# Passivity of Memristive BAM Neural Networks with Probabilistic and Mixed Time-Varying Delays

Authors: Weiping Wang; Meiqi Wang; Xiong Luo; Lixiang Li; Wenbing Zhao
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper is concerned with the passivity problem of memristive bidirectional associative memory neural networks (MBAMNNs) with probabilistic and mixed time-varying delays. By applying random variables with Bernoulli distribution, the information of probability time-varying delays is taken into account. Furthermore, we consider the probability distribution of the variation and the extent of the delays; therefore, the results derived are less conservative than in the existing papers. In particular, the leakage delays as well as distributed delays are all taken into consideration. Based on appropriate Lyapunov-Krasovskii functionals (LKFs) and some useful inequalities, several conditions for passive performance are established in linear matrix inequalities (LMIs). Finally, numerical examples are given to demonstrate the feasibility of the presented theories, and the results reveal that the probabilistic and mixed time-varying delays have an unstable influence on the system and should not be ignored.


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## Body

## 1. Introduction

Bidirectional associative memory neural networks (BAMNNs) are a class of two-layer neural systems, which were first introduced by Kosko in 1987. The neurons in the first layer are connected to another layer, and in the same layer, the neurons are not interconnected [1–3]. Owing to their special structure, BAMNNs have displayed many good features in various areas such as signal processing, image processing, and optimization problems [4–6]. In 2015, the stability of inertial BAMNNs with time-varying delay via impulsive control was discussed in [7]. Zhang et al. considered the exponential stability of BAMNNs with time-varying delays in [8]. Wang et al. addressed the global asymptotic stability of impulsive fractional-order BAMNNs with time delay in [9].Memristor, a combination of a resistor and memory, has received increasing attention in many fields [10–13]. By applying the nonvolatile feature of the memristor, researchers were able to develop MBAMNN models. Because of the pinched hysteresis effects, MBAMNNs have a memory function, which can be used to imitate the human brain [14, 15]. In 2015, nonfragile synchronization of MBAMNNs with random feedback gain fluctuations was investigated in [16]. Based on functional differential inclusions, Jiang et al. obtained the dynamic behaviors for MBAMNNs with time-varying delays in [17].Moreover, passivity is a special case of a broader theory of dissipativity, which plays a significant role in the stability analysis of dynamical systems, nonlinear control, and other areas. The main innate character of passivity theory is that the passive characteristics can make the system internally stable [18–20]. In recent years, many researchers have proposed passivity analysis for memristive neural networks (MNNs). Liu and Xu investigated the passivity analysis of MNNs with different state-dependent memductance functions and mixed time-varying delays in [21]. In 2016, the passivity of MBAMNNs with uncertain delays and different memductance was investigated in [22]. Nevertheless, there are few people to study the passivity of MBAMNNs, which encourages our idea.In the human brain, the transmission of information in neurons is often accompanied by a time delay, so time delay is inevitable in the neural networks, which is the origin of oscillation, divergence, and so forth [23–33]. Sometimes, the value of delay may be very large, but the probability of such delay is very small. Therefore, we use the probability distribution of time delay in the interval to reflect an actual situation better. Furthermore, it is clearer to describe the probabilistic time-varying delays through introducing random variables with Bernoulli distribution. In recent years, some researchers have discussed the probabilistic time-varying delays in the neural networks [34, 35]. In 2016, Pradeep et al. investigated the robust stability analysis of stochastic neural networks with probabilistic time-varying delays in [36]. Li et al. considered passivity analysis of memristive neural networks with probabilistic time-varying delays in [37]. Hence, it is of great importance to research the passivity of MBAMNNs with probabilistic time-varying delays.In addition, there also exist two types of time-varying delays named leakage delays (or forgetting delays) and distributed time delays. The research of leakage time delay can be traced back to the early 90s of the last century; researchers found out that, due to the delay in switching time or signal transmission, there is a time delay in the negative feedback term of the network system; this delay is named leakage time delay. As is well known, leakage delays exist in many real systems such as population dynamics and neural networks [38, 39]. Moreover, leakage delay also has a significant influence on the dynamics of neural networks because it has been shown that such kind of time delay in the leakage term has a tendency to destabilize a system. Under the influence of leakage and additive time-varying delays, robust passivity analysis for neural networks was addressed in [40]. In 2016, the robust stability analysis for discrete-time neural networks with leakage delays was studied in [41].On the other hand, due to the presence of multiple parallel paths with a variety of neuronal synapses’ lengths and sizes, there is a spatial width of the network, and then there may exist either a distribution of the transmission voltage in these parallel paths or a distribution of transmission delays over a period of time. Hence, the distribution delay is used to describe this phenomenon [42, 43]. In 2015, Du et al. investigated the passivity of neural networks with discrete and distributed time-varying delays in [44]. In 2016, Yang et al. considered finite-time stabilization of uncertain neural networks with distributed time-varying delays in [45]. However, to the best of our knowledge, there are few results on the passivity of MBAMNNs with probabilistic, leakage, and distributed time-varying delays. Thus, it is significant to study the passivity of MBAMNNs with these time-varying delays.Motivated by the main points discussed above, the contribution of this paper lies in three aspects.(1) This is the first attempt to discuss the passivity analysis of MBAMNNs with probabilistic and mixed time-varying delays. In particular, the leakage delays as well as distributed delays are all taken into consideration.(2) The LKFs that we designed include double and triple integral terms, and by applying some helpful inequalities, the passivity analysis of MBAMNNs becomes less conservative than the existing results [19, 21].(3) After using MATLAB LMI control toolbox, all the derived results are expressed in LMIs, and a feasible solution can be easily obtained.The rest of the paper is structured as follows. In Section2, we introduce the model of the MBAMNNs with probabilistic time-varying delays. In Section 3, the main results on passivity analysis of MBAMNNs with probabilistic and mixed time-varying delays are derived. In Section 4, some numerical simulations are provided to demonstrate the feasibility of our results. In Section 5, the conclusion is shown.


## 2. Model Description and Preliminaries

In this paper, we propose the MBAMNN with probabilistic time-varying delays as follows:(1)x˙it=-Cixit+∑j=1majixitfjyjt+∑j=1mbjixitfjyjt-τt+uit,y˙jt=-Djyjt+∑i=1neijyjtgixit+∑i=1nhijyjtgixit-dt+vjt,i=1,…,n,j=1,…,m, or it can be rewritten as follows:(2)x˙t=-Cxt+Axtfyt+Bxtfyt-τt+ut,y˙t=-Dyt+Eytgxt+Hytgxt-dt+vt, where xi(t) and yj(t) denote the state variables related to the ith and jth neurons. A(x(t))=(aji(xi(t)))m×n and  E(y(t))=(eij(yj(t)))n×m are the connection weight matrices, fj(·) and gi(·) are the activation functions, and B(x(t))=(bji(xi(t)))m×n and  H(y(t))=(hij(yj(t)))n×m are the delayed connection weight matrices. The self-feedback connection weights Ci and Dj are positive diagonal matrices. ui(t) and vj(t) represent the continuous external inputs; the nonnegative continuous variables τ(t) and d(t) correspond to the time-varying delays.Assumption 1.
