [8], [44][48]. In [9], the authors proposed neural adaptive output feedback control based on passivity with adaptive set-point regulation of nonlinear uncertain non-negative and compartmental systems. In the last decade, great attention has been paid to the passivity analysis of RNNs with delay-independent [10], [49] and delay-dependent [11][14], [36]. In [13], the author studied the passivity analysis of Markovian jump neural network with leakage time varying delay, discrete and distributed time varying delays. We can see the discussion of the extended dissipativity of discrete-time neural networks NNs with time-varying delay in [55]. However, comparatively less interest has been shown towards the passivity analysis of mode dependent delay on neutral type neural networks (NNNs) with Markovian jumping (Mj). On the other hand, a Markov jump system is a special classes of a hybrid system. Indeed, they have great ability to model the dynamical systems and their application can be found in manufacturing systems, economic systems, network control systems, modeling production system, communication systems and so on. In the recent years, several results are reported on the stability analysis for neural networks with Markovian jump parameters, see the references [16][19], [41][43]. In [20], the authors studied global exponential estimates of delayed stochastic NNs with Markovian switching by constructing with positive definite matrices in stochastic Lyapunov functional which are dependent on the system mode and a triple-integral term. The ℋ synchronization issue for singularly perturbed coupled neural networks (SPCNNs) affected by both nonlinear constraints and gain uncertainties was explored in [57] using a novel double-layer switching regulation containing Markov chain and persistent dwell-time switching regulation (PDTSR). Convolutional Neural Networks (CNNs) are efficient tools for pattern recognition applications. More on this topic can be seen in [58], [60]. An exponential synchronization problem for the multi-weighted complex dynamical network (MCDN) with hybrid delays on a time scale is investigated in [61]. We can see the establishment of fixed-point and coincidence-point consequences in generalized metric spaces in [59]. The nonfragile ℋ synchronization issue for a class of discrete-time Takagi–Sugeno (T–S) fuzzy Markov jump systems was investigated in [56]. In [21], the authors have studied stochastic NNNs with mixed time-delays under adaptive synchronization. Additionally, the problem of state estimation of RNNs with Mj parameters and mixed delays based on mode-dependent approach was investigated in [22].

    However, some of the researchers discussed the robust passive filtering for NNNs with delays in [25]. In [26], the authors investigated the global asymptotic stability of NNNs with delays by utilizing the Lyapunov-Krasovskii functional (LKF) and the linear matrix inequality approach. While employing the method of Lyapunov–Krasovskii functional, we necessarily need these three steps for the derivation of a global asymptotic stability criterion: constructing a Lyapunov–Krasovskii functional, estimating the derivative of the Lyapunov–Krasovskii functional, and formulating a global asymptotic stability criterion. You will get an overview of recent developments in each of the above steps if you refer to [54].The author in [52] studied passivity and exponential passivity for NNNs with various delays. In [51], the authors investigated the robust passivity analysis of mixed delayed NNs with distributed time-varying delays. The exponential passivity of discrete-time switched NNs with transmission delay was studied in [53]. Recently, the authors in [5] studied passivity analysis for NNNs with Mj parameters and time delay in the leakage term. New delay-dependent passivity conditions are derived in terms of LMIs with a proper construction of LKF, and it can be checked easily via standard numerical packages. However, triple and quadruple integrals have not been taken into account to derive the passivity conditions and, moreover, the mode-dependent time delays have not been included in [5]. Recently, a novel Lyapunov functional with some terms involving triple or quadruple integrals are taken into account to study the state estimation problem with mode-dependent approach in [22]. Motivated by the above discussion, the main purpose of this paper is to study the global passivity of Mj for NNNs with leakage and mode-dependent delay terms. By construction of a new LKF involving mode-dependent Lyapunov matrices, some sufficient conditions are derived in terms of LMIs. For the sake of illustration, a numerical example is given to demonstrate the usefulness and effectiveness of the presented results. Unlike previous results, we will introduce an improved Lyapunov–Krasovskii functional with triple and quadruple integrals for deriving the reported stability results in this paper. Based on this discussion, our technique not only provides different approach but also gives less conservative conditions than those studied in [5], [22]. The rest of this paper is organized as follows. The problem and some preliminaries are introduced in Section 2. In Section 3, the main results are stated and proved. Some sufficient conditions for global passivity results are developed here. In Section 4, an illustrative example is provided to demonstrate the effectiveness of the proposed criteria. We conclude the results of this paper in Section 5.

