# Passivity Analysis of Fractional-Order Neutral-Type Fuzzy Cellular BAM Neural Networks with Time-Varying Delays

Authors: M. Syed Ali; M. Hymavathi; Hamed Alsulami; Tareq Saeed; Bashir Ahmad
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, passivity analysis of fractional-order neutral-type fuzzy cellular bidirectional associative memory (BAM) neural networks with time-varying delays is investigated. Based on the Lyapunov–Krasovskii functional, delay-dependent sufficient conditions for solvability of the passive problem are obtained in terms of linear matrix inequalities (LMIs), which can be easily checked by using the MATLAB LMI toolbox. Finally, numerical examples are provided to show the effectiveness of the main results.


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## Body

## 1. Introduction

The study of fractional calculus can be dated back to 1695, and the fractional operator concept was put forward by Leibnitz, which did not acquire sufficient attention for a long period since it is complicated. Many actual systems can be described by fractional-order differential equations, making the slowly developed fractional calculus be a renewal of interest [1–5]. Generally speaking, fractional calculus is a generalization of classical calculus and is more accurate to describe reality models compared to the corresponding integer-order calculus in different research communities, such as particle physics, wave mechanics, electrical systems, and computational methods for mathematical physics, and many references cited therein [6–9]. Because of the nonlocality of fractional calculus, it is presented to explain actual systems on the basis of standard integral calculus. Furthermore, fractional calculus is widely used in domains such as statistical mechanics, medical imaging, combinatorial optimization, and viscoelastic systems [10–13]. However, fractional neural networks are still a hot research topic and have been extensively studied in the recent decades [14–16]. Recently, the synchronization of fractional neural networks has attracted increasing interest [17–20].Fractional-order neural networks with neutral-type delays were also very rarely studied in the available literature. The neural network model having time delays in the time derivatives of states is called delayed neutral-type neural networks. The neutral-type system not only characterizes the dynamic property of the system state but also describes the dynamic varying rule of the delay state of the system. Such a phenomenon constantly encountered in the area of chemical reactors, transmission lines in electrical engineering, heat exchangers, and population dynamic systems. Because of this component, neutral systems have turned into the subject of broad look into by numerous researchers [21–28]. Many neural networks can be regarded as special cases of neutral neural networks, and most of the neural networks can be transformed into neutral neural networks for research [29, 30]. Furthermore, stability criteria for neutral-type BAM neural networks with time delays were derived in [31–34].In this paper, we would like to integrate fuzzy operations into BAM neural networks. Combining the advantages of the fuzzy operation theory and cellular neural network, T. Yang and L. B. Yang firstly proposed the fuzzy cellular neural networks in 1996. Owing to the contribution of Song, in the establishment of fuzzy logic [35], the fuzzy logic has been paid more and more attention. Studies have shown that fuzzy cellular neural networks have their potential in image processing and pattern recognition, and a few outcomes have been discussed [36–41]. Associative memory is one amongst the foremost significant behaviors of the human brain, which might be applied in the study of brain-like systems, intelligent thinking for intelligent robots, and so on. The bidirectional associative memory (BAM) neural network models were first proposed and studied by Kosko [42], which are made up of two neuron layers, i.e., U-layer and V-layer, which have no interconnection among neurons within the same layer. The neurons in one layer are fully interconnected to the neurons in the second layer and have the features of heteroassociative, content-addressable memory. In real life, BAM neural networks have powerful information processing abilities and some good application fields, such as information associative memory, image processing, and artificial intelligence. Subsequently, different results on the stability and other behaviors of delayed fuzzy BAM neural networks have been derived (see [43–46] and references cited therein).As pointed out, discrete-time models are more better to describe the dynamical behaviors than continuous ones since the populations have nonoverlapping generations. In different engineering systems, an origin of in stability is time delay exists [47–49]. Based on the dependence of delay, stability criteria have been classified into 2 types: delay-dependent and delay-independent criteria. Thus, delay-dependent criteria play an important role for assuring the addressed neural networks to be stable to get maximum delay bounds. Different methods were imported for these types of problems [50, 51]. The methods were as follows: (i) free-weighting matrix method, (ii) reciprocally convex approach, (iii) augmented Lyapunov–Krasovskii functional.Passivity theory was first proposed in the circuit analysis [52] and then connected with various distinctive structures, including high-order nonlinear systems and electrical framework [53]. It can be extensively associated with play out the stability analysis, signal processing, fuzzy control, sliding mode control, and networked control. The main aim of passivity theory is that the passive properties of the system can keep the system internally stable. In reality, many systems require to be passive in order to rarefy noise effectively. Recently, several authors have studied the passivity of delay neural networks; see [54–59].Inspired by the above discussions, the problem of passivity-based fractional-order neutral-type fuzzy cellular BAM neural networks with time-varying delays is examined in this paper. More precisely, in this work, by employing a Lyapunov functional with augmented elements and the linear matrix inequality technique, a set of delay-dependent passivity conditions is derived for obtaining the required result. Finally, the numerical example is presented.


