# Passivity Analysis of Markov Jumping Delayed Reaction-Diffusion Neural Networks under Different Boundary Conditions

Authors: Ziwei Li; Xuelian Wang; Qingkai Kong; Jing Wang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

This work analyzes the passivity for a set of Markov jumping reaction-diffusion neural networks limited by time-varying delays under Dirichlet and Neumann boundary conditions, respectively, in which Markov jumping is used to describe the variations among system parameters. To overcome some difficulties originated from partial differential terms, the Lyapunov–Krasovskii functional that introduces a new integral term is proposed and some inequality techniques are also adopted to obtain the delay-dependent stability conditions in terms of linear matrix inequalities, which ensures that the designed neural networks satisfy the specified performance of passivity. Finally, the advantages and effectiveness of the obtained results are verified via displaying two illustrated examples.


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## Body

## 1. Introduction

In the last few decades, the research of neural networks (NNs) has received widespread attention, and abundant works have been obtained. NNs have been successfully used in all kinds of areas, such as signal processing, voiceprint recognition, nonlinear dynamics, and image compression [1–3]. Additionally, time-varying delays (TVDs) are omnipresent in reality such that their effects on the NNs cannot be ignored, and mainly in that the existence of time delays (TDs) may cause instability and poor performance of systems in practical situations [4–10]. Correspondingly, not introducing TDs will cause the inaccuracy of values when NNs are mathematically modeled. Therefore, much research attention has been concentrated on the issue of stability analysis for the NNs with TDs, and accordingly a great number of works have been published in the literature (see, e.g., [11] and the references therein).It should be remarkable that the behavior of the networks depends not only on time but also on the spatial position from the perspective of engineering, such as liquid flow and diffusion processes. In reality, reaction-diffusions exist in many systems, such as chemical systems and NNs [12]. It should be remarkable that the node states of such systems should consider the variables of space and time, which can be described by partial differential equations with reaction-diffusion terms (RDTs). Thus, it is of great essential to introduce the RDT into NNs to achieve a better approximation effect of the spatiotemporal actions. To name a few, for the complex spatiotemporal networks, the synchronization problem of space-varying coefficients was concerned in [13]. For the coupled partial differential systems, the synchronization issue with space-independent coefficients was discussed in [14]. However, finding a suitable method to deal with the RDT of neural networks is a difficult challenge. Although it is very tricky, this challenge has driven the development of RDNNs. Up to now, a mass of outstanding works on the stability analysis of NNs with RDT have been published [15, 16]. For example, the synchronization issue of hybrid-coupled RDNNs with TDs was discussed in [17]. In [18], the passivity of coupled RDNNs with nonlinear coupling was considered. Via detailed understanding, it is found that most of the existing methods are used to handle the RDT under the framework of Friedrichs inequality. In this case, there is a certain degree of conservatism, which is difficult to be extended to the general matrix form. To tackle this problem, using appropriate matrix inequalities to better reduce the conservatism was investigated. Based on the abovementioned analysis, it prompts us to take the stability problem of RDNNs into consideration by using proper matrix inequality.On the contrary, NNs often show a special switching characteristic of the network mode, and this case will make that the RDNNs cannot be handily modeled [19, 20]. Fortunately, the Markov process has been proved to be able to depict the switching (or jumping) cases effectively [16, 21–25]. Hence, taking the Markov jumping model applied into RDNNs with TVDs is practical and reasonable, in which each state represents a jumping mode. Recently, lots of works for RDNNs with the Markov jumping parameters have been published [26–28]. For instance, the stochastic delayed NNs with RDT and Markov jumping parameters were studied in [29]. In [30], the state estimation of Markov jump delayed NNs with RDT was discussed. In this case, the study of RDNNs with TVDs and Markov jumping is of significance. Besides, passivity theory is recognized as a powerful tool and plays a critical role in diverse areas such as energy management and flow control [31–35]. In early days, the passivity was widely used in a system where the variables of input and output merely depend on time and rarely in a system with RDT. Thereby, it is extremely necessary to explore the passivity of RDNNs with TVDs. As far as we know, only a small number of investigators have studied the passivity analysis of Markov jumping RDNNs, which powerfully motivates the present study.Based on the above discussions, this paper concentrates on the stability and passivity analysis of Markov jumping RDNNs with TVDs. In addition, the main contributions include the following three sides:(1)
Compared with some existing literature [11, 36], in this paper, Jensen’s inequality and the Wirtinger-type integral inequality, a new inequality dealing with the reaction-diffusion terms and the reciprocally convex method, are introduced to derive criteria based on linear matrix inequalities (LMIs), which is beneficial for reducing the conservativeness of the results.(2)
Furthermore, by using some inequality techniques, several passive criteria for RDNNs with TVDs and Markov jumping parameters are obtained, and the passivity of Markov jumping RDNNs with TVDs is analyzed.(3)
The Markov jumping model is employed to the research of RDNNs, and the passivity of Markov jumping RDNNs is investigated under the different boundary conditions (BCs), which determines the common internal stability of Markov jumping RDNNs.


