# Analysis of Adaptive Synchronization for Stochastic Neutral-Type Memristive Neural Networks with Mixed Time-Varying Delays

Authors: Desheng Hong; Zuoliang Xiong; Cuiping Yang
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

Linear feedback control and adaptive feedback control are proposed to achieve the synchronization of stochastic neutral-type memristive neural networks with mixed time-varying delays. By applying the stochastic differential inclusions theory, Lyapunov functional, and linear matrix inequalities method, we obtain some new adaptive synchronization criteria. A numerical example is given to illustrate the effectiveness of our results.


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## Body

## 1. Introduction

During the last few years, as we know neural networks have been widely researched in control, image processing, associative memory design, pattern recognition, information science, and so on (see [1–3]). Chua firstly predicted the memristor as the fourth fundamental electrical circuit element in 1971 [4]. In 2008, Hewlett-Packard research team [5] obtained a practical memristor device and exhibited its characteristic, such as nanoscale and the memory ability. It has been shown that memristors can be used to work as biological synapses in artificial neural network and replace resistor to simulate the human brain in memristor-based neural networks (MNNs) model, which would benefit many practical applications (see [6, 7]).It is well know that time delays present complex and unpredictable behaviors in practice often caused by finite switching speeds of the amplifiers, which may affect the stability of the system and even results in oscillation, divergence, and instability phenomena. Therefore, much effort has been devoted to analyze dynamic behaviors of MNNs with various types of times delays (see [8, 9]); constant time delays and the time-varying delays have been studied in [10–12]. The investigations of MNNs discussed consider the discrete delays in [13]. However, since the neural signal propagation is often distributed during a certain time period in the presence of an amount of parallel pathways with a variety of axon sizes and lengths, hence, the authors in [14, 15] have concentrated on the mixed delays.On the other hand, in reality, the fluctuations from the release of neurotransmitters or other probabilistic causes may affect the stability property in the nervous system and synaptic transmission. So the stability analysis with stochastic perturbation has aroused great interest of many researchers (see [16, 17]). It is natural and important that systems containing some information are not only related to the derivative of the current state, but also have a great relationship with the previous derivative, which is called neural-type neural networks (see [9, 18, 19]).Recently synchronization and antisynchronization of memristor-based neural networks have received great attention due to their potential, such as secure communication information science and biological technology [20]. But the networks are not always able to synchronize by themselves. Then, various effective control approaches and techniques have been proposed for synchronization, such as impulsive control, feedback control, adaptive control, and intermittent control (see [21, 22]). And a lot of achievements have been made in the stability and synchronization problem of MNNs, including exponential synchronization, lag synchronization, and finite time synchronization (see [23–26]).Motivated by the above discussion, even though the synchronization problem of stochastic MNNs has been studied, there are few studies on the synchronization problem of stochastic neutral-type MNNs. So in this paper we focus our minds on the adaptive synchronization for neutral-type MNNs with mixed time-varying delays to bridge the gap. By applying the stochastic differential inclusions theory, Lyapunov functional, and linear matrix inequalities method, we obtain some new adaptive synchronization criteria.This paper is organized as follows. In Section2, we introduce the model and some preliminaries. The main theoretical results are derived in Section 3. In Section 4, a numerical simulation is presented to verify our obtained results. Finally, conclusion is given in Section 5.Throughout this paper, solutions of all the systems considered are intended in the Filippov’s sense.Rn and Rn×n denote the n-dimensional Euclidean space and the set of all n×n real matrices, respectively. The superscript T denotes matrix transposition, tr· denotes the trace of the corresponding matrix, and I denotes the identity matrix. λmax and λmin denote the maximum and minimum eigenvalues of a real symmetric matrix. diag⋯ stands for the block diagonal matrix. CF02-τ,0;Rn denote the family of all F0 measurable, C-τ,0;Rn-valued stochastic variables ξ=ξθ:-τ≤θ≤0, such that ∫–τ0Eξs2ds<∞, where E· stands for the correspondent expectation operator with respect to the given probability measure P.cou,v denotes the closure of a convex hull generated by real numbers u and v or real matrices u and v.