## 2. Preliminaries and Model Description

In this section, we recall definitions of fractional calculus and several lemmas which will be used later.Definition 1 (see [60]).
The Caputo fractional derivative of orderβ for a function f(t) is defined as(1)Dt0Ctβft=1Γn−β∫0tt−τn−β−1fnτdτ,where β>0 and n is an integer satisfying n−1<β<n.Definition 2 (see [61]).
System (4) is called passive if there exists a scalar γ>0 such that(2)2∫0tkhTvsgTusθsϕsds≥−γ∫0tkθTsϕTsθsϕsds,for all solutions of (12) with u0=0 and v0=0 and for all tk≥0.Lemma 1 (see [62]).
Letut⊂ℝn be a continuous and derivable function. Then, for any t>0, the following inequality holds:(3)120CDtβuTtut≤uTt0CDtβut.Lemma 2 (see [63]).
For any positive definite matrixQ>0, scalar τ>0, and vector function f⋅:0,τ⟶ℝn, the following inequality holds:(4)∫0τfsdsTH∫0τfsds≤τ∫0τfTsHfsds.Lemma 3 (see [64]).
Given constant matricesΥ1, Υ2, and Υ3, where Υ1=Υ1T, Υ2=Υ2T, and Υ2>0; then,(5)Υ1+Υ3TΥ2−1Υ3<0if and only if(6)Υ1Υ3TΥ3−Υ2<0.Lemma 4 (see [65]).
Suppose thatx,y∈ℝn are the two states of system (1). Then, one has(7)∧j=1mγijhjy¯j−∧j=1mγijhjyj≤∑j=1mγijhjy¯j−hjyj,∨j=1mδijhjy¯j−∨j=1mδijhjyj≤∑j=1mδijhjy¯j−hjyj,∧i=1nγ˜jigix¯i−∧i=1nγ˜jigixi≤∑i=1nγ˜jigix¯i−gixi,∨i=1nδ˜jigix¯i−∨i=1nδ˜jigixi≤∑i=1nδ˜jigix¯i−gixi.Assumption 1.
σ(t) and τt are time-varying functions satisfying(8)1.0≤σ1t≤σ1,σ1.t≤μ1<1,2.0≤τ1t≤σ1,τ1.t≤μ2<1,3.0≤ht≤h,h˙t≤h1<1,4.0≤dt≤d,d˙t≤d1<1.