    Notations: Throughout this paper the following notations are used: n - n-dimensional Euclidean space; n×n - the set of all n×n real matrices; diag() - a block diagonal matrix; I -the identity matrix with compatible dimensions; CT - the transpose of C; X and Y are symmetric matrices, where XY (similarly X>Y)- XY is a positive semi-definite (similarly positive definite); (Ω,,𝒫)- a complete probability space with a natural filtration {t}t0; E[] - expectation operator with respect to the given probability measure 𝒫; 𝒞([d,0];n) - the family of continuously differentiable function; φ=max{maxτθ0|φ(θ)|,maxdθ0|φ(θ)|}; 𝒞20([d,0];n) - the family of bounded 0-measurable; 𝒞([d,0];n)-valued stochastic variables ξ={ξ(θ):dθ0} such that 0dE|ξ(θ)|2ds<; *-the symmetric block in one symmetric matrix.

    Let {(t),t+} be right-continuous on (Ω,,𝒫). Here (Ω,,𝒫) is a complete probability space with Markov chain, (t) takes the values from a finite state space 𝒮={1,2,,N} with generator Γ=(πij)N×N given by

    P{(t+Δt)=j|(t)=i}={πijΔt+o(Δt),ij,1+πiiΔt+o(Δt),i=j,

    where Δt>0 and limΔt0o(Δt)Δt=0, πij0 (ij) is the transition rate from i to j, and πii=Nj=1,jiπij.

    We consider Mode-dependent Markov jump NNNs with mixed time-delays:

    {˙x(t)=𝒜((t))x(tσ((t)))+((t))(x(t))+𝒞((t))(x(tτ(t,(t))))+𝒟((t))˙x(th(t,(t)))+((t))ttd(t,(t))(x(s))ds+u(t)y(t)=(x(t))

    where x(t)=[x1(t),x2(t),,xn(t)]Tn is the state vector linked with n neurons. The diagonal matrix 𝒜((t))=diag(A1((t)),A2((t)),,An((t))) has positive entries Ai((t))>0 (i=1,2,,n). ((t)), 𝒞((t)), 𝒟((t)), ((t)) are known appropriate dimensional constant matrices. Here the neuron activation function is (x(t))=[1(x1(t)),2(x2(t)),,n(xn(t))]T. u(t) denotes a constant input. τ(t,(t)), h(t,(t)), d(t,(t)) are mode dependent discrete, neutral and distributed delays, respectively and σ((t)) is the mode dependent leakage delay.

    Throughout this paper, we assume the following.

    Assumption 1. For any j=1,2,,n, j(0)=0 and there exist constants ˆlj and ˆl+j such that

    ˆljj(γ1)j(γ2)γ1γ2ˆl+j,

    where γ1,γ2, and γ1γ2.

    For the sake of convenience, we denote 𝒜((t)=i)=𝒜i,((t)=i)=i,𝒞((t)=i)=𝒞i, 𝒟((t)=i)=𝒟i, ((t)=i)=i, respectively.

    System (2.1) can be rewritten as

    {˙x(t)=𝒜ix(tσi)+i(x(t))+𝒞i(x(tτi(t)))+𝒟i˙x(thi(t))+ittdi(t)(x(s))ds+u(t)y(t)=(x(t))

    and the parameters associated with time delays are assumed to satisfy following:

    0τi(t)τi,  ˙τi(t)τµi,0hi(t)hi,  ˙hi(t)hµi,  0di(t)di,  ˙di(t)dµi,  σi>0

    where τi,hi,di,τµi,hµi and dµi are some real constants and τ=maxiS{τi}, h=maxiS{hi}, d=maxiS{di}, σ=maxiS{σi}.

    Now we can see a few necessary lemmas and a definition.

    (ba)[bay(s)TMy(s)ds][bay(s)ds]TM[bay(s)ds].

    holds.

    Proof. By Schur complement,

    [y(s)TMy(s)y(s)Ty(s)M1]0,s[a,b].

    On integration from a to b yields, [bay(s)TMy(s)dsbay(s)Tdsbay(s)ds(ba)M1] 0,  s[a,b]. Now using the Schur complement on this inequality, we obtain our desired result.    □

    ±2xTyxTx+yT1y.