The functionsfj(t)(j=1,2,…,m) and gi(t)(i=1,2,…,n) are bounded and continuous and satisfy the conditions as follows:(3)ςj-≤fjb-fjab-a≤ςj+,σi-≤gib-giab-a≤σi+, with ς-=diag⁡(ς1-,ς2-,…,ςm-),  ς+=diag⁡(ς1+,ς2+,…,ςm+),  σ-=diag⁡(σ1-,σ2-,…,σn-), and  σ+=diag⁡(σ1+,σ2+,…,σn+),  a,b∈R,a≠b.Based on the current-voltage characteristic and the feature of memristor, the memristive connection weightsaji(xi(t)),  bji(xi(t)),  eij(yj(t)), and hji(xi(t)) will change with time. Then, we let (4)ajixit=a^ji,xit>Θi,aˇji,xit≤Θi,bjixit=b^ji,xit>Θi,bˇji,xit≤Θi,eijyjt=e^ij,yjt>Ψj,eˇij,yjt≤Ψj,hijyjt=f^ij,yjt>Ψj,fˇij,yjt≤Ψj, in which aji,  bji,  eij, and hij are constants and the switching jumps Θi>0,  Ψj>0.Based on Figures1 and 2, it is clear that aji(xi(t)),  bji(xi(t)),  eij(yj(t)), and hij(yj(t)) are piecewise continuous functions; the solutions of the systems are indicated in Filippov’s sense and the interval is represented by [·,·]. Set (5)a¯ji=max⁡a^ji,aˇji,a_ji=min⁡a^ji,aˇji,b¯ji=max⁡b^ji,bˇji,b_ji=min⁡b^ji,bˇji,e¯ij=max⁡e^ij,eˇij,e_ij=min⁡e^ij,eˇij,h¯ij=max⁡h^ij,hˇij,h_ij=min⁡h^ij,hˇij, for i=1,2,…,n,  j=1,2,…,m. co⁡[r¯,r_] indicates the convex closure of [r¯,r_]. Obviously, the set-valued maps are defined as (6)co⁡ajixit=a^ji,xit>Θi,a_ji,a¯ji,xit=Θi,aˇji,xit<Θi,co⁡bjixit=b^ji,xit>Θi,b_ji,b¯ji,xit=Θi,bˇji,xit<Θi,co⁡eijyjt=e^ij,yjt>Ψj,e_ij,e¯ij,yjt=Ψj,eˇij,yjt<Ψj,co⁡hijyjt=h^ij,yjt>Ψj,h_ij,h¯ij,yjt=Ψj,hˇij,yjt<Ψj.Figure 1
The typical current-voltage characteristic of a memristor.Figure 2
The characteristic of a piecewise linear charge-controlled memristor.As a matter of convenience, we make the following assumptions.Assumption 2.
We define the probability distribution of time delaysτ(t) and d(t) as follows:(7)Prob⁡τt∈0,τ1=μ0,Prob⁡τt∈τ1,τ2=1-μ0,Prob⁡dt∈0,d1=ω0,Prob⁡dt∈d1,d2=1-ω0, where 0≤μ0≤1 and 0≤ω0≤1 are constants.Thus, the random variablesμ(t) and ω(t) can be defined as(8)μt=1,τt∈0,τ1,0,τt∈τ1,τ2,ωt=1,dt∈0,d1,0,dt∈d1,d2.Then, it can be derived thatμ(t) and ω(t) are Bernoulli distributed sequences with(9)Prob⁡μt=1=μ0,Prob⁡μt=0=1-μ0,Prob⁡ωt=1=ω0,Prob⁡ωt=0=1-ω0.According to Assumption2, it is easy to see that(10)Eμt-μ0=0,Eμt-μ02=μ01-μ0,Eωt-ω0=0,Eωt-ω02=ω01-ω0.Now, time-varying delaysτ1(t),  τ2(t),  d1(t), and d2(t) are introduced as(11)τt=τ1t,τt∈0,τ1,τ2t,τt∈τ1,τ2,dt=d1t,dt∈0,d1,d2t,dt∈d1,d2.Assumption 3.
Here, constantsτ1,  τ2,  d1,  d2,  μ1,  μ2,  ω1, and ω2 exist, such that(12)0≤τ1t≤τ1,τ˙1t≤μ1,0≤τ2t≤τ2,τ˙2t≤μ2,0≤d1t≤d1,d˙1t≤ω1,0≤d2t≤d2,d˙2t≤ω2.Assumption 4.
The leakage delaysδ(t) and  ρ(t) and distributed delays α(t) and  β(t) satisfy(13)0≤δt≤δ,δ˙t≤δ0,0≤ρt≤ρ,ρ˙t≤ρ0,0≤αt≤α,0≤βt≤β.By employing the theories of set-valued maps, differential inclusions, stochastic variablesμ0, ω0, and new functions τ1(t),  τ2(t),  d1(t), and  d2(t), system (1) becomes(14)x˙it∈-Cixit+∑j=1mco⁡ajixitfjyjt+μt∑j=1mco⁡bjixitfjyjt-τ1t+1-μt∑j=1mco⁡bjixitfjyjt-τ2t+uit,y˙jt∈-Djyjt+∑i=1nco⁡eijyjtgixit+ωt∑i=1nco⁡hijyjtgixit-d1t+1-ωt∑i=1nco⁡hijyjtgixit-d2t+vjt. Or it can be rewritten as follows:(15)x˙t=-Cxt+Axtfyt+μtBxtfyt-τ1t+1-μtBxtfyt-τ2t+ut,y˙t=-Dyt+Eytgxt+ωtHytgxt-d1t+1-ωtHytgxt-d2t+vt; equivalently,(16)x˙t=-Cxt+Axtfyt+μ0Bxtfyt-τ1t+1-μ0Bxtfyt-τ2t+μt-μ0Bxtfyt-τ1t-fyt-τ2t+ut,y˙t=-Dyt+Eytgxt+ω0Hytgxt-d1t+1-ω0Hytgxt-d2t+ωt-ω0Hytgxt-d1t-gxt-d2t+vt.Definition 5.
System (15) is called passive if there exists a scalar γ>0 such that(17)2∫0tzfTysgTxsusvsds≥-γ∫0tzuTsvTsusvsds, for all solutions of (15) with x(0)=0 and y(0)=0 and for all tz≥0.Remark 6.
Passivity analysis originates from circuit theory and it uses the input-output description method based on energy to design and analyze a system. The physical meaning of passivity is reflected in Definition5 where the energy growth of the system is always less than or equal to the total energy of the external inputs; this means that the passive system is always accompanied by the loss of energy. In fact, the storage function of the passive system can be used as a Lyapunov function under certain conditions. Thus, both Lyapunov stability theory and passivity theory can be used to research the stability of the system.Lemma 7.
For any scalarsp2>p1>0, matrix R∈Rn×n,  R=RT>0, and vector function ω:[p1,p2]↦Rn, the inequalities hold as follows:(18)p2-p1∫p1p2ωTsRωsds≥∫p1p2ωsdsTR∫p1p2ωsds,(19)p2-p122∫p1p2∫t+θtωTsRωsdsdθ≥∫p1p2∫t+θtωsdsTR∫p1p2∫t+θtωsds.


## 3. Main Results

For derivation convenience, we denote(20)A=max⁡a_ji,a¯ji,B=max⁡b_ji,b¯ji,E=max⁡e_ij,e¯ij,H=max⁡h_ij,h¯ij.Theorem 8.