## 2. Preliminaries

In this paper, the following stochastic neutral-type memristive neural network with mixed time-varying delays is described by (i=1,2,…,n) (1)
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,with initial conditions xi(t)=ϕi(t),t∈[-τ,0], where xi(t) is the voltage of the capacitor Ci, f~j(·), g~j(·), and h~j(·) are neuron activation functions, and Ii is the external constant input. C=diag(c1,c2,…,cn) and D=diag(d1,d2,…,dn) are self-feedback connection matrices and ci>0, di>0 (i=1,2,…,n), aij(xi(t)), bij(xi(t)), and wij(xi(t)) represent memristor-based weights: (2)aijxit=W1ijCi×sginij,bijxit=W2ijCi×sginij,wijxit=W3ijCi×sginij,sginij=1i≠j,-1i=j.Here W(k)ij denote the memductances of memristors R(k)ij, k=1,2,3. According to the pinched hysteretic loops of property of memristors, we set(3)aijxit=a^ijxit≤γi,aˇijxit>γi,bijxit=b^ijxit≤γi,bˇijxit>γi,wijxit=w^ijxit≤γi,wˇijxit>γi,where the switching jumps γi>0, a^ij, aˇij, b^ij, bˇij, w^ij, wˇij(i,j=1,2,…,n) are constants. A(xi(t))=aij(xi(t))n×n, B(xi(t))=bij(xi(t))n×n, and W(xi(t))=wij(xi(t))n×n are memristive connection weights, which represent the neuron interconnection matrix, respectively. If A(xi(t))=aij(xi(t))n×n, B(xi(t))=bij(xi(t))n×n, and W(xi(t))=wijxitn×n are constants, system (1) will reduce to a general network. Let a_ij=min⁡{a^ij,aˇij}, a¯ij=max⁡{a^ij,aˇij}, b_ij=min⁡{b^ij,bˇij}, b¯ij=max{b^ij,bˇij}, w_ij=min⁡{w^ij,wˇij}, w¯ij=max⁡{w^ij,w˘ij}, A¯=a¯ijn×n, B¯=b¯ijn×n, W¯=w¯ijn×n, A_=a_ijn×n, B_=b_ijn×n, W_=w_ijn×n, x(t)=x1(t),x2(t),…,xn(t)T, I=I1,I2,…,InT, f~xt=f~(x1(t)),f~(x2(t)),…,f~(xn(t))T, g~(x(t))=g~(x1(t)),g~(x2(t)),…,g~(xn(t))T, h~(x(t))=h~(x1(t)),h~(x2(t)),…,h~(xn(t))T, for i=1,2,…,n. τk(t)(k=1,2,3) represent the time-varying transmission delays. Since aij(xi(t)), bij(xi(t)), wij(xi(t)) are discontinuous, in this paper, the solutions of all the following systems are illustrated in Filippov’s sense. By applying theory of differential inclusions and set-valued maps in system (1), this can be written as follows: (4)dxt-Dxt-τ1t∈-Cxt+∑j=1ncoaijxitf~xt+∑j=1ncobijxitg~xt-τ2t+∑j=1ncowijxit∫t-τ3tth~xsds+Idt,where the set-valued maps are defined as follows:(5)coaijxit=a^ijxit<γi,a_ij,a¯ijxit=γi,aˇijxit>γi,cobijxit=b^ijxit<γi,b_ij,b¯ijxit=γi,bˇijxit>γi,cowijxit=w^ijxit<γi,w_ij,w¯ijxit=γi,wˇijxit>γi,or equivalently, there exist aij(t)∈co[aij(xi(t))], bij(t)∈co[bij(xi(t))], wij(t)∈co[wij(xi(t))], and A~(t)=aij(t)n×n, B~(t)=bij(t)n×n, W~(t)=wij(t)n×n, such that (6)dxt-Dxt-τ1t=-Cxt+A~tf~xt+B~tg~xt-τ2t+W~t∫t-τ3tth~xsds+Idt.We consider system (6) as the drive system. Similarly, the response system is (7)dyt-Dyt-τ1t=-Cyt+A~tf~yt+B~tg~yt-τ2t+W~t∫t-τ3tth~ysds+I+utdt+σt,yt-xt,yt-τ1t-xt-τ1t,yt-τ2t-xt-τ2t,yt-τ3t-xt-τ3tdωt,where