    Lemma 3. [31] (Schur complement) Given Ω1,Ω2 and Ω3 are constant matrices with appropriate dimensions, where Ω1,Ω2>0 are symmetric matrices, then

    Ω1+ΩT3Ω12Ω3<0[Ω1ΩT3Ω2]<0,  or  [Ω2Ω3Ω1]<0.

    Definition 1. [32] If there exists a scalar ν0 such that tp0 and for all solutions of (2.1), the following inequality holds under zero initial conditions,

    2tp0E{y(s)Tu(s)}dsγtp0E{u(s)Tu(s)}ds,

    then the system (2.3) is said to be passive.

    Now, we denote

    ˆL1=diag{ˆl1ˆl+1,ˆl2ˆl+2,,ˆlmˆl+m},  and  ˆL2=diag{ˆl1+ˆl+12,ˆl2+ˆl+22,,ˆlm+ˆl+m2}.

    [XiNiXi]0,

    Nj=1πijVjV0,

    Nj=1,jiπij𝒢j𝒢0,

    Φ=[ΩΓT1πijKj]<0,

    where 𝒢j in (3.3) respectively represents Qj , Wj, Rj, Sj, Uj, Xj, Yj, Tj, Zj, Lj and correspondingly 𝒢 represents Q, W, R, S, U, X, Y, T, Z, (e.q.,when 𝒢j is Qj, 𝒢 is Q) and

    Ω=(ϑi,j)15×15,

    ϑ1,1=Pi𝒜i𝒜TiPi+πiiPi+jiπijPj+jiπijKj+Q1i+τQ1+Wi+τW+Ri+σR

    +τiUi+τ22U1τiXi1hiYi2Ti2Zi2LiˆL1H1iˆL1H3iˆL1H4i

    ˆL1H5iˆL1H6i2σ2M,ϑ1,2=1τiNTi+1τiXi,ϑ1,3=1τiNTi+ˆL2H3i,

    ϑ1,4=2τiTi,ϑ1,5=Pii+Q2i+Q2+ˆL2H1i+J1i,ϑ1,6=Pi𝒞i+J1𝒞i,

    ϑ1,7=J1𝒜i𝒜TiJT1+ˆL2H5i,ϑ1,8=𝒜TiPi𝒜iπiiPi𝒜i2σiLi,ϑ1,9=2σM,

    ϑ1,10=Pi,ϑ1,11=Pi𝒟i+J1𝒟i,ϑ1,12=1hiYi+ˆL2H4i,ϑ1,13=2hiZi,

    ϑ1,14=J1i,ϑ1,15=J1+ˆL2H6i,ϑ2,2=(1τµi)Q1iˆL1H2i2τiXi+1τiNTi,

    ϑ2,3=1τiXi1τiNTi,ϑ2,6=(1τµi)Q1i+ˆL2H2i,ϑ3,3=Wi1τiXiH3i,

    ϑ4,4=1τiUi2τ2iTi,ϑ5,5=Q3i+Q3+diVi+d22VH1i,ϑ5,8=TiPi𝒜i,

    ϑ5,15=TiJT2,ϑ6,6=(1τµi)Q3iH2i,ϑ6,8=𝒞TiPi𝒜i,ϑ6,15=𝒞TiJT2,

    ϑ7,7=RiH5i,ϑ7,15=𝒜TiJT2,ϑ8,8=πii𝒜TiPi𝒜i2σ2iLi,ϑ8,10=𝒜TiPi,

    ϑ8,11=𝒜TiPi𝒟i,ϑ8,14=𝒜TiPii,ϑ9,9=jiπij𝒜TjPj𝒜j2M,ϑ10,10=γI,

    ϑ10,15=JT2,ϑ11,11=Si(1hµi),ϑ11,15=𝒟TiJT2,ϑ12,12=1hiYiH4i,

    ϑ13,13=2h2iZi,ϑ14,14=(1dµi)diVi,ϑ14,15=TiJT2,ϑ15,15=Si+hS+τiXi+τ22X

    +hiYi+h22Y+τ2i2Ti+τ36T+h2i2Zi+h36Z+σ2i2Li+σ36L+σ42Z(J2+JT2)H6i,

    ΓT=[0.....08elementsjiπij(𝒜TjPj)T0.....015elements]T,

    and the other coefficients are zero.