Under Assumptions1–3, system (16) is passive, if there exist any appropriately dimensional matrices Mm(m=1,2,…,4),  Ww(w=1,2,…,6), a scalar γ>0, and symmetric positive definite matrices Ri(i=1,2),  Qj(j=3,4,…,6),  Oo(o=3,4,…,6),  Pk(k=2,3,6,7),  Zl(l=1,2,…,4), and  Nn(n=1,2), such that the following LMIs hold:(21)Ωi,j=Ωi,j14×14<0,Φi,j=Φi,j14×14<0, where(22)Ω1,1=-R1C-CTR1+Q3+Q4-P2-2Z1-2Z2-M1C-CTM1-2ς-W1ς+,Ω1,4=P2,Ω1,6=-M1-M2C,Ω1,7=R1A+M1A+W1ς-+ς+,Ω1,8=μ0R1+M1B,Ω1,9=1-μ0M1+R1B,Ω1,12=2τ1Z1,Ω1,13=2τ2-τ1Z2,Ω1,14=R1+M1,Ω2,2=-1-μ1Q3-2ς-W2ς+,Ω2,8=-1-μ1Q3+W2ς-+ς+,Ω3,3=-1-μ2Q5-2ς-W3ς+,Ω3,9=-1-μ2Q5+W3ς-+ς+,Ω4,4=Q5+Q6-Q4-P2-P3,Ω4,5=P3,Ω4,10=Q5+Q6-Q4,Ω5,5=-Q6-P3,Ω5,11=-Q6,Ω6,6=τ12P2+τ2-τ12P3+τ122Z1+τ22-τ122Z2-2M2,Ω6,7=M2A,Ω6,8=μ0M2B,Ω6,9=1-μ0M2B,Ω6,14=M2,Ω7,7=-2W1,Ω7,14=-I,Ω8,8=-1-μ1Q3-2W2,Ω9,9=-1-μ2Q5-2W3,Ω10,10=Q5+Q6-Q4,Ω11,11=-Q6,Ω12,12=-2τ12Z1,Ω13,13=-2τ2-τ12Z2,Ω14,14=-γI,(23)Φ1,1=-R2D-DTR2+O3+O4-P6-2Z3-2Z4-M3D-DTM3-2σ-W4σ+,Φ1,4=P6,Φ1,6=-M3-M4D,Φ1,7=R2E+M3E+W4σ-+σ+,Φ1,8=ω0R2+M3H,Φ1,9=1-ω0M3+R2H,Φ1,12=2d1Z3,Φ1,13=2d2-d1Z4,Φ1,14=R2+M3,Φ2,2=-1-ω1O3-2σ-W5σ+,Φ2,8=-1-ω1O3+W5σ-+σ+,Φ3,3=-1-ω2O5-2σ-W6σ+,Φ3,9=-1-ω2O5+W6σ-+σ+,Φ4,4=O5+O6-O4-P6-P7,Φ4,5=P7,Φ4,10=O5+O6-O4,Φ5,5=-O6-P7,Φ5,11=-O6,Φ6,6=d12P6+d2-d12P7+d122Z3+d22-d122Z4-2M4,Φ6,7=M4E,Φ6,8=ω0M4H,Φ6,9=1-ω0M4H,Φ6,14=M4,Φ7,7=-2W4,Φ7,14=-I,Φ8,8=-1-ω1O3-2W5,Φ9,9=-1-ω2O5-2W6,Φ10,10=O5+O6-O4,Φ11,11=-O6,Φ12,12=-2d12Z3,Φ13,13=-2d2-d12Z4,Φ14,14=-γI.Proof.
Consider the following LKF candidate:(24)Vt=∑i=18Vit, where(25)V1t=xTtR1xt,V2t=yTtR2yt,V3t=∫t-τ1ttxsfysTQ3xsfysds+∫t-τ1txsfysTQ4xsfysds+∫t-τ2tt-τ1xsfysTQ5xsfysds+∫t-τ2t-τ1xsfysTQ6xsfysds,V4t=∫t-d1ttysgxsTO3ysgxsds+∫t-d1tysgxsTO4ysgxsds+∫t-d2tt-d1ysgxsTO5ysgxsds+∫t-d2t-d1ysgxsTO6ysgxsds,V5t=τ1∫-τ10∫t+θtx˙TsP2x˙sdsdθ+τ2-τ1∫-τ2-τ1∫t+θtx˙TsP3x˙sdsdθ,V6t=d1∫-d10∫t+θty˙TsP6y˙sdsdθ+d2-d1∫-d2-d1∫t+θty˙TsP7y˙sdsdθ,V7t=∫-τ10∫θ0∫t+λtx˙TsZ1x˙sdsdλdθ+∫-τ2-τ1∫θ0∫t+λtx˙TsZ2x˙sdsdλdθ,V8t=∫-d10∫θ0∫t+λty˙TsZ3y˙sdsdλdθ+∫-d2-d1∫θ0∫t+λty˙TsZ4y˙sdsdλdθ.
Then, we define the infinitesimal generatorL of V(t) as (26)LVt=limΔ→0+⁡1ΔEVt+Δt-Vt.
Taking the mathematical expectation ofV(t), we get(27)ELV1t=E2xTtR1-Cxt+Axtfyt+μ0Bxtfyt-τ1t+1-μ0Bxtfyt-τ2t+μt-μ0Bxtfyt-τ1t-fyt-τ2t+ut≤E2xTtR1-Cxt+Afyt+μ0Bfyt-τ1t+1-μ0Bfyt-τ2t+ut,ELV2t=E2yTtR2-Dyt+Eytgxt+ω0Hytgxt-d1t+1-ω0Hytgxt-d2t+ωt-ω0Hytgxt-d1t-gxt-d2t+vt≤E2yTtR2-Dyt+Egxt+ω0Hgxt-d1t+1-ω0Hgxt-d2t+vt.
According to Assumption3, we obtain(28)ELV3t≤ExTtQ3+Q4xt-1-μ1xt-τ1tfyt-τ1tTQ3xt-τ1tfyt-τ1t+xt-τ1fyt-τ1TQ5+Q6-Q4xt-τ1fyt-τ1-1-μ2xt-τ2tfyt-τ2tTQ5xt-τ2tfyt-τ2t-xt-τ2fyt-τ2TQ6xt-τ2fyt-τ2,ELV4t≤EyTtO3+O4yt-1-ω1yt-d1tgxt-d1tTO3yt-d1tgxt-d1t+yt-d1gxt-d1TO5+O6-O4yt-d1gxt-d1-1-ω2yt-d2tgxt-d2tTO5yt-d2tgxt-d2t-yt-τ2gxt-τ2TO6yt-τ2gxt-τ2.
It is obvious that(29)ELV5t=Ex˙Ttτ12P2+τ2-τ12P3x˙t-τ1∫t-τ1tx˙TsP2x˙sds-τ2-τ1∫t-τ2t-τ1x˙TsP3x˙sds,ELV6t=Ey˙Ttd12P6+d2-d12P7y˙t-d1∫t-d1ty˙TsP6y˙sds-d2-d1∫t-d2t-d1y˙TsP7y˙sds.
Using Lemma7, we have(30)-τ1∫t-τ1tx˙TsP2x˙sds≤-∫t-τ1tx˙sdsTP2∫t-τ1tx˙sds=-xt-xt-τ1TP2xt-xt-τ1,-τ2-τ1∫t-τ2t-τ1x˙TsP3x˙sds≤-∫t-τ2t-τ1x˙sdsTP3∫t-τ2t-τ1x˙sds=-xt-τ1-xt-τ2TP3xt-τ1-xt-τ2.
Similarly,(31)-d1∫t-d1ty˙TsP6y˙sds≤-∫t-d1ty˙sdsTP6∫t-d1ty˙sds=-yt-yt-d1TP6yt-yt-d1,(32)-d2-d1∫t-d2t-d1y˙TsP7y˙sds≤-∫t-d2t-d1y˙sdsTP7∫t-d2t-d1y˙sds=-yt-d1-yt-d2TP7yt-d1-yt-d2.