u(t)=u1(t),u2(t),…,un(t)T∈Rn is the controller, ω(t)=ω1(t),ω2(t),…,ωn(t)T is an n-dimensional Brownian motion defined on the complete probability space (Ω,F,P) with a natural filtration Ftt≥0, and σ:R+×Rn×Rn×Rn×Rn→Rn×n is the noise intensity matrix, where σ satisfies σ(t,0,0,0,0)≡0. Let e(t)=e1(t),e2(t),…,en(t)T be the synchronization error, where ei(t)=yi(t)-xi(t), ei(t-τ2(t))=yi(t-τ2(t))-xi(t-τ2(t)), ei(t-τ3(t))=yi(t-τ3(t))-xi(t-τ3(t)). From (6) and (7), we can get the following synchronization error system: (8)det-Det-τ1t=-Cet+A~tfet+B~tget-τ2t+W~t∫t-τ3tthesds+utdt+σt,et,et-τ1t,et-τ2t,et-τ3tdωt,where f(e(t))=f~(y(t))-f~(x(t)), g(e(t-τ2(t)))=g~(y(t-τ2(t)))-g~(x(t-τ2(t))), h(e(t-τ3(t)))=h~(y(t-τ3(t)))-h~(x(t-τ3(t))). To prove our main results, the following assumptions and lemmas are needed.Assumption 1 (see [27]).
There exist diagonal matricesLi-=diag⁡(li1-,li2-,…,lin-) and Li+=diag⁡(li1+,li2+,…,lin+),i=1,2,3, satisfying l1j-≤f~j(u)-f~j(v)/u-v≤l1j+,l2j-≤g~j(u)-g~j(v)/u-v≤l2j+, l3j-≤h~j(u)-h~j(v)/u-v≤l3j+, for all u,v∈R,u≠v,j=1,2,…,n.Assumption 2.
There exist positive constantsτ1,τ2,τ3,μ1,μ2, and μ3, such that(9)0≤τ1t≤τ1,0≤τ2t≤τ2,0≤τ3t≤τ3,τ˙1t≤μ1<1,τ˙2t≤μ2<1,τ˙3t≤μ3<1.Remark 3.
The assumption strong condition can be weaken; please refer to [28, 29] Assumptions 2 and 1.Assumption 4.
∀ a , b ∈ R N, there exist positive constants L_i,L¯i,M_i,M¯i,N_i,N¯i, such that(10)f~i·≤L_i,g~i·≤M_i,h~i·≤N_i,f~ia-f~ib≤L¯ia-b,g~ia-g~ib≤M¯ia-b,h~ia-h~ib≤N¯ia-b,where f~i(0)=g~i(0)=h~i(0)=0 and i,j=1,2,…,n.Assumption 5 (see [30]).
There exist positive matricesR1,R2,R3, and R4, such that(11)trσTt,x1,x2,x3,x4σt,x1,x2,x3,x4≤x1TR1x1+x2TR2x2+x3TR3x3+x4TR4x4,for all x1,x2,x3,x4∈Rn and t∈R+.Assumption 6.
The matrixD satisfies ρ(D)<1, where ρ(D) is the spectral radius of D.Definition 7 (see [31]).
The two coupled memristive neural networks (6) and (7) are said to be stochastic synchronization for almost every initial data if for every ξ∈CF02([τ,0];Rn),limx→∞⁡e(t;ξ)=0a.s.Lemma 8 (see [32]).
For any vectorsa,b∈Rn, the inequality ±2aTb≤aTSa+bTS-1b holds, in which S is any matrix with S>0.Lemma 9 (see [33]).
For any positive definite matrixM∈Rn×n, scalar γ>0, and vector function η:[0,γ]→Rn such that the integration concerned is well defined, then(12)∫0γηsdsTM∫0γηsds≤γ∫0γηsTMηsds.Lemma 10 (see [34]).
Ifa1≥a2≥a3,b1≥b2≥b3, then 2(a1b1+a2a2+b3b3)≥a1b2+a1b3+a2b1+a2b3+a3b1+a3b2,∀ai,bi(i=1,2,3)∈R.Remark 11.
There are some other convenient and useful inequality techniques; refer to [1] Lemmas 2, 3, 7, and 8.Lemma 12 (see [35]).
Given matricesΩ1,Ω2,Ω3, where Ω1=Ω1T and Ω2>0, then Ω1+Ω3TΩ2-1Ω3<0 if and only if(13)Ω1Ω3TΩ3-Ω2<0or-Ω2Ω3Ω3TΩ1<0.