    Proof. Here, we consider LKF candidate:

    V(xt,i,t)=13κ=1Vκ(xt,i,t),

    where

    V1(xt,i,t)=[x(t)𝒜ittσix(s)ds]TPi[x(t)𝒜ittσix(s)ds],

    V2(xt,i,t)=ttτi(t)ζT(s)Qiζ(s)ds+0τtt+θζT(s)Qζ(s)dsdθ,

    V3(xt,i,t)=ttτixT(s)Wix(s)ds+0τtt+θxT(s)Wx(s)dsdθ,

    V4(xt,i,t)=ttσixT(s)Rix(s)ds+0σtt+θxT(s)Rx(s)dsdθ,

    V5(xt,i,t)=tthixT(s)Six(s)ds+0htt+θxT(s)Sx(s)dsdθ,

    V6(xt,i,t)=0di(t)tt+θT(x(s))Vi(x(s))dsdθ+0d0θtt+βT(x(s))V(x(s))dsdβdθ,

    V7(xt,i,t)=0τitt+θxT(s)Uix(s)dsdθ+0τ0θtt+βxT(s)Ux(s)dsdβdθ,

    V8(xt,i,t)=0τitt+θ˙xT(s)Xi˙x(s)dsdθ+0τ0θtt+β˙xT(s)X˙x(s)dsdβdθ,

    V9(xt,i,t)=0hitt+θ˙xT(s)Yi˙x(s)dsdθ+0h0θtt+β˙xT(s)Y˙x(s)dsdβdθ,

    V10(xt,i,t)=0τi0θtt+β˙xT(s)Ti˙x(s)dsdβdθ+0τ0θ0βtt+α˙xT(s)T˙x(s)dsdαdβdθ,

    V11(xt,i,t)=0hi0θtt+β˙xT(s)Zi˙x(s)dsdβdθ+0h0θ0βtt+α˙xT(s)Z˙x(s)dsdαdβdθ,

    V12(xt,i,t)=0σi0θtt+β˙xT(s)Li˙x(s)dsdβdθ+0σ0θ0βtt+α˙xT(s)L˙x(s)dsdαdβdθ,

    V13(xt,i,t)=σ20σ0θtt+β˙xT(s)M˙x(s)dsdβdθ,

    where ζT(t)=[xT(t),T(x(t))]T.

    From (2.1), we get

    V(xt,i,t)=13κ=1Vκ(xt,i,t),

    where

    V1(xt,i,t)=2[x(t)𝒜ittσix(s)ds]TPiddt[x(t)𝒜ittσix(s)ds]

    +Nj=1πij[x(t)𝒜jttσjx(s)ds]TPj[x(t)𝒜jttσjx(s)ds],

    2[x(t)𝒜ittσix(s)ds]TPi[𝒜ix(t)+i(x(t))+𝒞i(x(tτi(t)))

    +𝒟i˙x(thi(t))+ittdi(t)(x(s))ds+u(t)]

    +πii[x(t)𝒜ittσix(s)ds]TPi[x(t)𝒜ittσix(s)ds]

    +jiπij[xT(t)Pjx(t)+ttσjxT(s)ds𝒜jPjK1jPj𝒜jttσjx(s)ds

    +ttσxT(s)ds𝒜TjPj𝒜jttσx(s)ds],

    V2(xt,i,t)ζT(t)Qiζ(t)ζT(tτi(t))Qiζ(tτi(t))(1τµi)+Nj=1πijttτj(t)ζT(s)Qjζ(s)ds

    +τζT(t)Qζ(t)ttτζT(s)Qζ(s)ds,

    V3(xt,i,t)=xT(t)Wix(t)xT(tτi)Wix(tτi)+Nj=1πijttτjxT(s)Wjx(s)ds

    +τxT(t)Wx(t)ttτxT(s)Wx(s)ds,

    V4(xt,i,t)=xT(t)Rix(t)xT(tσi)Rix(tσi)+Nj=1πijttσjxT(s)Rjx(s)ds

    +σxT(t)Rx(t)ttσxT(s)Rx(s)ds,

    V5(xt,i,t)˙xT(t)Si˙x(t)˙xT(thi(t))Si˙x(thi(t))(1hµi)+Nj=1πijtthj(t)˙xT(s)Sj˙x(s)ds