Moreover, we obtain(33)ELV7t=Ex˙Ttτ122Z1+τ22-τ122Z2x˙t-∫-τ10∫t+θtx˙TsZ1x˙sdsdθ-∫-τ2-τ1∫t+θtx˙TsZ2x˙sdsdθ,ELV8t=Ey˙Ttd122Z3+d22-d122Z4y˙t-∫-d10∫t+θty˙TsZ3y˙sdsdθ-∫-d2-d1∫t+θty˙TsZ4y˙sdsdθ.
Using (18), we have (34)-∫-τ10∫t+θtx˙TsZ1x˙sdsdθ≤-2τ12∫-τ10∫t+θtx˙sdsdθTZ1∫-τ10∫t+θtx˙sdsdθ=-2xTtZ1xt+4τ1xTtZ1∫t-τ1txsds-2τ12∫t-τ1txsdsTZ1∫t-τ1txsds,-∫-τ2-τ1∫t+θtx˙TsZ2x˙sdsdθ≤-2τ2-τ12∫-τ2-τ1∫t+θtx˙sdsdθT×Z2∫-τ2-τ1∫t+θtx˙sdsdθ=-2xTtZ2xt+4τ2-τ1xTtZ2∫t-τ2t-τ1xsds-2τ2-τ12∫t-τ2t-τ1xsdsTZ2∫t-τ2t-τ1xsds.
Similarly,(35)-∫-d10∫t+θty˙TsZ3y˙sdsdθ≤-2d12∫-d10∫t+θty˙sdsdθTZ3∫-d10∫t+θty˙sdsdθ=-2yTtZ3yt+4d1yTtZ3∫t-d1tysds-2d12∫t-d1tysdsTZ3∫t-d1tysds,(36)-∫-d2-d1∫t+θtyTsZ4ysdsdθ≤-2d2-d12∫-d2-d1∫t+θtysdsdθT×Z4∫-d2-d1∫t+θtysdsdθ=-2yTtZ4yt+4d2-d1yTtZ4∫t-d2t-d1ysds-2d2-d12∫t-d2t-d1ysdsTZ4∫t-d2t-d1ysds.
To derive a less conservative criterion, we add the following inequations with any matricesM1,  M2,  M3, and M4 of appropriate dimensions:(37)0≤2xTtM1+x˙TtM2-x˙-Cxt+Afyt+μ0Bfyt-τ1t+1-μ0Bfyt-τ2t+ut,0≤2yTtM3+y˙TtM4-y˙-Dyt+Egxt+ω0Hgxt-d1t+1-ω0Hgxt-d2t+vt.
Based on Assumption1 [20], we have (38)ς-xt-fyt≤0,fyt-ς+xt≤0,σ-yt-gxt≤0,gxt-σ+yt≤0, and there exist positive diagonal matrices Wi>0(i=1,2,…,8); the following inequalities hold:(39)0≤2ς-xt-fytTW1fyt-ς+xt≤-2fTytW1fyt+2xTtW1ς-+ς+fyt-2xTtς-W1ς+xt.
Similarly,(40)0≤-2fTyt-τ1W2fyt-τ1+2xtTt-τ1W2ς-+ς+fyt-τ1-2xTt-τ1ς-W2ς+xt-τ1,0≤-2fTyt-τ2W3fyt-τ2+2xTt-τ2W3ς-+ς+fyt-τ2-2xTt-τ2ς-W3ς+xt-τ2,0≤2σ-yt-gxtTW4gxt-σ+yt≤-2gTxtW4gxt+2yTtW4σ-+σ+gxt-2yTtσ-W4σ+yt,0≤-2gTxt-d1W5gxt-d1+2ytTt-d1W5σ-+σ+gxt-d1-2yTt-d1σ-W5σ+yt-d1,0≤-2gTxt-d2W6gxt-d2+2yTt-d2W6σ-+σ+gxt-d2-2yTt-d2σ-W6σ+yt-d2.
Define(41)ξt=∫t-τ1txTsdsxTtxTt-τ1txTt-τ2txTt-τ1xTt-τ2x˙tfTytfTyt-τ1tfTyt-τ2tfTyt-τ1fTyt-τ2∫t-τ1txTsds∫t-τ2t-τ1xTsdsuTt,ζt=∫t-d1tyTsdsyTtyTt-d1tyTt-d2tyTt-d1yTt-d2y˙tgTxtgTxt-d1tgTxt-d2tgTxt-d1gTxt-d2∫t-d1tyTsds∫t-d2t-d1yTsdsvTt.
Combining (23), (27)–(37), and (40), we have(42)ELVt-2fTysgTxsusvs-γuTtvTtutvtds≤EξTtΩξt+ζTtΦζt.
Thus, we conclude that if (42) holds, then(43)ELVt-2fTysgTxsusvs-γuTtvTtutvtds≤0.
By integrating (43) aboutt from 0 to tz, we get (44)2E∫0tzfTysgTxsusvsds≥EVt-EV0-γE∫0tzuTsvTsusvsds≥-γE∫0tzuTsvTsusvsds.
Therefore, from Definition5, we know that the MBAMNN is passive. This completes the proof.Considering the MBAMNN with leakage, probabilistic, and distributed time-varying delays, then model (1) becomes(45)x˙it=-Cixit-δt+∑j=1majixitfjyjt+∑j=1mbjixitfjyjt-τt+∑j=1mkjixit∫t-αttfjyjsds+uit,y˙jt=-Djyjt-ρt+∑i=1neijyjtgixit+∑i=1nhijyjtgixit-dt+∑i=1nlijyjt∫t-βttgixisds+vjt, where the nonnegative continuous variables δ(t),  ρ(t) and α(t),  β(t) correspond to the leakage and distributed time-varying delays and K(x(t))=(kji(xi(t)))m×n and L(x(t))=(lij(yj(t)))n×m are the delayed connection weight matrix functions. Then, we let(46)kjixit=b^ji,xit>Θi,bˇji,xit≤Θi,lijyjt=f^ij,yjt>Ψj,fˇij,yjt≤Ψj, in which kji and lij are known constants and kji(xi(t)) and lij(yj(t)) are piecewise continuous functions; we set(47)k¯ji=max⁡k^ji,kˇji,k_ji=min⁡k^ji,kˇji,l¯ij=max⁡l^ij,lˇij,l_ij=min⁡l^ij,lˇij.Then, the set-valued maps are defined as(48)co⁡kjixit=k^ji,xit>Θi,k_ji,k¯ji,xit=Θi,kˇji,xit<Θi,co⁡lijyjt=l^ij,yjt>Ψj,l_ij,l¯ij,yjt=Ψj,lˇij,yjt<Ψj.By applying the theories of set-valued maps, differential inclusions, stochastic variables, and new functions, system (44) becomes(49)x˙it∈-Cixit-δt+∑j=1mco⁡ajixitfjyjt+μt∑j=1mco⁡bjixitfjyjt-τ1t+1-μt∑j=1mco⁡bjixitfjyjt-τ2t+∑j=1mco⁡kjixit∫t-αttfysds+uit,y˙jt∈-Djyjt-ρt+∑i=1nco⁡eijyjtgixit+ωt∑i=1nco⁡hijyjtgixit-d1t+1-ωt∑i=1nco⁡hijyjtgixit-d2t+∑i=1nco⁡lijyjt∫t-βttgxsds+vjt; equivalently,(50)x˙t=-Cxt-δt+Axtfyt+μ0Bxtfyt-τ1t+1-μ0Bxtfyt-τ2t+μt-μ0×Bxtfyt-τ1t-fyt-τ2t+Kxt∫t-αttfysds+ut,y˙t=-Dyt-ρt+Eytgxt+ω0Hytgxt-d1t+1-ω0Hytgxt-d2t+ωt-ω0×Hytgxt-d1t-gxt-d2t+Lyt∫t-βttgxsds+vt.For presentation convenience, we denote(51)K=max⁡k_ji,k¯ji,L=max⁡l_ij,l¯ij.Theorem 9.