## 3. Main Results

Consider the following fractional-order neutral-type fuzzy cellular BAM neural networks with time-varying delays as(9)Dαxit−fiDαxit−ht=−aixit+∑j=1mbijhjyjt+∑j=1mcijhjyjt−τ1t+∑j=1mdijlj+∧j=1mγij∫−∞tkjt−shjyjsds+∨j=1mδij∫−∞tkjt−shjyjsds+∧j=1mRijlj+∨j=1mSijlj+∑j=1mtij∫t−τ2tthjyjsds+Ii,t≥0,i=1,2,…,n,Dαyjt−yjDαyjt−dt=−a˜jyjt+∑i=1nb˜jigixjt+∑i=1nc˜jigixjt−σ1t+∑i=1nd˜jili+∧i=1nγ˜ji∫−∞tkit−sgixisds+∨i=1nδ˜ji∫−∞tkit−sgixisds+∧i=1nR˜jili+∨i=1nS˜jili+∑i=1nt˜ji∫t−σ2ttgixisds+Ij,t≥0,j=1,2,…,m,where Dα is Caputo’s fractional derivative, xit and yjt are the state of the ith neuron and jth neuron at time t, respectively, gi and hj denote the functions of the ith neuron and the jth neuron at time t, respectively, ai and a˜j are the diagonal matrices with positive entries, ai and a˜j represent the rates with which the ith neuron and the jth neuron will reset their potential to the resting state in isolation when disconnected from the networks and external inputs, respectively, bij,cij,b˜ji,c˜ji denote the connection weight matrices of the feedback template and feedforward template, dij,d˜ji denote the connection weight matrices of the feedback template and feedforward template, respectively, γij,γ˜ji and δij,δ˜ji denote the connection weights of the delay fuzzy feedback MIN template and the delay fuzzy feedback MAX template, respectively, li and lj denote the inputs of the ith neuron and the jth neuron at time t, respectively, and ∨ and ∧ denote the fuzzy AND and fuzzy OR operation, respectively.In this paper, we refer to model (9) as the drive system; the response system is given as follows:(10)Dαx¯it−fiDαx¯it−ht=−aix¯it+∑j=1mbijhjy¯jt+∑j=1mcijhjy¯jt−τ1t+∑j=1mdijlj+∧j=1mγij∫−∞tkjt−shjy¯jsds+∨j=1mδij∫−∞tkjt−shjy¯jsds+∧j=1mRijlj+∨j=1mSijlj+∨j=1mδij∫−∞tkjt−shjy¯jsds+∧j=1mRijlj+∨j=1mSijlj+∑j=1mtij∫t−τ2tthjy¯jsds+Ii+θit,Dαy¯jt−ejDαy¯jt−dt=−a˜jy¯jt+∑i=1nb˜jigix¯jt+∑i=1nc˜jigix¯jt−σ1t+∑i=1nd˜jili+∧i=1nγ˜ji∫−∞tkit−sgix¯isds+∨i=1nδ˜ji∫−∞tkit−sgix¯isds+∧i=1nR˜jili+∨i=1nS˜jili+∑i=1nt˜ji∫t−σ2ttgix¯isds+Ij+ϕjt,where θit and ϕjt indicate (the control input, and let uit=x¯it−xit and vjt=y¯jt−yjt for i∈I,j∈J; then, from systems (9) and (10), the error dynamical system can be derived as(11)Dαuit−fiDαuit−ht=−aiuit+∑j=1mbijhjvjt+∑j=1mcijhjvjt−τ1t+∧j=1mγij∫−∞tkjt−shjvjsds+∨j=1mδij∫−∞tkjt−shjvjsds+∑j=1mtij∫t−τ2tthjvjsds+θit,Dαvjt−ejDαvjt−dt=−a˜jvjt+∑i=1nb˜jigiuis+∑i=1nc˜jigiujt−σ1t+∧i=1nγ˜ji∫−∞tkit−sgiuisds+∨i=1nδ˜ji∫−∞tkit−sgiuisds+∑i=1nt˜ji∫t−σ2ttgiuisds+ϕjt.It is sufficiently rewritten as(12)Dαut−Fut−ht=−Aut+Bhvt+Chvt−τ1t+γ|+δ∫−∞tkt−shvsds+T∫t−τ2tthvsds+θt,Dαvt−Evt−ht=−A˜vt+B˜gut+C˜gut−τ1t+γ˜+δ˜∫−∞tkt−sgusds+T˜∫t−σ2ttgusds+ϕt.Theorem 1.