## 3. Main Results

In this section, the stochastic synchronization for the two coupled memristive neural networks (6) and (7) is investigated under Assumptions 1–6.


### 3.1. Stochastic Adaptive Synchronization for the Two Coupled Memristive Neural Networks via the Adaptive Feedback Control

Theorem 13.
Under Assumptions1–6, the two coupled memristive neural networks (6) and (7) can be synchronized for almost every initial data, if there exist positive diagonal matrices H1,H2,H3,P=diag⁡(p1,p2,…,pn), positive definite matrices Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8,S1,S2, and a positive scalar λ such that the LMIs hold:(14)P≤λI,(15)τ3S1+S2≤Q8,(16)Ψ=Ψ11Ψ1200Ψ1500Ψ180∗Ψ2200Ψ2500Ψ280∗∗Ψ33000000∗∗∗Ψ4400000∗∗∗∗Ψ550000∗∗∗∗∗Ψ66000∗∗∗∗∗∗Ψ7700∗∗∗∗∗∗∗Ψ880∗∗∗∗∗∗∗∗Ψ99,<0, where(17)Ψ11=-2PC+2Pα+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q1+Q2+Q3+λR1+R4,Ψ12=DPC-αPD,Ψ15=PA¯,Ψ18=PB¯,Ψ22=λR2+μ1-1Q1+DPW¯S2-1W¯TPTDT,Ψ25=-DPA¯,Ψ28=-DPB¯,Ψ33=λR3+μ2-1Q2,Ψ44=μ3-1Q3,Ψ55=-H1+τ1Q6+τ2Q7,Ψ66=-H2+Q4,Ψ77=τ3Q8-H3+Q5,Ψ88=μ2-1Q4,Ψ99=μ3-1Q5.And the adaptive feedback controller is designed as(18)ut=-Ket, where the feedback strength K=diag⁡(k1,k2,…,kn) is updated by the following law:(19)k˙i=φiei2t-φidieiteit-τ1t, with arbitrary constant φi>0(i=1,2,…,n).Proof.
We consider the following Lyapunov-Krasovskii functions:(20)Vt,et=∑i=110Vit,eit, where (21)V1t,et=et-Det-τ1tTPet-Det-τ1t,V2t,et=∫t-τ1ttesTQ1esds,V3t,et=∫t-τ2ttesTQ2esds,V4t,et=∫t-τ3ttesTQ3esds,V5t,et=∫t-τ2ttgTesQ4gesds,V6t,et=∫t-τ3tthTesQ5hesds,V7t,et=∫-τ1t0∫t+rtfTesQ6fesdsdr,V8t,et=∫-τ2t0∫t+rtfTesQ7fesdsdr,V9t,et=∫-τ3t0∫t+rthTesQ8hesdsdr,V10t,et=∑i=1npiki+α2φi. By Itô formula, it follows that(22)LVt,et=Vtt,et+Vet,et-Cet+A~tfet+B~tget-τ2t+W~t∫t-τ3tthesds+ut+12trσTt,et,et-τ1t,et-τ2t,et-τ3tVeeσt,et,et-τ1t,et-τ2t,et-τ3t,where Vt(t,e(t))=∂V(t,e(t))/∂t, Vet,et=∂Vt,et/∂e1,…,∂Vt,et/∂en, Vee(t,e(t))=∂2V(t,e(t))/∂ei∂ejn×n, and (23)LV1t,et=2et-Det-τ1tTP-Cet+A~tfet+B~tget-τ2t+W~t∫t-τ3tthesds-Ket+trσTt,et,et-τ1t,et-τ2t,et-τ3tP×σt,et,et-τ1t,et-τ2t,et-τ3t=eTt-2PCet+eTt2PA~tfet+eTt2PB~tget-τ2t+eTt2PW~t∫t-τ3tthesds+eTt-τ1t2DPCet+eTt-τ1t-2DPA~tfet+eTt-τ1t-2DPB~tget-τ2t+eTt-τ1t-2DPW~t∫t-τ3tthesds-2∑i=1npikiei2t+2∑i=1ndipikieiteit-τ1t+trσTt,et,et-τ1t,et-τ2t,et-τ3tPσt,et,et-τ1t,et-τ2t,et-τ3t. From