Under Assumptions1–4, system (50) is passive, if there exist any appropriately dimensional matrices Mm(m=1,2,…,4),Ww(w=1,2,…,6), a scalar γ>0, and symmetric positive definite matrices Ri(i=1,2),Qj(j=1,2,…,6),Oo(o=1,2,…,6),Pk(k=1,2,…,8), Zl(l=1,2,…,4), and  Nn(n=1,2), such that (52)Ωi,j=Ωi,j18×18<0,Φi,j=Φi,j18×18<0, where (53)Ω1,1=-R1C-CTR1+Q1+Q2+Q3+Q4+δ2P4-P1-P2-2Z1-2Z2-2ς-W1ς+,Ω1,2=-M1C+1-δ02P1,Ω1,6=P2,Ω1,8=-M1,Ω1,9=R1A+M1A+W1ς-+ς+,Ω1,10=μ0R1+M1B,Ω1,11=1-μ0M1+R1B,Ω1,14=CTR1C,Ω1,15=2τ1Z1,Ω1,16=2τ2-τ1Z2,Ω1,17=M1K+R1K,Ω1,18=R1+M1,Ω2,2=-1-δ0Q1-1-δ02P1,Ω2,8=-M2C,Ω3,3=-Q2,Ω4,4=-1-μ1Q3-2ς-W2ς+,Ω4,10=-1-μ1Q3+W2ς-+ς+,Ω5,5=-1-μ2Q5-2ς-W3ς+,Ω5,11=-1-μ2Q5+W3ς-+ς+,Ω6,6=Q5+Q6-Q4-P2-P3,Ω6,7=P3,Ω6,12=Q5+Q6-Q4,Ω7,7=-Q6-P3,Ω7,13=-Q6,Ω8,8=δ2P1+τ12P2+τ2-τ12P3+τ122Z1+τ22-τ122Z2-2M2,Ω8,9=M2A,Ω8,10=μ0M2B,Ω8,11=1-μ0M2B,Ω8,17=M2K,Ω8,18=M2,Ω9,9=-2W1+α2N1,Ω9,14=-CTR1A,Ω9,18=-I,Ω10,10=-1-μ1Q3-2W2,Ω10,14=-μ0CTR1B,Ω11,11=-1-μ2Q5-2W3,Ω11,14=-1-μ0CTR1B,Ω12,12=Q5+Q6-Q4,Ω13,13=-Q6,Ω14,14=-P4,Ω14,17=-CTR1K,Ω14,18=-CTR1,Ω15,15=-2τ12Z1,Ω16,16=-2τ2-τ12Z2,Ω17,17=-N1,Ω18,18=-γI,(54)Φ1,1=-R2C-CTR2+O1+O2+O3+O4+ρ2P8-P5-P6-2Z3-2Z4-2σ-W4σ+,Φ1,2=-M3C+1-ρ02P5,Φ1,6=P6,Φ1,8=-M3,Φ1,9=R2A+M3A+W4σ-+σ+,Φ1,10=ω0R2+M3B,Φ1,11=1-ω0M3+R2B,Φ1,14=CTR2C,Φ1,15=2d1Z3,Φ1,16=2d2-d1Z4,Φ1,17=M3K+R2K,Φ1,18=R2+M3,Φ2,2=-1-ρ0O1-1-ρ02P5,Φ2,8=-M4C,Φ3,3=-O2,Φ4,4=-1-ω1O3-2σ-W5σ+,Φ4,10=-1-ω1O3+W5σ-+σ+,Φ5,5=-1-ω2O5-2σ-W6σ+,Φ5,11=-1-ω2O5+W6σ-+σ+,Φ6,6=O5+O6-O4-P6-P7,Φ6,7=P7,Φ6,12=O5+O6-O4,Φ7,7=-O6-P7,Φ7,13=-O6,Φ8,8=ρ2P5+d12P6+d2-d12P7+d122Z3+d22-d122Z4-2M4,Φ8,9=M4A,Φ8,10=ω0M4B,Φ8,11=1-ω0M4B,Φ8,17=M4K,Φ8,18=M4,Φ9,9=-2W4+α2N2,Φ9,14=-CTR2A,Φ9,18=-I,Φ10,10=-1-ω1O3-2W5,Φ10,14=-ω0CTR2B,Φ11,11=-1-ω2O5-2W6,Φ11,14=-1-ω0CTR2B,Φ12,12=O5+O6-O4,Φ13,13=-O6,Φ14,14=-P8,Φ14,17=-DTR2L,Φ14,18=-CTR2,Φ15,15=-2d12Z3,Φ16,16=-2d2-d12Z4,Φ17,17=-N2,Φ18,18=-γI.Proof.
Consider LKF candidate as follows:(55)Vt=∑i=110Vit, where(56)V1t=xt-C∫t-δttxsdsTR1xt-C∫t-δttxsds,V2t=yt-D∫t-ρttysdsTR2yt-D∫t-ρttysds,V3t=∫t-δttxTsQ1xsds+∫t-δtxTsQ2xsds+∫t-τ1ttxsfysTQ3xsfysds+∫t-τ1txsfysTQ4xsfysds+∫t-τ2tt-τ1xsfysTQ5xsfysds+∫t-τ2t-τ1xsfysTQ6xsfysds,V4t=∫t-ρttyTsO1ysds+∫t-ρtyTsO2ysds+∫t-d1ttysgxsTO3ysgxsds+∫t-d1tysgxsTO4ysgxsds+∫t-d2tt-d1ysgxsTO5ysgxsds+∫t-d2t-d1ysgxsTO6ysgxsds,V5t=δ∫-δ0∫t+θtx˙TsP1x˙sdsdθ+τ1∫-τ10∫t+θtx˙TsP2x˙sdsdθ+τ2-τ1∫-τ2-τ1∫t+θtx˙TsP3x˙sdsdθ+δ∫-δ0∫t+θtxTsP4xsdsdθ,V6t=ρ∫-ρ0∫t+θty˙TsP5y˙sdsdθ+d1∫-d10∫t+θty˙TsP6y˙sdsdθ+d2-d1∫-d2-d1∫t+θty˙TsP7y˙sdsdθ+ρ∫-δ0∫t+θtyTsP8ysdsdθ,V7t=∫-τ10∫θ0∫t+λtx˙TsZ1x˙sdsdλdθ+∫-τ2-τ1∫θ0∫t+λtx˙TsZ2x˙sdsdλdθ,V8t=∫-d10∫θ0∫t+λty˙TsZ3y˙sdsdλdθ+∫-d2-d1∫θ0∫t+λty˙TsZ4y˙sdsdλdθ,V9t=α∫-α0∫t+θtfTysN1fysdsdθ,V10t=β∫-β0∫t+θtgTxsN2gxsdsdθ.
Then, we define the infinitesimal generatorL of V(t) as(57)LVt=limΔ→0+⁡1ΔEVt+Δt-Vt.