Under Assumption1, system (4) satisfies the passivity analysis if there exist matrices G1>0, G2>0, H1>0, H2>0, H3>0, H4>0, H5>0, H6>0, Y1>0, Y2>0, Y3>0, Y4>0, Z1>0, and Z2>0 and positive diagonal matrices L1>0, L2>0, R1>0, and R2>0 such that the following LMI holds:(13)π=π11π12π13π14∗π22π23π24∗∗π33π34∗∗∗π44<0,where(14)π11=1,1L1F210G1F0∗H3+σ2H5+Z2−L1000∗∗−H31−μ1−L200∗∗∗−Y31−h10∗∗∗∗−Z2,π12=0000G1B000B˜TG200F21L20C˜TG2000000000γ˜+δ˜TG20,π13=G1C0G1γ+δG1T000000000000000000000,π14=0G10AT000−I0B˜T0000C˜T000FT00000γ˜+δ˜T,π22=−H5σ200T˜TG20∗−H11−μ1−L2F11000∗∗−Y1σ100∗∗∗−2G2A˜+H2+τ1Y2−R1F12R1F22∗∗∗∗10,10,π23=0000000000000000G2E00000000,π24=0000T˜T000000000000G20A˜T0−I0BT0,π33=−H41−μ2−R2000F22R2∗−Y41−d1000∗∗−Z100∗∗∗−H6τ20∗∗∗∗−H21−μ2−R2F12,π34=000CT00000ET000|γ|+|δ|T0000TT000000,π44=−Y2τ10000∗−γI0I0∗∗−γI0I∗∗∗−Y30∗∗∗∗−Y4,where 10,10=H4+τ2H6+Z1−R1.Proof.
Let(15)Vt,ut,vt=∑i=17Vt,ut,vti,where(16)V1t,ut,vt=D−1−αuTtG1ut+D−1−αvTtG2vt,V2t,ut,vt=∫t−σ1ttuTsH1usds+∫t−τ1ttvTsH2vsds,V3t,ut,vt=∫t−σ1ttgTusH3gusds+∫t−τ1tthTvsH4hvsds,V4t,ut,vt=∫−σ20∫t+θtgTusH5gusdsdθ+∫−τ20∫t+θthTvsH6hvsdsdθ,V5t,ut,vt=∫−σ10∫t+θtuTsY1usdsdθ+∫−τ10∫t+θtvTsY2vsdsdθ,V6t,ut,vt=∫t−httDαusTY3Dαusds+∫t−dttDαusTY4Dαusds,V7t,ut,vt=∑j=1mz1j∫0∞kjθ∫t−θthj2vjsdsdθ+∑i=1nz2i∫0∞kiθ∫t−θtgi2ujsdsdθ.
Taking the time derivative along the trajectories of system (12),(17)V˙1t,ut,vt≤2uTtG1Dαut+2vTtG2Dαvt,(18)V˙2V˙2t,ut,vt≤uTtH1ut−uTt−σ1tH1ut−σ1t1−μ1+vTtH2vt−vTt−τ1tH2vt−τ1t1−μ1,(19)V˙3t,ut,vt≤gTutH3gut−gTut−σ1tH3gut−σ1t1−μ1+hTvtH4hvt−hTvt−τ1tH4hvt−τ1t1−μ2,(20)V˙4t,ut,vt=σ2gTutH5gut−∫t−σ2tgTusH5gusds+τ2hTvtH6hvt−∫t−τ2thTvsH6hvsds,(21)V˙5t,ut,vt=σ1uTtY1ut−∫t−σ1tuTsY1usds+τ1vTtY2vt−∫t−τ1tvTsY2vsds,(22)V˙6t,ut,vt≤DαutTY3Dαut−Dαut−htTY3Dαut−ht1−h1+DαutTY4Dαut−Dαut−dtTY4Dαut−dt1−d1,(23)V˙7t,ut,vt=hTvtZ1hvt−∫−∞tkt−shvsdsTZ1∫−∞tkt−shvsds+gTutZ2gut−∫−∞tkt−sgusdsTZ2∫−∞tkt−sgusds.