## 2. Preliminaries

Consider the RDNNs with Markov jumping parameters and TVDs as follows:(1)∂ηit,x∂t=∑k=1r∂∂xkaik∂ηit,x∂xk−biαtηit,x+∑j=1ϱcijαtfjηit,x+∑j=1ϱdijαtgjηit−τt,x+νit,x,ηi0,x=εix,t,x∈−τ2,0×Ω,where x=x1,x2,…,xrT∈ℝr, i=1,2,…,ρ;ηit,x denotes the state variable at time t and in space s; aik>0 means the transmission diffusion coefficient of the ith neuron; biαt>0 is the neuron charging time constants; cijαt and dijαt denote the connection weight coefficients of the jth neuron on the ith neuron; fj⋅ and gj⋅ represent the neuron activation functions of the jth neuron at time t; νit,x is the control input; εis,x is bounded and continuous on −τ2,0×Ω, i=1,2,…,ρ; τt indicates the transmission delay and satisfies 0<τt≤τ2 and τ˙t≤σ<1, in which τ2 and σ are some given constants; αtt∈ℤ+ is a right-continuous Markov process taking values in a finite state space ℳ=1,2,…,ρ with the transition probability matrix Φ≜πυκ given by(2)Prαt+△=καt=υ=πυκ△+o△,πυ≠κ,1+πυυ△+o△,υ=κ,where lim△⟶0o△/△=0,△>0, and πυκ≥0 for υ≠κ is the transition rate from mode υ at time t to mode κ at time t+△.In our paper, two kinds of BCs are introduced:(1)
Dirichlet BCs:(3)ηit,x=0,t,x∈−τ2,+∞×∂Ω.(2)
Neumann BCs:(4)∂ηit,x∂n¯=∂ηit,x∂x1,∂ηit,x∂x2,…,∂ηit,x∂xrT=0,t,x∈−τ2,+∞×∂Ω.Besides,fi and gi are defined as the activation functions, which are assumed to meet the listed condition as below.Assumption 1.
(see [30]). The activation functions fi and gi satisfy the bounded conditions and there exist positive matrices Ϝ0=diagϜ01,Ϝ02,…,Ϝ0ρ and Ϝ1=diagϜ11,Ϝ12,…,Ϝ1ρ such that(5)0≤fia1−fia2a1−a2≤Ϝ0i,0≤gia1−gia2a1−a2≤Ϝ1i,where i=1,2,…,ρ, fi0=0, and gi0=0, for any a1, a2∈ℝl and a__1≠a__2.
In this paper, for eachαt=α∈ℳ, system (1) can be rewritten as the vector matrix form as follows:(6)∂ηt,x∂t=Ak∘∇2ηt,x−Bαηt,x+Cαfηt,x+Dαgηt−τt,x+νt,x,η0,x=εx,t,x∈−τ2,0×Ω,where(7)Ak≜diaga1k,a2k,…,aρk,Bα≜diagb1α,b2α,…,bρα,j=1,2,…,ρ,Cα≜cijαρ×ρ,Dα≜dijαρ×ρ,εx=ε1x⋯ε2xT,gη⋅≜g1η1⋅⋯gρηρ⋅T,νt,x≜ν1t,x⋯νρt,xT,ηt,x≜η1t,x⋯ηρt,xT,fη⋅≜f1η1⋅⋯fρηρ⋅T,with(8)Ak∘∇2ηt,x≜∑k=1r∂∂xka1k∂η1t,x∂xk⋯∑k=1r∂∂xkaρk∂ηρt,x∂xkT.
The output vectoryt,x∈ℝρ of system (6) is chosen as(9)yt,x=Eηt,x+Hνt,x,where E, F∈ℝρ×ρ are known constant matrices.Definition 1.
(see [37]). For t,x∈0,+∞×Ω, if the input νt,x and the output yt,x satisfy the following inequality:(10)∫t1t2∫ΩyTt,xMνt,x−νTt,xWνt,xdxdt≥St2−St1,in which M, W∈ℝρ×ρ, W>0, and λmM+λmW>0, ∀t2,t1∈0,+∞ and t2>t1, where St:0,+∞⟶0,+∞ is the storage function, then network (6) is said to be input-strictly passive.Lemma 1 (see [38]).
For the compact setΩ and system (6), if ηit,x=ηit,x1,x2,…,xr∈C1Ω satisfies ηit,x∂Ω=0, that is, it vanishes on the boundary ∂Ω of Ω, then(11)∫Ωηit,x∂2ηit,x∂xk2dx≤−π2lk2∫Ωηi2t,xdx.Remark 1.
It is worth noting that compared to other methods of coping with RDT, such as Lemma1 in [39], in our paper, it is validated that utilizing (11) can better reduce conservativeness, with regards to deriving of our main results which is very significant. The proof is omitted therein but can be found in the literature [38].