Taking the mathematical expectation ofV(t), we get(58)ELV1t=E2xt-C∫t-δttxsdsTR1×-Cxt+Axtfyt+μ0Bxt×fyt-τ1t+1-μ0Bxtfyt-τ2t+μt-μ0Bxtfyt-τ1t-fyt-τ2t+Kxt∫t-αttfysds+ut≤E2xt-C∫t-δttxsdsTR1×-Cxt+Afyt+μ0Bfyt-τ1t+1-μ0Bfyt-τ2t+K∫t-αttfysds+ut,ELV2t=E2yt-D∫t-ρttysdsTR2×-Dyt+Eytgxt+ω0Hyt×gxt-d1t+1-ω0Hytgxt-d2t+ωt-ω0Hytgxt-d1t-gxt-d2t+Lyt∫t-βttgxsds+vt≤E2yt-D∫t-ρttysdsTR2×-Dyt+Egxt+ω0Hgxt-d1t+1-ω0Hgxt-d2t+L∫t-βttgxsds+vt.
According to Assumptions3 and 4, we get (59)ELV3t=ExTtQ1+Q2+Q3+Q4xt-1-δ0xTt-δtQ1xt-δt-xTt-δQ2xt-δ-1-μ1xt-τ1tfyt-τ1tTQ3xt-τ1tfyt-τ1t-1-μ2xt-τ2tfyt-τ2tTQ5xt-τ2tfyt-τ2t+xt-τ1fyt-τ1TQ5+Q6-Q4xt-τ1fyt-τ1+xt-τ2fyt-τ2TQ6xt-τ2fyt-τ2,ELV4t=EyTtO1+O2+O3+O4yt-1-ρ0yTt-ρtO1yt-ρt-yTt-ρO2yt-ρ-1-ω1yt-d1tgxt-d1tTO3yt-d1tgxt-d1t-1-ω2xt-d2tgxt-d2tTO5yt-d2tgxt-d2t+xt-d1gxt-d1TO5+O6-O4yt-d1gxt-d1+yt-d2gxt-d2TO6yt-d2gxt-d2.
It is obvious that(60)ELV5t=Ex˙Ttδ2P1+τ12P2+τ2-τ12P3x˙t+δ2xTtP4xt-δ∫t-δtx˙TsP1x˙sds-τ1∫t-τ1tx˙TsP2x˙sds-τ2-τ1∫t-τ2t-τ1x˙TsP3x˙sds-δ∫t-δtxTsP4xsds,ELV6t=Ey˙Ttρ2P5+d12P6+d2-d12P7y˙t+ρ2yTtP8yt-ρ∫t-ρty˙TsP5y˙sds-d1∫t-d1ty˙TsP6y˙sds-d2-d1∫t-d2t-d1y˙TsP7y˙sds-ρ∫t-ρtyTsP8ysds.
Using Lemma7, we have(61)-δ∫t-δtx˙TsP1x˙sds≤-∫t-δttx˙sdsTP1∫t-δttx˙sds=-xt-1-δ0xt-δTP1×xt-1-δ0xt-δ,-δ∫t-δtxTsP4xsds≤-∫t-δttxsdsTP4∫t-δttxsds.
Similarly, we have(62)-ρ∫t-ρty˙TsP4y˙sds≤-∫t-ρtty˙sdsTP4∫t-ρtty˙sds=-yt-1-ρ0yt-ρTP4×yt-1-ρ0yt-ρ,-ρ∫t-ρtyTsP8ysds≤-∫t-ρttysdsTP8∫t-ρttysds.
Then, the same as (32)–(35), we can obtain E{LV7(t)} and E{LV8(t)}.
Moreover,(63)ELV9t=Eα2fTytN1fyt-α∫t-αtfTysN1fysds,(64)ELV10t=Eβ2gTxtN2gxt-β∫t-βtgTxsN2gxsds.
Also, we get(65)-α∫t-αtfTysN1fysds≤-∫t-αttfysdsTN1∫t-αttfysds,-β∫t-βtgTxsN2gxsds≤-∫t-βttgxsdsTN2∫t-βttgxsds.
Now, we add the following inequations with any matricesM1,  M2,  M3, and  M4 of appropriate dimensions to derive a less conservative criterion:(66)0≤2xTtM1+x˙TtM2-x˙-Cxt-δt+Afyt+μ0Bfyt-τ1t+1-μ0Bfyt-τ2t+K∫t-αttfysds+ut,0≤2yTtM3+y˙TtM4-y˙-Dyt-ρt+Egxt+ω0Hgxt-d1t+1-ω0Hgxt-d2t+L∫t-βttgxsds+vt.
Define(67)ξt=∫t-τ1txTsdsxTtxTt-δtxTt-δxTt-τ1txTt-τ2txTt-τ1xTt-τ2x˙tfTytfTyt-τ1tfTyt-τ2tfTyt-τ1fTyt-τ2∫t-δttxTsds∫t-τ1txTsds∫t-τ2t-τ1xTsds∫t-αttfTysdsuTt,ζt=yTtyTt-ρtyTt-ρyTt-d1tyTt-d2tyTt-d1yTt-d2y˙tgTxtgTxt-d1tgTxt-d2tgTxt-d1gTxt-d2∫t-ρttyTsds∫t-d1tyTsds∫t-d2t-d1yTsds∫t-βttgTxsdsvTt.
Combining (40) and (58)–(66), we have(68)ELVt-2fTysgTxsusvs-γuTtvTtutvtds≤EξTtΩξt+ζTtΦζt.
Thus, we conclude that if (68) holds, then(69)ELVt-2fTysgTxsusvs-γuTtvTtutvtds≤0.
By integrating (69) about t from 0 to tz, we have(70)2E∫0tzfTysgTxsusvsds≥EVt-EV0-γE∫0tzuTsvTsusvsds≥-γE∫0tzuTsvTsusvsds.
Therefore, from Definition5, we know that the MBAMNN is passive. This completes the proof.Remark 10.
In this paper, model (50) contains the leakage, probabilistic, and distributed time-varying delays. In particular, we treat some of these delays as a special case of some existing results.Remark 11.
In the corollary, we define leakage delaysδ and  ρ as constant delays. According to the preceding conditions, we consider the MBAMNN in the following:(71)x˙t=-Cxt-δ+Axtfyt+μ0Bxtfyt-τ1t+1-μ0Bxtfyt-τ2t+μt-μ0×Bxtfyt-τ1t-fyt-τ2t+ut,y˙t=-Dyt-ρ+Eytgxt+ω0Hytgxt-d1t+1-ω0Hytgxt-d2t+ωt-ω0×Hytgxt-d1t-gxt-d2t+vt.Corollary 12.