Lemma 8, we get(24)eTt2PW~t∫t-τ3tthesds≤eTtPW¯S1-1W¯TPTet+∫t-τ3tthesdsTS1∫t-τ3tthesds,eTt-τ1t-2DPW~t∫t-τ3tthesds≤eTt-τ1tDPW¯S2-1W¯TPTDTet-τ1t+∫t-τ3tthesdsTS2∫t-τ3tthesds. Utilizing Lemma 9 yields(25)∫t-τ3tthesdsTS1+S2∫t-τ3tthesds≤τ3t∫t-τ3tthTesS1+S2hesds≤∫t-τ3tthTesτ3S1+S2hesds. It follows from Assumption 5 and (14) that(26)trσTt,et,et-τ1t,et-τ2t,et-τ3tP×σt,et,et-τ1t,et-τ2t,et-τ3t≤λmaxPtrσTt,et,et-τ1t,et-τ2t,et-τ3t×σt,et,et-τ1t,et-τ2t,et-τ3t≤λeTtR1tet+eTt-τ1tR2tet-τ1t+eTt-τ2tR3tet-τ2t+eTt-τ3tR4tet-τ3t. By Itô formula, we have(27)LV2t,et=eTtQ1et-1-τ˙1teTt-τ1tQ1et-τ1t≤eTtQ1et+μ1-1eTt-τ1tQ1et-τ1t.LV3t,et=eTtQ2et-1-τ˙2teTt-τ2tQ2et-τ2t≤eTtQ2et+μ2-1eTt-τ2tQ2et-τ2t,LV4t,et=eTtQ3et-1-τ˙3teTt-τ3tQ3et-τ3t≤eTtQ3et+μ3-1eTt-τ3tQ3et-τ3t,LV5t,et=gTetQ4get-1-τ˙2tgTet-τ2tQ4get-τ2t≤gTetQ4get+gTet-τ2tμ2-1Q4get-τ2t,LV6t,et=hTetQ5het-1-τ˙3thTet-τ2tQ5het-τ3t≤hTetQ5het+hTet-τ3tμ3-1Q5het-τ3t,LV7t,et=τ1tfTetQ6fet-∫t-τ1ttfTesQ6fesds≤fTetτ1Q6fet,LV8t,et=τ2tfTetQ7fet-∫t-τ2ttfTesQ7fesds≤fTetτ2Q7fet, From Assumption 1, it follows that(28)fTetH1fet≤eTtL1H1L1et,gTetH2get≤eTtL2H2L2et,hTetH3het≤eTtL3H3L3et, where H1,H2,H3 are positive diagonal matrices and Lj=diag⁡(lj1,lj2,…,ljn),lji=max⁡{lji-,lji+}(j=1,2,3) for i=1,2,…,n.(29)LV9t,et=τ3thTetQ8het-∫t-τ3tthTesQ8hesds≤hTetτ3Q8het-∫t-τ3tthTesQ8hesds,LV10t,et=2∑i=1npiφiki+αk˙i=2∑i=1npiki+αei2t-dieiteit-τ1t.Condition (15) yields(30)∫t-τ3tthTesτ3S1+S2hesds-∫t-τ3tthTesQ8hesds≤0. Substituting inequalities (23)–(30) into (22), we obtain(31)LVt,et=eTt-2PC+2Pα+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q1+Q2+Q3+λR1+R4et+eTt2PA¯fet+eTt2PB¯get-τ2t+eTt-τ1t2DPC-2αPDet+eTt-τ1t-2DPA¯fet+eTt-τ1t-2DPB¯get-τ2t+eTt-τ1tλR2+μ1-1Q1+DPW¯S2-1W¯TPTDTet-τ1t+eTt-τ2tλR3+μ2-1Q2et-τ2t+eTt-τ3tμ3-1Q3et-τ3t+hTetτ3Q8-H3+Q5het+fTetτ1Q6-H1+τ2Q7fet+gTet-H2+Q4get+hTet-τ3tμ3-1Q5het-τ3t+gTet-τ2tμ2-1Q4get-τ2t-eTtλR4et+eTt-τ3tλR4et-τ3t=ΦTtΨΦt-eTtλR4et+eTt-τ3tλR4et-τ3t, where (32)Φt=et,et-τ1t,et-τ2t,et-τ3t,fet,get,het,get-τ2t,het-τ2tT,Ψ=Ψ11Ψ1200Ψ1500Ψ180∗Ψ2200Ψ2500Ψ280∗∗Ψ33000000∗∗∗Ψ4400000∗∗∗∗Ψ550000∗∗∗∗∗Ψ66000∗∗∗∗∗∗Ψ7700∗∗∗∗∗∗∗Ψ880∗∗∗∗∗∗∗∗Ψ99, with(33)Ψ11=-2PC+2Pα+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q1+Q2+Q3+λR1+R4,Ψ12=DPC-αPD,Ψ15=PA¯,Ψ18=PB¯,Ψ22=λR2+μ1-1Q1+DPW¯S2-1W¯TPTDT,Ψ25=-DPA¯,Ψ28=-DPB¯,Ψ33=λR3+μ2-1Q2,Ψ44=μ3-1Q3,Ψ55=-H1+τ1Q6+τ2Q7,Ψ66=-H2+Q4,Ψ77=τ3Q8-H3+Q5,Ψ88=μ2-1Q4,Ψ99=μ3-1Q5.Using