From the Cauchy–Schwarz inequality and (23), the following inequality holds:(24)∑j=1mz1j∫0∞kjθ∫t−θthj2vjsdsdθ=∑j=1mz1j∫0∞kjθhj2vjtdθ−∑j=1mz1j∫0∞kjθhj2vjt−θdθ=hTvtZ1hvt−∑j=1mz1j∫0∞kjθhj2vjt−θdθ≤hTvtZ1hvt−∑j=1mz1j∫0∞kjθhjvjt−θdθ2=hTvtZ1hvt−∫−∞tkt−shvsdsTZ1∫−∞tkt−shvsds,∑i=1nz2i∫0∞kiθ∫t−θtgi2uisdsdθ=∑i=1nz2i∫0∞kiθgi2uitdθ−∑i=1nz2i∫0∞kiθgi2uit−θdθ=gTutZ2gut−∑i=1nz2i∫0∞kiθgi2uit−θdθ≤gTutZ2gut−∑i=1nz2i∫0∞kiθgiuit−θdθ2=gTutZ2gut−∫−∞tkt−sgusdsTZ2∫−∞tkt−sgusds.
According to Lemma2, we get(25)−σ2∫t−σ2tgTusH5gusds≤∫t−σ2ttgusdsTH5∫t−σ2ttgusds,(26)−τ2∫t−τ2thTvsH6hvsds≤∫t−τ2tthvsTH6∫t−τ2tthvsds.
In addition, for anyn×n diagonal matrices L1>0, L2>0, R1>0, and R2>0, the following inequality holds:(27)utgutT−L1F11L1F21∗−L1utgut+ut−σ1tgut−σ1tT−L2F11L2F21∗−L2ut−σ1tgut−σ1t≥0.(28)vthvtT−R1F12R1F22∗−R1t+vt−τ1thvt−τ1tT−R2F12R2F22∗−R2vt−τ1thvt−τ1t≥0.
From (17)–(28) and passive definition, we have(29)V˙t,ut,vt−2hTvtgTutθtϕt−γθTtϕTtθtϕt≤ℑTtπℑt,where(30)ℑTt=uTtgTutgTut−σ1tDαvt−h1tT∫−∞tkt−sgusdsT∫t−σ2ttgusdsTut−σ1t∫t−σ1tuTsdsTvTthTvthTvt−τ1tDαvt−dtT∫−∞tkt−shvsdsT∫t−τ2tthvsdsTvTt−τ1t∫t−τ1tvTsdsTθTtϕTt.
Therefore, we conclude that(31)V˙t,ut,vt−2hTvtgTutθtϕt−γθTtϕTtθtϕt≤0.
By integrating (31) with respect to t over the time period 0,tk, we have(32)2∫0tkhTvsgTusθsϕsds≥Vtk,utk,vtk−V0,u0,v0−γ∫0tkθTsϕTsds≥−γ∫0tkθTsϕTsds.
Foru(0) = 0 and v(0) = 0, we have V (0, u(0), v(0)) = 0 and Vtk,utk,vtk>0. Thus, it follows from Definition 2 that the neutral-type fuzzy BAM neural network is passive. The proof is completed.Remark 1.
Equation (9) proposes an improved passivity-based fractional-order neutral-type fuzzy cellular BAM neural network with time-varying delays. Based on Theorem 1, we have the following result for neutral delay is not equal to discrete delay of fuzzy cellular BAM neural networks with time-varying delays, without neutral term then system (12) becomes(33)Dαut=−Aut+Bhvt+Chvt−τ1t+γ+δ∫−∞tkt−shvsds+T∫t−τ2tthvsds+θt,Dαvt=−A˜vt+B˜gut+C˜gut−σ1t+γ+|δ˜|∫−∞tkt−sgusds+T˜∫t−σ2ttgusds+ϕt.Theorem 2.