    +h˙xT(t)S˙x(t)tth˙xT(s)S˙x(s)ds,

    V6(xt,i,t)=di(t)T(x(t))Vi(x(t))(1dµi)ttdi(t)T(x(s))Vi(x(s))ds

    +Nj=1πij0dj(t)tt+θT(x(s))Vj(x(s))dsdθ

    +d22T(x(t))V(x(t))0dtt+θT(x(s))V(x(s))dsdθ,

    V7(xt,i,t)=τixT(t)Uix(t)ttτixT(s)Uix(s)ds+Nj=1πij0τjtt+θxT(s)Ujx(s)dsdθ

    +τ22xT(t)Ux(t)0τtt+θxT(s)Ux(s)dsdθ,

    V8(xt,i,t)=τi˙xT(t)Xi˙x(t)ttτi˙xT(s)Xi˙x(s)ds+Nj=1πij0τjtt+θ˙xT(s)Xj˙x(s)dsdθ

    +τ22˙xT(t)X˙x(t)0τtt+θ˙xT(s)X˙x(s)dsdθ,

    V9(xt,i,t)=hi˙xT(t)Yi˙x(t)tthi˙xT(s)Yi˙x(s)ds+Nj=1πij0hjtt+θ˙xT(s)Yj˙x(s)dsdθ

    +h22˙xT(t)Y˙x(t)0htt+θ˙xT(s)Y˙x(s)dsdθ,

    V10(xt,i,t)=τ2i2˙xT(t)Ti˙x(t)0τitt+θ˙xT(s)Ti˙x(s)dsdθ+τ36˙xT(t)T˙x(t)

    +Nj=1πij0τj0θtt+β˙xT(s)Tj˙x(s)dsdβdθ0τ0θtt+β˙xT(s)T˙x(s)dsdβdθ,

    V11(xt,i,t)=h2i2˙xT(t)Zi˙x(t)0hitt+θ˙xT(s)Zi˙x(s)dsdθ+h36˙xT(t)Z˙x(t)

    +Nj=1πij0hj0θtt+β˙xT(s)Zj˙x(s)dsdβdθ0h0θtt+β˙xT(s)Z˙x(s)dsdβdθ,

    V12(xt,i,t)=σ2i2˙xT(t)Li˙x(t)0σitt+θ˙xT(s)Li˙x(s)dsdθ+σ36˙xT(t)L˙x(t)

    +Nj=1πij0σj0θtt+β˙xT(s)Lj˙x(s)dsdβdθ0σ0θtt+β˙xT(s)L˙x(s)dsdβdθ,

    V13(xt,i,t)=σ42˙xT(t)M˙x(t)σ20σtt+θ˙xT(s)M˙x(s)dsdθ.

    Using the upper bounds of discrete, neutral, distributed time-varying delays, leakage delays, Lemma 2 and with πii<0, the following relationship is obtained

    ji(2πijxT(t)Pj𝒜jttσjx(s)ds)jiπij(xT(t)Kjx(t)+ttσjxT(s)ds𝒜TjPjK1jPj𝒜jttσjx(s)ds)jiπij(xT(t)Kjx(t)+ttσxT(s)ds𝒜TjPjK1jPj𝒜jttσx(s)ds),

    jiπij(ttσjxT(s)ds)𝒜TjPj𝒜j(ttσjx(s)ds)jiπij(ttσxT(s)ds)𝒜TjPj𝒜j(ttσx(s)ds).

    Similarly,

    Nj=1πijttτj(t)ζT(s)Qjζ(s)dsNj=1,jiπijttτj(t)ζT(s)Qjζ(s)dsNj=1,jiπijttτζT(s)Qjζ(s)dsttτζT(s)Qζ(s)ds,

    Nj=1πijttτjxT(s)Wjx(s)dsttτxT(s)Wx(s)ds,

    Nj=1πijttσjxT(s)Rjx(s)dsttσxT(s)Rx(s)ds,

    Nj=1πijtthj(t)˙xT(s)Sj˙x(s)dstth˙xT(s)S˙x(s)ds,

    Nj=1πij0dj(t)tt+θT(x(s))Vj(x(s))dsdθ0dtt+θT(x(s))V(x(s))dsdθ,

    Nj=1πij0τjtt+θxT(s)Ujx(s)dsdθ0τtt+θxT(s)Ux(s)dsdθ,

    Nj=1πij0τjtt+θ˙xT(s)Xj˙x(s)dsdθ0τtt+θ˙xT(s)X˙x(s)dsdθ,

    Nj=1πij0hjtt+θ˙xT(s)Yj˙x(s)dsdθ0htt+θ˙xT(s)Y˙x(s)dsdθ,

    Nj=1πij0τj0θtt+β˙xT(s)Tj˙x(s)dsdβdθ0τ0θtt+β˙xT(s)T˙x(s)dsdβdθ,

    Nj=1πij0hj0θtt+β˙xT(s)Zj˙x(s)dsdβdθ0h0θtt+β˙xT(s)Z˙x(s)dsdβdθ,

    Nj=1πij0σj0θtt+β˙xT(s)Lj˙x(s)dsdβdθ0σ0θtt+β˙xT(s)L˙x(s)dsdβdθ.