## 3. Main Results

In this section, we consider dynamical NNs, which includesr identical with spatial diffusion, and each node is a ρ-dimensional for Markov jumping RDNNs. Additionally, sufficient condition for assuring the asymptotic stability of Markov jumping RDNNs is acquired. Besides, the passivity criteria for RDNNs (6) under the different BCs are achieved.Theorem 1.
Given scalarsτ2>0 and 1>σ>0, the solution of RDNNs (6) is input-strictly passive under the Dirichlet boundary condition if there exist some diagonal matrices Pα>0, Γ0>0, Γ1>0, and X>0, real positive matrices Q1, Q2, Z, U, R, and W, and matrix Y of appropriate dimensions, such that ∀τ∈0,τ2, the following inequalities hold:(12)RˇϒϒTRˇ≥0,(13)Π≜Π0+∑i=14Πi+Π5τ+Π6−2ω1TETMω11−2ω11THTMω11−ω11TWω11<0,where(14)Π1≜2ω1T−π24PαKAkKT−PαBαω1+ω1T∑κ=1ρπυκPκω1+ω1TPαCαω4+ω1TPαDαω5+ω1TPαHαω11−ω1TBαTXω6+ω4TCαTXω6+ω5TDαTXω6+ω6TXω11−ω6TXω6,Π0≜ω1TΓ0Ϝ0ω4−2ω4TΓ0ω4,Π2≜ω1TQ1ω1−1−σω3TQ1ω3,Π3≜ω1TQ2ω1−ω2TQ2ω2,Π4≜ω1TϜ1TZϜ1ω1−1−σω5TZω5,Π51≜−τ2τ2−τω7TUω7−τ2τω9TUω9,ωi≜0ρ×i−1ρIn0ρ×ρ−iρ,i=1,2,…,11,Π5≜τ22ω1TUω1+Π51,K≜η1l1⋯η1lr⋯ηρl1⋯ηρlrρ×ρr,Rˇ≜diagR,3R,5R,τˇ≜τ2−τt,Λ2≜ω1T−ω3Tω1T+ω3T−2ω9Tω1T−ω3T+6ω9T−12ω10T,Λ1≜ω3T−ω2Tω3T+ω2T−2ω7Tω3T−ω2T+6ω7T−12ω8T,Π6≜τ22ω6TRω6−Λ1Λ2RˇΥΥTRˇΛ1Λ2T,ξt,x≜ηTt,xηTt−τ2,xηTt−τt,xfTηt,xgTηt−τt,x∂ηTt,x∂t1τˇ∫t−τ2t−τtηTs,xds1τ⌣2∫t−τ2t−τt∫δt−τtηTs,xdsdδ1τt∫t−τttηTs,xds1τ2t∫t−τtt∫δtηTs,xdsdδνTt,xT.Proof.
We firstly choose the following Lyapunov–Krasovskii functional:(15)Vk=∑i=16Viηt,x,α,t,where(16)V1ηt,x,α,t=∫ΩηTt,xPαηt,xdx+∑k=1r∫Ω∂ηTt,x∂xkXAk∂ηt,x∂xkdx,V2ηt,x,α,t=∫Ω∫t−τttηTs,xQ1ηs,xdsdx,V3ηt,x,α,t=∫Ω∫t−τ2tηTs,xQ2ηs,xdsdx,V4ηt,x,α,t=∫Ω∫t−τttgTηs,xZgηs,xdsdx,V5ηt,x,α,t=τ2∫Ω∫−τ20∫t+wtηTs,xUηs,xdsdwdx,V6ηt,x,α,t=τ2∫Ω∫−τ20∫t+wt∂ηTs,x∂sR∂ηs,x∂sdsdwdx.