The fuzzy BAM neural networks in (20) satisfy the passivity analysis if there exist matrices G1>0, G2>0, H1>0, H2>0, H3>0, H4>0, H5>0, H6>0, Y1>0, Y2>0, Z1>0, and Z2>0, positive diagonal matrices L1>0, L2>0, R1>0, and R2>0, and positive scalars F11 and F21 such that the following LMI holds:(34)Θ=Θ11Θ12Θ13∗Θ22Θ23∗∗Θ33<0,where(35)Θ11=1,1L1F210000∗2,20000∗∗−H31−μ1−L200F21L2∗∗∗−Z200∗∗∗∗−H5σ20∗∗∗∗∗6,6,Θ12=00G1BG1CG1γ+δG1T0B˜TG200000C˜TG200000γ˜+δ˜TG200000T˜TG20000000000,Θ13=00G10000−I0000000000000000,Θ22=−Y1σ100000∗8,8R1F22000∗∗H4+τ2H6+Z1−R1000∗∗∗−R2−H41−μ200∗∗∗∗−Z10∗∗∗∗∗−H6τ2,Θ23=0000000G200−I0F22R200000000000,Θ33=−H21−μ2−R2F12000∗−Y2τ100∗∗−γI0∗∗∗−γI,where (1, 1) = −2G1A+H1+σ1Y1−L1F11, (2, 2) = H3+σ2H5+Z2−L1, (6, 6) = −H11−μ1−L2F11, and (8, 8) = −2G2A˜+H2+τ1Y2−R1F12.Proof.
Let(36)Vt,ut,vt=∑i=16Vit,ut,vt,where(37)V1t,ut,vt=D−1−αuTtG1ut+D−1−αvTtG2vt,V2t,ut,vt=∫t−σ1ttuTsH1usds+∫t−τ1ttvTsH2vsds,V3t,ut,vt=∫t−σ1ttgTusH3gusds+∫t−τ1tthTvsH4hvsds,V4t,ut,vt=∫−σ20∫t+θtgTusH5gusdsdθ+∫−τ20∫t+θthTvsH6hvsdsdθ,V5t,ut,vt=∫−σ10∫t+θtuTsY1usdsdθ+∫−τ10∫t+θtvTsY2vsdsdθ,V6t,ut,vt=∑j=1mz1j∫0∞kjθ∫t−θthj2vjsdsdθ+∑i=1nz2i∫0∞kiθ∫t−θtgi2uisdsdθ.
The proof of this theorem immediately follows from Theorem1.Remark 2.
Consider the following BAM neural network (12), and let T=0andT˜=0; then,(38)Dαut−Fut−ht=−Aut+Bhvt+Chvt−τ1t+γ+δ∫−∞tkt−shvsds+θt,Dαvt−Evt−ht=−A˜vt+B˜gut+C˜gut−σ1t+γ˜+δ˜∫−∞tkt−sgutds+ϕt.Theorem 3.
The fuzzy BAM neural networks in (21) satisfy the passivity analysis if there exist matrices G1>0, G2>0, H1>0, H2>0, H3>0, H4>0, Q1>0, Q2>0, Y3>0, Y4>0, Z1>0, and Z2>0, positive diagonal matrices L1>0, L2>0, R1>0, and R2>0, and positive scalars F11 and F21 such that the following LMI holds:(39)Φ=Φ11Φ12Φ13∗Φ22Φ23∗∗Φ33<0,Φ11=1,1L1F210000∗Z2−L1+H3000B˜TG2∗∗−H31−μ1−L20F21L2C˜TG2∗∗∗−Z20γ˜T+δ˜TG2∗∗∗∗−H1−μ1−L2F110∗∗∗∗∗6,6,Φ12=G1BG1CG1γ+δ000000000000000000000000000R1F2200000,Φ13=G10G1F0AT00−I000B˜T00000C˜T00000γ˜+δ˜T0000000G20G2E0A˜T,Φ22=7,700000∗−R2−H41−μ10F22R200∗∗−Z1000∗∗∗10,1000∗∗∗∗−Q11−h10∗∗∗∗∗−Q21−d1,Φ23=−I000BT00000CT00000γ+δT0000000000000000000,Φ33=−γI000I0∗−γI000I∗∗−Y31−h10FT0∗∗∗−Y41−d10ET∗∗∗∗−Y30∗∗∗∗∗−Y4,where 1,1=−2G1A+Q1+H1−L1F11, 6,6=−2G2A˜+Q2+H2−R1F12, 7,7=H4+Z1−R1, and 10,10=−H21−μ2−R2F12.Proof.