    By using Lemma 1, we get

    (1dµi)ttdi(t)T(x(s))Vi(x(s))ds(1dµi)dittdi(t)T(x(s))dsVittdi(t)(x(s))ds,

    ttτixT(s)Uix(s)ds1τittτixT(s)dsUittτix(s)ds,

    tthi˙xT(s)Yi˙x(s)ds1hitthi˙xT(s)dsYitthi˙x(s)ds,

    0τitt+θ˙xT(s)Ti˙x(s)dsdθ2τ2i0τitt+θ˙xT(s)dsdθTi0τitt+θ˙x(s)dsdθ,

    0hitt+θ˙xT(s)Zi˙x(s)dsdθ2h2i0hitt+θ˙xT(s)dsdθZi0hitt+θ˙x(s)dsdθ,

    0σitt+θ˙xT(s)Zi˙x(s)dsdθ2σ2i0σitt+θ˙xT(s)dsdθZi0σitt+θ˙x(s)dsdθ,

    0σtt+θ˙xT(s)M˙x(s)dsdθ2σ20σtt+θ˙xT(s)dsdθZi0σtt+θ˙x(s)dsdθ.

    Note that from (3.1) and using the reciprocally convex technique in [34], we obtain

    ttτi˙xT(s)Xi˙x(s)dstτi(t)tτi˙xT(s)Xi˙x(s)dsttτi(t)˙xT(s)Xi˙x(s)ds

    1τiϖT(t)[RiXiRi]ϖ(t),

    where ϖ(t)=[xT(tτi(t))xT(tτi),xT(t)xT(tτi(t))]. For positive diagonal matrices H1i,H2i,H3i,H4i,H5i,H6i, by Assumption 1, we get

    [x(t)(x(t))]T[ˆL1H1iˆL2H1iH1i][x(t)(x(t))]0,

    [x(tτi(t))(x(tτi(t)))]T[ˆL1H2iˆL2H2iH2i][x(tτi(t))(x(tτi(t)))]0,

    [x(t)x(tτi)]T[ˆL1H3iˆL2H3iH3i][x(t)x(tτi)]0,

    [x(t)x(thi)]T[ˆL1H4iˆL2H4iH4i][x(t)x(thi)]0,

    [x(t)x(tσi)]T[ˆL1H5iˆL2H5iH5i][x(t)x(tσi)]0,

    [x(t)˙x(t)]T[ˆL1H6iˆL2H6iH6i][x(t)˙x(t)]0.

    Hence, for any matrices J1, J2 of appropriate dimensions, we get

    0=2[xT(t)J1+˙xT(t)J2][𝒜ix(tσi)+i(x(t))+𝒞i(x(tτi(t)))  +𝒟i˙x(thi(t))+ittdi(t)(x(s))ds+u(t)˙x(t)].

    Using (3.6) and adding (3.20)-(3.26), we have

    V(xt,i,t)2yT(t)u(t)γuT(t)u(t){ηT(t)Φη(t)},

    where

    ηT(t)=[xT(t)  xT(tτi(t))  xT(tτi)ttτixT(s)ds  T(x(t))  T(x(tτi(t)))

    xT(tσi)ttσixT(s)ds  ttσxT(s)dsuT(t)  ˙xT(thi(t))  xT(thi)

    tthixT(s)ds  ttdi(t)T(x(s))ds  ˙xT(t)].

    Hence from equation (3.4) we have,

    V(xt,i,t)2y(t)Tu(t)γu(t)Tu(t)0.

    Now, to show the passivity of the delayed NNs in (2.3), we take

    J(tp)=𝔼{tp0[γu(t)Tu(t)2y(t)Tu(t)]dt}

    where tp0.

    From Dynkin's formula, we get

    𝔼[tp0V(xt,i,t)dt]=𝔼[V(xtp,i,tp)]𝔼[V(x0,(0),0)].