Then, calculating time derivatives ofViηt,x,α,t, one has(17)∂∂tV1ηt,x,α,t=2∫ΩξTt,xω1TPαA∘∇2ηt,x−Bαω1+Cαω4+Dαω5+ω11ξt,xdx+∑κ=1ρπυκ∫ΩξTt,xω1TPκω1ξt,xdx+2∑k=1r∫Ω∂ηTt,x∂xkXAk∂∂xk∂ηt,x∂tdx,(18)∂∂tV2ηt,x,α,t≤∫ΩξTt,xω1TQ1ω1−1−σω3TQ1ω3ξt,xdx=∫ΩξTt,xΠ2ξt,xdx,(19)∂∂tV3ηt,x,α,t=∫ΩξTt,xω1TQ2ω1−ω2TQ2ω2ξt,xdx=∫ΩξTt,xΠ3ξt,xdx,(20)∂∂tV4ηt,x,α,t≤∫ΩgTηt,xZgηt,x−1−σξTt,xω5TZω5ξt,xdx,≤∫ΩξTt,xΠ4ξt,xdx,(21)∂∂tV5ηt,x,α,t=τ22∫ΩξTt,xω1TUω1ξt,xdx−τ2∫Ω∫t−τ2tηTs,xUηs,xdsdx,(22)∂∂tV6ηt,x,α,t=τ22∫ΩξTt,xω6TRω6ξt,xdx−τ2∫Ω∫t−τ2t∂ηTs,x∂sR∂ηs,x∂sdsdx.
From (11), we can obtain(23)2∫ΩηTt,xPαA∘∇2ηt,xdx≤2∫ΩηTt,x−π24PαKAKTηt,xdx.
For diagonal matricesX>0, from (6), we can find that(24)0=2∫Ω∂ηTt,x∂tXAk∘∇2ηt,x−Bαηt,x+Cαgηt,x+Dαgηt−τt,x+νt,x−∂ηt,x∂tdx.
On the basis of Lemma 3 in [37], we obtain(25)2∫Ω∂ηTt,x∂tXAk∘∇2ηt,xdx=−2∑k=1r∫Ω∂ηTt,x∂xkXAk∂∂xk∂ηt,x∂tdx.
From (24) and (25), one has(26)∂∂tV1ηt,x,α,t≤2∫ΩξTt,xω1TPα−π24KAKTω1−Bαω1+Cαω4+Dαω5+ω11ξt,xdx−2∫ΩξTt,xω6TXBαω1−Cαω4−Dαω5−ω11+ω6ξt,xdx+∑κ=1ρπυκ∫ΩξTt,xω1TPκω1ξt,xdx≤∫ΩξTt,xΠ1ξt,xdx.
By Jensen’s inequality, we can obtain(27)−τ2∫Ω∫t−τ2tηTs,xUηs,xdsdx≤−τ2τˇ∫Ω∫t−τ2t−τtηTs,xdsU∫t−τ2t−τtηs,xdsdx−τ2τt∫Ω∫t−τttηTs,xdsU∫t−τttηs,xdsdx≤∫ΩξTt,xΠ51ξt,xdx.
From conditions (21) and (27), it can be obtained that(28)∂∂tV5ηt,x,α,t≤∫ΩξTt,xτ22ω1TUω1+Π51ξt,xdx=∫ΩξTt,xΠ5ξt,xdx.
According to the Wirtinger-type integral inequality in [40] and Lemma 5 of [41], we have(29)−τ2∫Ω∫t−τ2t∂ηTs,x∂sR∂ηs,x∂sdsdx≤−∫ΩξTt,xΛ1Λ2RˇΥΥTRˇΛ1TΛ2Tξt,xdx.
By (23) and (29), one has(30)∂∂tV6ηt,x,α,t≤∫ΩξTt,xΠ6ξt,xdx.
Furthermore, for positive diagonal matricesΓ0 and Γ1, from (5), one obtains(31)2ηTt,xΓ0Ϝ0fηt,x−2fTηt,xΓ0fηt,x≥0,ηTt,xϜ1TZϜ1ηt,x−gTηt,xZgηt,x≥0.
By (15)-(31), for each α∈ℳ, we can get(32)∑i=16∂Viηt,x,α,t∂t−∫Ω2yTt,xMνt,xdx+∫ΩνTt,xWνt,xdx≤∫ΩξTt,xΠξt,xdx,which combining with (13) means(33)∫ΩyTt,xMνt,xdx−12∫ΩvTt,xWνt,xdx≥S˙t,in which St=Vηt,x,α,t/2.