When Assumptions1–3 hold, system (71) is passive, if there exist any appropriately dimensional matrices Mm(m=1,2,…,4),  Ww(w=1,2,…,6), a scalar γ>0, and symmetric positive definite matrices Ri(i=1,2),  Qj(j=2,…,6),  Oo(o=2,…,6),  Pk(k=1,2,…,8), and Zl(l=1,2,…,4), such that(72)Ωi,j=Ωi,j17×17<0,Φi,j=Φi,j17×17<0, where(73)Ω1,1=-R1C-CTR1+Q2+Q3+Q4+δ2P4-P1-P2-2Z1-2Z2-2ς-W1ς+,Ω1,2=P1-M1C,Ω1,5=P2,Ω1,7=-M1,Ω1,8=R1A+M1A+W1ς-+ς+,Ω1,9=μ0R1+M1B,Ω1,10=1-μ0M1+R1B,Ω1,13=CTR1C,Ω1,14=2τ1Z1,Ω1,15=2τ2-τ1Z2,Ω1,16=M1K,Ω1,17=R1+M1,Ω2,2=-Q2-P1,Ω2,7=-M2C,Ω3,3=-1-μ1Q3-2ς-W2ς+,Ω3,9=-1-μ1Q3+W2ς-+ς+,Ω4,4=-1-μ2Q5-2ς-W3ς+,Ω4,10=-1-μ2Q5+W3ς-+ς+,Ω5,5=Q5+Q6-Q4-P2-P3,Ω5,6=P3,Ω5,11=Q5+Q6-Q4,Ω6,6=-Q6-P3,Ω6,12=-Q6,Ω7,7=δ2P1+τ12P2+τ2-τ12P3+τ122Z1+τ22-τ122Z2-2M2,Ω7,8=M2A,Ω7,9=μ0M2B,Ω7,10=1-μ0M2B,Ω7,16=M2K,Ω7,17=M2,Ω8,8=-2W1+α2N1,Ω8,13=-CTR1A,Ω8,17=-I,Ω9,9=-1-μ1Q3-2W2,Ω9,13=-μ0CTR1B,Ω10,10=-1-μ2Q5-2W3,Ω10,13=-1-μ0CTR1B,Ω11,11=Q5+Q6-Q4,Ω12,12=-Q6,Ω13,13=-P4,Ω13,17=-CTR1,Ω14,14=-2τ12Z1,Ω15,15=-2τ2-τ12Z2,Ω16,16=-N1,Ω17,17=-γI,(74)Φ1,1=-R2C-CTR2+O2+O3+O4+ρ2P8-P5-P6-2Z3-2Z4-2σ-W4σ+,Φ1,2=P5-M3D,Φ1,5=P6,Φ1,7=-M3,Φ1,8=R2A+M3A+W4σ-+σ+,Φ1,9=ω0R2+M3B,Φ1,10=1-ω0M3+R2B,Φ1,13=CTR2C,Φ1,14=2d1Z3,Φ1,15=2d2-d1Z4,Φ1,16=M3K,Φ1,17=R2+M3,Φ2,2=-O2-P5,Φ2,7=-M4C,Φ3,3=-1-ω1O3-2σ-W5σ+,Φ3,9=-1-ω1O3+W5σ-+σ+,Φ4,4=-1-ω2O5-2σ-W6σ+,Φ4,10=-1-ω2O5+W6σ-+σ+,Φ5,5=O5+O6-O4-P6-P7,Φ5,6=P7,Φ5,11=O5+O6-O4,Φ6,6=-O6-P7,Φ6,12=-O6,Φ7,7=ρ2P5+d12P6+d2-d12P7+d122Z3+d22-d122Z4-2M4,Φ7,8=M4A,Φ7,9=ω0M4B,Φ7,10=1-ω0M4B,Φ7,16=M4K,Φ7,17=M4,Φ8,8=-2W4+α2N2,Φ8,13=-CTR2A,Φ8,17=-I,Φ9,9=-1-ω1O3-2W5,Φ9,13=-ω0CTR2B,Φ10,10=-1-ω2O5-2W6,Φ10,13=-1-ω0CTR2B,Φ11,11=O5+O6-O4,Φ12,12=-O6,Φ13,13=-P8,Φ13,17=-CTR2,Φ14,14=-2d12Z3,Φ15,15=-2d2-d12Z4,Φ16,16=-N2,Φ17,17=-γI.Remark 13.
In numerical simulations, we use the models of MBAMNNs with mixed time-varying delays; the models that we used are novel. In the existing papers, we can see that the most of the papers are about passivity analysis of memristive single-layer neural networks and there are few results about MBAMNNs [17–19]. MBAMNNs are a class of two-layer systems which can realize heteroassociation memory and autoassociation memory; hence, the MBAMNNs can be able to imitate the human brain better, and the limitation of traditional neural networks can be broken. The LKFs that we designed include double and triple integral terms, and using some effective methods, we can obtain several conditions about MBAMNNs with leakage, probabilistic, and distributed time-varying delays. Hence, our research results are less conservative than the existing ones in the models and the treatment methods [19, 21]. In particular, these delays are all taken into consideration separately. Thus, our paper not only improves the models but also considers a variety of time-varying delays [18–21].


## 4. Numerical Simulation

In this section, we provide several numerical examples to illustrate the usefulness of our stability results.Example 1.
We consider a 2-dimensional MBAMNN of (16) with the parameters as follows:(75)a11x1t=0.3,x1t>Φ1,0.7,x1t≤Φ1,a12x1t=0.4,x1t>Φ1,0.5,x1t≤Φ1,a21x2t=-0.5,x2t>Φ2,-0.6,x2t≤Φ2,a22x2t=-0.76,x2t>Φ2,-0.7,x2t≤Φ2,b11x1t=0.33,x1t>Φ1,0.3,x1t≤Φ1,b12x1t=0.25,x1t>Φ1,0.2,x1t≤Φ1,b21x2t=0.6,x2t>Φ2,0.56,x2t≤Φ2,b22x2t=0.63,x2t>Φ2,0.6,x2t≤Φ2,e11y1t=1.3,y1t>Ψ1,0.4,y1t≤Ψ1,e12y1t=1.28,y1t>Ψ1,0.38,y1t≤Ψ1,e21y2t=-1.05,y2t>Ψ2,-1.1,y2t≤Ψ2,e22y2t=-0.88,y2t>Ψ2,-0.8,y2t≤Ψ2,h11y1t=0.38,y1t>Ψ1,0.4,y1t≤Ψ1,h12y1t=0.25,y1t>Ψ1,0.2,y1t≤Ψ1,h21y2t=0.5,y2t>Ψ2,0.87,y2t≤Ψ2,h22y2t=0.38,y2t>Ψ2,-0.5,y2t≤Ψ2.
Here, we take the activation functions asfi(s)=gi(s)=tanh⁡(s),  i=1,2. Then, let τ1=0.5, τ2=0.6,  μ1=0.3,  μ2=0.5,  μ0=0.4,  d1=1.5,  d2=2,  ω1=0.6,  ω2=0.8,  and ω0=0.3. Probabilistic time-varying delays τ1(t)=0.7+0.2sin⁡(t), τ2(t)=0.65+0.6cos⁡(0.5t), d1(t)=0.4+0.4sin⁡(t), and  d2(t)=0.3+0.3cos⁡(t). According to Assumption 1, we verify that ς-=σ-=0, and ς+=σ+=1. Then, there exist the matrices as follows: (76)C=1.52001.33,D=1.21001.76,A=0.70.5-0.6-0.76,B=0.330.250.60.63,E=1.31.28-1.1-0.88,H=0.40.250.87-0.5.
Figure3 shows the state trajectories of system (16); it is actually unstable. By solving the LMIs (21), we get the following feasible solutions:(77)R1=10.34782.37312.37319.8579,Q3=3.16340.64230.64233.7317,Q4=7.15041.47501.47508.3565,Q5=4.04251.12301.12304.5925,Q6=1.96720.20930.20932.3559,P2=4.31590.10230.10234.0132,P3=4.7189-0.0015-0.00154.6924,Z1=0.31100.05410.05410.3985,Z2=0.0457-0.0004-0.00040.0450,R2=14.01103.79323.793214.2218,O3=3.65950.99580.99584.4985,O4=8.01282.08852.08859.9776,O5=5.39122.23612.23616.2646,O6=1.6776-0.0893-0.08932.3607,P6=7.48490.10760.10767.7770,P7=8.7529-0.0353-0.03538.8750,Z3=0.3312-0.0060-0.00600.4389,Z4=0.22620.00040.00040.2189,M1=1.74880.16000.16001.7944,M2=4.36670.41390.41394.0400,W1=5.40823.02133.02136.4308,W2=3.54460.50530.50533.7780,W3=3.53910.80500.80503.7611,M3=2.00550.25790.25792.2772,M4=8.17161.60201.60207.4034,W4=7.94014.16414.16419.2674,W5=5.07071.23611.23615.9315,W6=5.38252.04052.04056.1309,γ=20.4722.