Lemma 12, if Ψ<0, let ζ=λmin(-Ψ), and, clearly, the constant ζ>0. This fact together with (31) gives(34)LVt,et≤-eTtλR4+ζIet+eTt-τ3tλR4-ζIet-τ3t=-ω1et+ω2et-τ3t, where ω1(e(t))=eT(t)(λR4+ζI)e(t) and ω2(e(t))=eT(t)(λR4-ζI)e(t).
It is obvious thatω1(e(t))>ω2(e(t)) for any e(t)≠0. Therefore, applying LaSalle-type invariance principle (see [27, 36]) for the stochastic differential delay equations, we can conclude that the coupled memristive neural networks (6) and (7) can be synchronized for almost every initial data. This completes the proof.WhenD=0, from Theorem 13, we obtain the following corollary.Corollary 14.
Under Assumptions1–5, the two coupled memristive neural networks (6) and (7) with D=0 can be synchronized for almost every initial data, if there exist positive diagonal matrices H1,H2,H3,P=diag⁡(p1,p2,…,pn), positive definite matrices Q2,Q3,Q4,Q5,Q7,Q8,S1, and a positive scalar λ such that the LMIs hold:(35)P≤λI,τ3S1≤Q8,Θ=Θ1100Ψ1500Ψ180∗Ψ33000000∗∗Ψ4400000∗∗∗Θ550000∗∗∗∗Ψ66000∗∗∗∗∗Ψ7700∗∗∗∗∗∗Ψ880∗∗∗∗∗∗∗Ψ99<0, where(36)Θ11=-2PC+2Pα+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q2+Q3+λR1+R4.Θ55=-H1+τ2Q7,And the adaptive feedback controller is designed as(37)ut=-Ket, where the feedback strength K=diag⁡(k1,k2,…,kn) is updated by the following law:(38)k˙i=φiei2t, with arbitrary constant φi>0(i=1,2,…,n).Remark 15.
When we remove the stochastic perturbations, our models become the model in [37], so our models are the extension of the model in [37]. Because the stochastic perturbations are unavoidable in practice, our models are more general and useful in practice.Remark 16.
IfA(xi(t))=aij(xi(t))n×n,B(xi(t))=bij(xi(t))n×n and W(xi(t))=wij(xi(t))n×n are constants, system (1) will reduce to a general network. What is more, when we remove the neutral terms, our models become the model in [10, 27], so our models are the extension of the model in [10, 27]. Because the neutral terms are important and complicated, our models are more general and useful in practice.