Let(40)Vt,ut,vt=∑i=15Vit,ut,vt,where(41)V1t,ut,vt=D−1−αuTtG1ut+D−1−αvTtG2vt,V2t,ut,vt=∫t−σ1ttuTsH1usds+∫t−τ1ttvTsH2vsds+∫t−httuTsQ1usds+∫t−dttvTsQ2vsds,V3t,ut,vt=∫t−σ1ttgTusH3gusds+∫t−τ1tthTvsH4hvsds,V4t,ut,vt=∫t−httDαusTY3Dαusds+∫t−dttDαusTY4Dαusds,V5t,ut,vt=∑j=1mz1j∫0∞kjθ∫t−θthj2vjsdsdθ+∑i=1nz2i∫0∞kiθ∫t−θtgi2uisdsdθ.
By Lemma1, we obtain the time derivative along the trajectories of system (38):(42)V˙1t,ut,vt≤2uTtG1Dαut+2vTtG2Dαvt,V˙2t,ut,vt≤uTtH1ut−uTt−σ1tH1ut−σ1t1−μ1+vTtH2vt−vTt−τ1tH2vt−τ1t1−μ2+uTtQ1ut−uTt−htQ1ut−ht1−h2+vTtQ2vt−vTt−dtQ2vt−dt1−d2,V˙3t,ut,vt≤gTutH3gut−gTut−σ1tH3gut−σ1t1−μ1+hTvtH4hvt−hTvt−τ1tH4hvt−τ1t1−μ2,V˙4t,ut,vt≤DαutTY3Dαut−Dαut−htTY3Dαut−ht1−h1+DαutTY4Dαut−Dαvt−dtTY4Dαvt−dt1−d1,V˙5t,ut,vt=hTvtZ1hvt−∫−∞tkt−shvsdsTZ1∫−∞tkt−shvsds+gTutZ2gut−∫−∞tkt−sgusdsTZ2∫−∞tkt−sgusds.
The proof of this theorem immediately follows from Theorem1.


## 4. Numerical Example

In this section, we provide numerical examples to illustrate the usefulness of our stability results. Consider the fractional-order neutral-type fuzzy cellular BAM neural network(43)Dαut−Fut−ht=−Aut+Bhvt+Chvt−τ1t+γ+δ∫−∞tkt−shvsds+T∫t−τ2tthvsds+θt,Dαvt−Evt−ht=−A˜vt+B˜gut+C˜gut−σ1t+γ˜+δ˜∫−∞tkt−sgusds+T˜∫t−σ2ttgusds+ϕt,with the following system parameter values:(44)A=0.9100000.9100000.9100000.91,A˜=0.6700000.6700000.6700000.67,B˜=0.980.650.230.120.090.450.670.760.050.880.990.560.770.650.870.80,C˜=0.120.340.450.980.560.440.990.340.550.760.450.780.870.450.980.67,T=0.980.090.070.560.060.870.040.870.990.880.470.560.860.650.880.89,E=0.990.760.560.880.090.650.620.570.450.570.670.890.560.760.440.76,F=0.090.670.870.660.780.780.560.980.980.480.560.870.660.870.790.99,B=0.440.550.760.440.560.980.090.870.550.450.650.890.980.880.450.87,C=0.670.780.980.650.870.880.890.660.340.980.660.980.440.090.760.65,T˜=0.880.690.970.560.050.870.940.860.980.680.470.360.860.690.780.59.Here, we take the activation functions ashjs=gis=tanhs,i=1,2. Then, let σ1=0.87, μ2=0.66, γ¯=0.98, δ¯=0.87,μ1=0.67,h1=0.12,d1=0.24,τ1=0.66,γ=0.56,andδ=0.54. By solving LMI (13), we get the following feasible