Using the feasible solution, we obtain the state trajectories which are described in Figure4, and we can see that both x(t) and y(t) are converging to the zero point; therefore, the proved MBAMNN is internally stable. Thus, it is globally passive.Figure 3
State trajectories of system (16).Figure 4
State trajectories of system (16).Example 2.
Consider a 2-dimensional MBAMNN of (50) with the parameters as follows.
Here, we take the activation functions asfi(s)=gi(s)=tanh⁡(s),  i=1,2. Then, let δ=0.2,  δ0=0.2,  τ1=0.6,  τ2=1,  μ1=0.3,  μ2=0.5,  μ0=0.4,  α=0.1,  ρ=0.18,  ρ0=0.15,  d1=1.5,  d2=2,  ω1=0.6,  ω2=0.8,  ω0=0.3, and  β=0.2. Leakage time-varying delays δ(t)=0.5+0.3sin⁡(t) and  ρ(t)=0.4+0.5cos⁡(t); probabilistic time-varying delays τ1(t)=0.4+0.3sin⁡(t),  τ2(t)=0.5+0.5cos⁡(0.5t),  d1(t)=0.3+0.3sin⁡(t), and  d2(t)=0.1+0.5cos⁡(t); distributed time-varying delays α(t)=β(t)=0.5+0.5cos⁡(t). After changing parameters, there exist the matrices (78)C=1.52001.33,D=1.21001.76,A=0.30.4-0.6-0.76,B=0.330.250.60.63,E=1.31.28-1.1-0.88,H=0.40.250.87-0.5,K=0.1200-0.1,L=-0.100-0.06.
From the state trajectories which are depicted in Figure5, we find that system (50) is actually unstable. By solving the LMIs (52), we have the following feasible solutions:(79)R1=7.04522.93612.936111.6063,Q1=6.5971-1.6049-1.60499.2078,Q2=0.55510.46010.46013.5493,Q3=1.71841.01711.01715.9298,Q4=3.55212.19462.194613.1443,Q5=2.67991.51751.51757.6941,Q6=0.57700.44020.44023.5446,P1=5.6725-11.6662-11.666276.6622,P2=0.08580.02720.02721.0798,P3=0.19190.05170.05172.2235,P4=478.917772.194072.1940122.7031,Z1=0.08350.05160.05160.6378,Z2=0.1446-0.0612-0.06121.8780,M1=1.17101.13661.13662.7731,M2=0.57250.07600.07605.8435,W1=2.92313.92083.920815.5396,W2=2.55161.34001.34008.4107,W3=2.75571.47381.47387.8878,N1=20.64550.16740.167425.5954,R2=22.162710.548310.548321.6007,O1=29.6665-0.0928-0.092817.2449,O2=2.13110.85880.85883.0198,O3=6.91601.20131.20137.3828,O4=20.70042.65702.657020.3705,O5=17.43261.32521.325215.6910,O6=2.16030.87460.87463.0879,P5=77.8766-12.7230-12.7230176.5013,P6=0.12600.04570.04570.1924,P7=0.87050.35580.35581.3869,P8=860.5842235.3961235.3961447.6840,Z3=0.23280.09010.09010.3687,Z4=0.32300.12920.12920.5292,M3=5.20952.39872.39873.2962,M4=5.14341.38331.38338.8319,W4=25.329714.260114.260134.2731,W5=5.97911.18791.18796.6651,W6=7.11610.60660.60666.5388,N2=40.52522.27842.278438.2262,γ=50.2752.
Using the feasible solution, we obtain the state trajectories in Figure6, and we find that both x(t) and y(t) are converging to the zero point; thus, the proved MBAMNN is internally stable. Therefore, it is globally passive.Figure 5
State trajectories of system (50).Figure 6
State trajectories of system (50).Example 3.
Consider a 2-dimensional MBAMNN of (71) with the parameters as follows.
Here, we take the activation functions asfi(s)=gi(s)=tanh⁡(s),  i=1,2. And let δ=0.3,  τ1=0.2,  τ2=0.3,  μ1=0.3,  μ2=0.4,  μ0=0.4,  d1=1.3,  d2=1.5,  ω1=0.6,  ω2=0.8, and  ω0=0.25. Probabilistic time-varying delays τ1(t)=0.6+0.4sin⁡(t),  τ2(t)=0.8+0.5cos⁡(0.5t),  d1(t)=0.5+0.3sin⁡(t), and  d2(t)=0.6+0.5cos⁡(t). After changing parameters, there exist the matrices (80)C=1.52001.33,D=1.21001.76,A=0.70.5-0.6-0.76,B=0.330.250.60.63,E=1.31.28-1.1-0.88,H=0.40.250.87-0.5.
Figure7 shows the state trajectories of system (71); it is actually unstable. By solving the LMIs in Corollary 12, we obtain the following feasible solutions:(81)R1=59.328330.380630.380656.1253,Q2=11.26924.81044.810413.1800,Q3=14.86958.87178.871714.8525,Q4=29.633117.869517.869530.7251,Q5=21.256512.766612.766619.9407,Q6=5.46453.25923.25926.9941,P1=36.13616.02486.024831.4429,P2=25.80853.84413.844127.0796,P3=33.11661.37141.371434.0596,P4=296.121093.882393.8823252.4336,Z1=0.11370.09190.09190.1535,Z2=0.7613-0.1194-0.11940.7099,M1=8.39250.66210.662110.4709,M2=12.43160.37750.377513.7920,W1=27.126617.250217.250233.8015,W2=19.636610.350810.350819.7140,W3=23.908013.143813.143822.5256,R2=25.570720.326620.326641.5693,O2=15.17794.90124.901243.6250,O3=13.77220.94400.94408.8489,M4=17.96611.52731.527315.4807,W4=55.141044.665344.665357.6623,O6=4.99031.57551.57553.5372,P5=104.096734.338534.338587.8965,P6=0.81590.36670.36670.4678,P7=24.13268.82188.821815.3127,P8=220.2420251.8009251.8009686.0176,Z3=2.27741.00531.00531.2605,Z4=38.71154.73024.730232.7023,M3=2.89590.67520.67528.0316,W5=12.30931.37101.37108.1296,W6=15.9416-1.5426-1.54269.4538,O4=46.4425-1.9760-1.976028.1517,O5=38.9043-4.3696-4.369622.8177,γ=183.3912.
Using the feasible solution, we obtain the state trajectories in Figure8, and we can see that both x(t) and y(t) are converging to the zero point; therefore, the proved MBAMNN is internally stable. Thus, it is globally passive.Figure 7
State trajectories of system (71).Figure 8
State trajectories of system (71).


## 5. Conclusion

We have studied the passivity problem of MBAMNNs with probabilistic and mixed time-varying delays in this paper. By introducing random variables with Bernoulli distribution, using some useful inequalities, and constructing appropriate LKFs, we have got new delay-dependent conditions in LMIs, which ensure the passivity criteria. Future work will focus on the passivity analysis of MBAMNNs with different types of time delays.



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*Source: 5830160-2018-04-01.xml*