From (33), one can see that(34)∫t1t2∫ΩyTt,xMνt,xdxdt−∫t1t2∫ΩνTt,xW2νt,xdxdt≥St2−St1,∀t2,t1∈0,+∞ and t2>t1. This completes the proof.Remark 2.
It is noted that the method of Lemma 2 in [42] to deal with the integrals to a certain degree has reduced the conservatism of the system, compared to the general Jensen’s inequalities. Therefore, in our paper, this technique is adopted to estimate the integral:(35)τ2∫Ω∫−τ20∫t+wt∂ηTs,x∂sR∂ηs,x∂sdsdwdx.Remark 3.
Compared with the traditional Wirtinger-based integral inequality in [42], Lemma 2 used in [40] is an improved method of the Wirtinger-based integral inequality that can better reduce conservativeness. Then, some delay-dependent stability criteria are obtained by applying the abovementioned inequalities and computed by using standard LMI techniques.Theorem 2.
Given scalarsτ2>0 and 1>σ>0, the solution of RDNNs (6) is input-strictly passive under the Neumann boundary conditions if there exist some diagonal matrices Pα>0, Γ0>0, Γ1>0, and X>0, real positive matrices Q1, Q2, Z, U, R, and W, and matrix Υ of appropriate size, such that ∀τ∈0,τ2, the following inequalities hold:(36)RˇΥΥTRˇ≥0,(37)Π=Π0+Π¯1+∑i=24Πi+Π5τ+Π6−2ω1TETMω11−2ω11THTMω11−ω11TWω11<0,where(38)Π¯1≜2−ω1TPαBαω1+ω1T∑κ=1ρπυκPqω1+ω1TPαCαω4+ω1TPαDαω5+ω1TPαω11−ω1TBαTXω6+ω4TCαTXω6+ω5TDαTXω6+ω6TXω11−ω6TXω6,and Π0, Π2, Π3, Π4, Π5τ, and Π6 are defined the same as in Theorem 1.Proof.
Considering the same Lyapunov–Krasovskii functional given in (15), afterwards, and calculating the derivative along the trajectory of (6) under the Neumann BCs (4), we can obtain that(39)2∫Ωηt,xPαAk∘∇2ηt,xdx=2∫∂Ω∑i=1ρηit,xPαAik∂ηit,x∂xkk=1r⋅n¯dS−∫Ω∑i=1ρPα∑k=1rAik∂ηit,x∂xk2dx=−2∫Ω∑i=1ρ∑k=1rPαAik∂ηit,x∂xk2dx,≤0.
Then, following the same procedure as in Theorem1, we can obtain the craved results immediately.