solutions:(45)G1=0.0086−0.0010−0.0012−0.0013−0.00100.0081−0.0012−0.0016−0.0012−0.00120.0083−0.0020−0.0013−0.0016−0.00200.0072,H1=−9.9204−0.0027−0.00430.0038−0.0027−9.92150.0002−0.0004−0.00430.0002−9.92440.00130.0038−0.00040.0013−9.9281,Y1=4.1743−0.00010.00030.0007−0.00014.17460.00010.00040.00030.00014.17440.00020.00070.00040.00024.1743,L1=10.6275000010.6279000010.6272000010.6279,H3=0.94450.00170.00660.00090.00170.94480.00030.00430.00660.00030.9479−0.00110.00090.0043−0.00110.9486,H5=2.22470.00030.00080.00080.00032.22350.00220.00230.00080.00222.22410.00260.00080.00230.00262.2231,Z2=1.5895−0.0036−0.0063−0.0115−0.00361.5933−0.0091−0.0135−0.0063−0.00911.5924−0.0132−0.0115−0.0135−0.01321.5934,L2=8.190400008.190700008.190300008.1908,Y3=7.4185−0.0013−0.0016−0.0016−0.00137.4179−0.0015−0.0020−0.0016−0.00157.4182−0.0024−0.0016−0.0020−0.00247.4169,G2=0.2911−0.0354−0.0505−0.0799−0.03540.3037−0.0622−0.0899−0.0505−0.06220.3014−0.0883−0.0799−0.0899−0.08830.2956,H2=−4.5412−0.0442−0.0520−0.0802−0.0442−4.5368−0.0596−0.0783−0.0520−0.0596−4.5360−0.0855−0.0802−0.0783−0.0855−4.5447,Y2=2.3446−0.0085−0.0121−0.0187−0.00852.3469−0.0138−0.0200−0.0121−0.01382.3472−0.0203−0.0187−0.0200−0.02032.3462,R1=13.4720000013.4666000013.4636000013.4756,H4=−0.03780.00900.00550.00970.0090−0.03620.00680.00810.00550.0068−0.03760.01030.00970.00810.0103−0.0353,(46)H6=1.4340−0.0017−0.0044−0.0063−0.00171.4354−0.0049−0.0072−0.0044−0.00491.4350−0.0069−0.0063−0.0072−0.00691.4350,Z1=3.2204−0.0030−0.0070−0.0104−0.00303.2227−0.0080−0.0118−0.0070−0.00803.2222−0.0115−0.0104−0.0118−0.01153.2221,R2=10.1925000010.17450.092900010.1710000010.1786,Y4=7.4064−0.00030.0018−0.0005−0.00037.4082−0.0007−0.00130.0018−0.00077.4090−0.0008−0.0005−0.0013−0.00087.4088,γ=7.8131.Therefore, it follows from Theorem1 that the fractional-order neutral-type fuzzy cellular BAM neural network with time-varying delay (43) is passive.


## 5. Conclusion

The problem of passivity-based fractional-order neutral-type fuzzy cellular BAM neural networks with time-varying delays has been presented in this paper. By constructing suitable LKFs and using Jensen’s inequalities, new delay-dependent conditions are established in terms of LMIs to guarantee the passivity criteria for the neural network. Future work will focus on the fractional-order complex-valued neural networks with delays and fuzzy complex-valued neural networks with delays. Finally, two examples are presented to illustrate the effectiveness of the proposed criteria.



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