    Therefore,

    J(tp)=𝔼{tp0[γu(t)Tu(t)2y(t)Tu(t)+V(xt,i,t)]dt}𝔼[tp0V(xt,i,t)dt]    =𝔼{tp0[γu(t)Tu(t)2y(t)Tu(t)+V(xt,i,t)]dt}  𝔼[V(xt,i,t)]+𝔼[V(x0,(0),0)].

    By applying lemma 3 to (3.4), we have

    Φ<0.

    Thus, if (3.30) holds, then 𝔼[V(xtp,i,tp)]0 and V(x0,(0),0)=0 holds with zero initial conditions. From (3.30), it follows that J(tp)0 for any tp0, which implies (2.5) is satisfied and hence the delayed NNs (2.3) is locally passive.

    Now, we prove the global passivity of the system.

    By taking expectation of (3.27) and then integration from 0 to t we get,

    t0𝔼[V(xs,r(s),s)]ds2t0𝔼[yT(s)u(s)]dsγt0𝔼[uT(s)u(s)]dst0𝔼[ηT(s)Φη(s)]ds.

    Then by Dynkin's formula,

    𝔼[V(xt,i,t)]𝔼[V(x0,(0),0)]2t0𝔼[yT(s)u(s)]dsγt0𝔼[uT(s)u(s)]ds

    t0𝔼[ηT(s)Φη(s)]ds.

    Hence,

    𝔼[V(xt,i,t)]t0𝔼[ηT(s)Φη(s)]ds𝔼[V(x0,(0),0)]+2t0𝔼[yT(s)u(s)]ds+γt0𝔼[uT(s)u(s)]ds<,t0.

    By Jenson's inequality and (3.6), we get

    E𝒜ittσix(s)ds2=E[𝒜ittσix(s)ds]T[𝒜ittσix(s)ds]λmax(𝒜2i)E[ttσix(s)ds]T[ttσix(s)ds]λmax(𝒜2i)λmin(Ri)[ttσiEx(s)ds]TRi[ttσiEx(s)ds]σiλmax(𝒜2i)λmin(Ri){ttσiExT(s)Rix(s)ds}σiλmax(𝒜2i)λmin(Ri)EV4(xt,i,t)σλmax(𝒜2i)λmin(Ri)EV(xt,i,t)σλmax(𝒜2i)λmin(Ri)EV(x0,𝒜(0),0),t0.

    Similarly, it follows from the definition of V1(xt,i,t) that

    Ex(t)𝒜ittσix(s)ds2=E[𝒜ittσix(s)ds]T[𝒜ittσix(s)ds]

    EV1(xt,i,t)λmin(Pi)

    EV(xt,i,t)λmin(Pi)

    EV(x0,(0),0)λmin(Pi),t0.

    Hence, it can be obtained that

    Ex(t)2=Ex(t)𝒜ittσix(s)ds+𝒜ittσix(s)ds22EAittσix(s)ds2+2Ex(t)𝒜ittσix(s)ds22σλmax(𝒜2i)λmin(Ri)EV(x0,(0),0)+2EV(x0,(0),0)λmin(Pi)<,t0,

    EV(x0,(0),0){2λmaxiS(Pi)(1+σ2imaxiS𝒜i)+τmaxiS{λmax(Qi)}+τ2λmax(Q)+τmaxiS{λmax(Wi)}+τ2λmax(W)+σmaxiS{λmax(Ri)}+σ2λmax(R)+hmaxiS{λmax(Si)}+h2λmax(S)+d2maxiS{λmax(Vi)}+d3λmax(V)+τ2maxiS{λmax(Ui)}+τ3λmax(U)+τ2maxiS{λmax(Xi)}+τ3λmax(X)+h2maxiS{λmax(Yi)}+h3λmax(Y)+τ3maxiS{λmax(Ti)}+τ4λmax(T)+h3maxiS{λmax(Zi)}+h4λmax(Z)+σ3maxiS{λmax(Li)}+σ4λmax(L)+σ5λmax(M)}<.

    From (3.33) and (3.34), we get that the solution of the system (2.3) is locally passive. Then the solutions x(t)=x(t,0,φ) of system (2.3) is bounded on [0,). The solution x(t) on [0,) is uniformly continuous because ˙x(t) is bounded on [0,). Further, from the equation (3.31), the following holds:

    λmin(Φ)t0E[xT(s)x(s)]ds𝔼[V(xt,i,t)]t0𝔼[ηT(s)Φη(s)]ds

    𝔼[V(x0,(0),0)]+2t0𝔼[yT(s)u(s)]ds

    +γt0𝔼[uT(s)u(s)]ds

    <,t0.