## 4. Numerical Examples

In this section, two numerical examples are provided to expound the effectiveness of the main results derived above.Example 1.
Consider continuous-time Markov jump RDNNs with TVDs under the Dirichlet BCs as follows:(40)∂ηt,x∂t=Ak∘∇2ηt,x−Bαηt,x+Cαfηt,x+Dαgηt−τt,x+νt,x,η0,x=0.5cosπx0.3cosπxT,t,x∈−0.4,0×Ω,in which r=1,2,Ω=x−0.5<x<0.5, and some system matrices in system (6) containing two modes can be given as follows:Mode1:(41)A=diag1.2,0.9,B1=diag0.9,0.8,C1=0.20.13−0.5−2.8,D1=0.10.4−0.30.5.Mode2:(42)B2=diag0.8,1.6,E=diag0.8,0.9,C2=0.20.3−1.1−0.8,D2=0.10.2−0.10.8,H=0.80.30.51.1,where fjη=0.5η+1−η−1 and gjη=arctanη, both satisfy assumption 1. Moreover, we set that σ=0.8, τ2=0.4, Ϝ0=I, and Ϝ1=2I. The transition probability matrix of Markov process αt is taken as(43)Φ=−0.50.50.24−0.24.
By solving the LMIs (12) and (13) in Theorem 1, it is obvious that there is a feasible solution. For the sake of simplicity, only partial solution matrices are listed:(44)P1=diag0.9878,0.7383,P2=diag0.9587,0.8063,Q1=0.9888−0.0434−0.04340.0081,Q2=0.5843−0.0273−0.02730.0247,U=2.2454−0.1032−0.10320.0814,Z=0.6341−0.1211−0.12110.7005.
Therefore, one can conclude that the Markov jump RDNNs (6) with TVDs under Dirichlet BCs is input-strictly passive according to Definition 1. Here, we assume that the input function ν1t,x=0.15tcosπx and ν2t,x=0.1tcosπx. Then, the state trajectories of variables η1t,x and η2t,x are plotted in Figures 1 and 2, respectively. On the other hand, Figures 3 and 4, respectively, describe the state trajectories of variables η1t,x and η2t,x, in which we choose ν1t,x=0 and ν2t,x=0, which means system (40) is globally stable under the zero input.Figure 1
State trajectories of variablesη1t,x with ν1t,x=0.15tcosπx in example 1.Figure 2
State trajectories of variablesη2t,x with ν2t,x=0.1tcosπx in example 1.Figure 3
State trajectories of variablesη1t,x with ν1t,x=0 in example 1.Figure 4
State trajectories of variablesη2t,x with ν2t,x=0 in example 1.Example 2.
Consider the RDNNs with Markov jumping parameters and TVDs under Neumann BCs in the following:(45)∂ηt,x∂t=Ak∘∇2ηt,x−Bαηt,x+Cαfηt,x+Dαgηt−τt,x+νt,x,η0,x=−0.4cosπx−0.8cosπxT,t,x∈−0.4,0×Ω,in which the rest of the values are the same as in example 1, and some system matrices containing two modes are given as follows:Mode1:(46)A=diag1.2,0.9,B1=diag0.9,0.8,C1=0.20.13−0.5−2.8,D1=0.70.4−0.30.5.Mode2:(47)B2=diag0.8,0.6,E=diag0.8,0.9,C2=0.20.3−1.1−0.8,D2=0.90.6−0.10.8,H=0.80.30.51.1.
In this case, we setσ=0.7 and τ2=0.2. Afterwards, the maximum allowable upper bound value τt is obtained via solving the LMIs of (36) and (37) in Theorem 2. Then, we can get the NNs (6) of RDT with Markov jumping parameters under Neumann BCs which are also input-strictly passive. Here, only some of the feasible solutions are shown in the following:(48)P1=10−2∗diag0.7,1.6,P2=10−2∗diag0.39,0.39,Q1=10−3∗2.0−0.5−0.52.2,Q2=10−3∗2.2−0.6−0.62.4,U=10−2∗5.88−1.45−1.456.53,Z=10−2∗1.360.430.431.87.
Here, Figures5 and 6, separately, depict the state trajectories of variables η1t,x and η2t,x under Neumann BCs, in which the input function is regarded as ν1t,x=0.15tcosπx and ν2t,x=0.1tcosπx. In the other case, consider the input function ν1t,x=ν2t,x=0, and the numerical simulations of system (45) are presented in Figures 7 and 8, respectively.
Furthermore, from Tables1 and 2, we can find that the upper bound of τ2 decreases with the value of σ increasing in the interval 0,1 from the Dirichlet BCs or the Neumann BCs, which means the increasing of σ has a negative impact on τ2. Moreover, from Table 3, compared with the different cases with obtained results, it is easy to discover that the maximum allowed value of time-varying delays in Theorem 1 is larger than the delay upper bound in Theorem 2, and one can get two observations:(i)
We can find that the assumptions about the BCs are limited to the Neumann zero-boundary condition. The main reason is that the results are obtained without consideration of the influence of RDT on the NNs, which increases the conservatism of the system.(ii)
Correspondingly, in order to avoid this problem, a RDNNs model with Dirichlet BCs introduces a new inequality, and the results consider the influence of the diffusion terms. Thus, the conservatism of the results obtained in this paper is reduced.Figure 5
State trajectories of variablesη1t,x with ν1t,x=0.15tcosπx in example 2.Figure 6
State trajectories of variablesη2t,x with ν2t,x=0.1tcosπx in example 2.Figure 7
State trajectories of variablesη1t,x with ν1t,x=0 in example 2.Figure 8
State trajectories of variablesη2t,x with ν2t,x=0 in example 2.Table 1
Upper bound ofτ2 under the Dirichlet BCs with different σ.
σ0.820.840.860.88τ20.41810.35510.30770.2688Table 2
Upper bound ofτ2 under the Neumann BCs with different σ.
σ0.700.750.800.85τ20.32670.29820.26680.2531Table 3
Upper bound ofτ2 under different cases of boundary conditions with different σ.
σ0.70.750.820.840.86Theorem1∞∞0.41810.35510.3077Theorem20.32670.29820.25310.23870.2311


## 5. Conclusions

In this paper, we have analyzed the passivity problem for Markov jumping reaction-diffusion neural networks, in which the time-varying delay phenomenon and reaction-diffusion terms have been taken into consideration. By introducing several novel Lyapunov–Krasovskii functionals and employing a new inequality, some sufficient conditions expressed as linear matrix inequalities have been established, which guarantees the passive stability of the considered reaction-diffusion neural networks. Two numerical examples have been provided to exhibit the effectiveness of the offered approaches. Finally, we will try to apply the above results to other practical systems in future works.The notations used throughout this paper are standard, which are presented in the following.



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*Source: 9369813-2020-07-06.xml*