    From Barbalats lemma [30], 𝔼[x(t)2]0 as t holds. Hence the proof is complete.

    Remark 1. Recently, studies on passivity analysis for neural networks of neutral type with Markovian jumping parameters and time delay in the leakage term were conducted in [5]. By constructing proper Lyapunov–Krasovskii functional, new delay-dependent passivity conditions are derived in terms of LMIs and it can easily be checked using standard numerical packages. Moreover, it is well known that the passivity behaviour of neural networks is very sensitive to the time delay in the leakage term. Triple and quadruple integrals have not been taken into account to derive the passivity conditions in [5]. Mode-dependent time delays were not included in [5]. Very recently, a mode-dependent approach is proposed by constructing a novel Lyapunov functional, where some terms involving triple or quadruple integrals are taken into account to study the state estimation problem in [22]. Motivated by this reason, we have introduced improved Lyapunov–Krasovskii functional with triple and quadruple integrals for deriving the reported stability results in this paper. Based on this discussion, our results will give less conservative results than those studied in [5], [22].

    In this section, a numerical example is provided to demonstrate the validity of the proposed theorems.

    Example 1. Consider a 2-D Mode-dependent Markov jump NNNs with mixed time-delays (2.3) with the following parameters

    𝒜1=[8.4009],   𝒜2=[7.8008.5],   1=[0.210.190.240.1],   2=[0.90.90.50.8],𝒞1=[0.090.20.20.1],   𝒞2=[0.10.10.20.3],   𝒟1=[0.200.20.09],   𝒟2=[0.100.50.1],1=[0.5000.5],   2=[0.10.020.20.07],   ˆL1=[0000],   ˆL2=[0.25000.25].

    Take 1(s)=2(s)=tanh(s), τ1(t)=τ2(t)=h1(t)=h2(t)=d1(t)=d2(t)=0.1cost+0.4, σ1=σ2=0.1, τµ1=τµ2=hµ1=hµ2=dµ1=dµ2=0.1. Γ=[7766].

    By using the MATLAB LMI toolbox, we can obtain the following feasible solution for the LMIs (3.1)–(3.4):

    P1=[0.01570.00080.00080.0143],   P2=[0.00680.00040.00040.0060],   Q11=[0.24010.00070.00070.2592],Q12=[0.37700.01920.01920.3992],   Q21=[0.41010.00430.02010.4457],   Q22=[0.64280.07730.02310.6959],Q31=[1.37860.01330.01331.4063],   Q32=[1.91010.02840.02841.9903],   W1=[0.10390.00880.00880.1018],W2=[0.15170.01210.01210.1487],   R1=[0.17580.01600.01600.1721],   R2=[0.56750.03330.03330.5563],S1=[1.08170.00750.00750.7204],   S2=[1.22310.00930.00930.9256],   V1=[10.71130.56920.56929.9973],V2=[10.84830.59530.595310.2619],   U1=[0.51200.04190.04190.5056],   U2=[0.62380.04920.04920.6133],X1=[21.04700.11530.115321.2070],   X2=[18.56420.31390.313918.6620],   Y1=[27.59460.00190.001926.6557],Y2=[28.41150.10960.109627.4131],   T1=[3.09280.09430.09433.0505],   T2=[3.81820.04310.04313.7941],Z1=[3.08670.10800.10803.0591],   Z2=[3.86470.05040.05043.8446],   L1=[0.00680.00010.00010.0068],L2=[0.00590.00040.00040.0062],   K1=[1.34610.14100.14101.3280],   K2=[0.09190.01350.01350.0907],Q1=[4.09130.18500.18504.4495],   Q2=[6.63320.76860.34447.3948],   γ=0.2272.

    Continuing in this way, the remaining feasible matrices are obtained. This shows that the given system (2.3) is globally passive in the mean square.

    In this paper, passivity analysis of Markovian jumping NNNs with time delays in the leakage term is considered. Delay-mode-dependent passivity conditions are derived by taking the inherent characteristic of such kinds of NNs into account. An improved LKF, with the triple integral terms and quadruple integrals, is constructed and the results are derived in terms of linear matrix inequalities. The information of the mode-dependent of all delays have been taken into account in the constructed LKF and derived novel stability criterion. Theoretical results are validated through a numerical example.



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