### 3.2. Stochastic Adaptive Synchronization for the Two Coupled Memristive Neural Networks via the Linear Feedback Control

Theorem 17.
Under Assumptions1–6, two coupled memristive neural networks (6) and (7) can be synchronized for almost every initial data, if there exist positive diagonal matrices H1,H2,H3,P=diag⁡(p1,p2,…,pn), positive definite matrices Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8,S1,S2, and a positive scalar λ such that the LMIs hold:(39)P≤λI,τ3S1+S2≤Q8,X=X11X1200X1500X180∗X2200X2500X280∗∗X33000000∗∗∗X4400000∗∗∗∗X550000∗∗∗∗∗X66000∗∗∗∗∗∗X7700∗∗∗∗∗∗∗X880∗∗∗∗∗∗∗∗X99<0, where(40)X11=-2PC-2PK+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q1+Q2+Q3+λR1+R4,X12=DPC+KPD,X15=PA¯,X18=PB¯,X22=λR2+μ1-1Q1+DPW¯S2-1W¯TPTDT,X25=-DPA¯,X28=-DPB¯,X33=λR3+μ2-1Q1,X44=μ3-1Q3,X55=-H1+τ1Q6+τ2Q7,X66=-H2+Q4,X77=τ3Q8-H3+Q5,X88=μ2-1Q4,X99=μ3-1Q5.And the linear feedback controller is designed as(41)ut=-Ket, where K=diag⁡(k1,k2,…,kn),ki>0 is the feedback gain.Proof.
We consider the following Lyapunov-Krasovskii functions:(42)Vt,et=∑i=19Vit,eit, where (43)V1t,et=et-Det-τ1tTPet-Det-τ1t,V2t,et=∫t-τ1ttesTQ1esds,V3t,et=∫t-τ2ttesTQ2esds,V4t,et=∫t-τ3ttesTQ3esds,V5t,et=∫t-τ2ttgTesQ4gesds,V6t,et=∫t-τ3tthTesQ5hesds,V7t,et=∫-τ1t0∫t+rtfTesQ6fesdsdr,V8t,et=∫-τ2t0∫t+rtfTesQ7fesdsdr,V9t,et=∫-τ3t0∫t+rthTesQ8hesdsdr. Then, the proof of Theorem 17 is similar to Theorem 13, so the proof process is omitted here.Corollary 18.
Under Assumptions1–5, two coupled memristive neural networks (6) and (7) with D=0 can be synchronized for almost every initial data, if there exist positive diagonal matrices H1,H2,H3,P=diag⁡(p1,p2,…,pn), positive definite matrices Q2,Q3,Q4,Q5,Q7,Q8,S1, and a positive scalar λ such that the LMIs hold:(44)P≤λI,τ3S1≤Q8,Z=Z1100Ψ1500Ψ180∗Ψ33000000∗∗Ψ4400000∗∗∗Z550000∗∗∗∗Ψ66000∗∗∗∗∗Ψ7700∗∗∗∗∗∗Ψ880∗∗∗∗∗∗∗Ψ99<0, where(45)Z11=-2PC-2PK+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q2+Q3+λR1+R4,Z55=-H1+τ2Q7.And the linear feedback controller is designed as(46)ut=-Ket, where K=diag⁡(k1,k2,…,kn),ki>0 is the feedback gain.Remark 19.
WhenD=0, the systems are no longer neutral-type neural networks. We find that adaptive synchronization of other types of neural networks model has been researched (see [38, 39]). We can also get the synchronization results from Theorem 17 when D=0.Remark 20.
WhenW~(t)=0, the systems no longer have distributed time-varying delays. Our models become the model in [40]; we can also get the synchronization results, so our models are the extension of the model in [40] and they are more general than that.


## 3.1. Stochastic Adaptive Synchronization for the Two Coupled Memristive Neural Networks via the Adaptive Feedback Control

Theorem 13.
Under Assumptions1–6, the two coupled memristive neural networks (6) and (7) can be synchronized for almost every initial data, if there exist positive diagonal matrices H1,H2,H3,P=diag⁡(p1,p2,…,pn), positive definite matrices Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8,S1,S2, and a positive scalar λ such that the LMIs hold:(14)P≤λI,(15)τ3S1+S2≤Q8,(16)Ψ=Ψ11Ψ1200Ψ1500Ψ180∗Ψ2200Ψ2500Ψ280∗∗Ψ33000000∗∗∗Ψ4400000∗∗∗∗Ψ550000∗∗∗∗∗Ψ66000∗∗∗∗∗∗Ψ7700∗∗∗∗∗∗∗Ψ880∗∗∗∗∗∗∗∗Ψ99,<0, where(17)Ψ11=-2PC+2Pα+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q1+Q2+Q3+λR1+R4,Ψ12=DPC-αPD,Ψ15=PA¯,Ψ18=PB¯,Ψ22=λR2+μ1-1Q1+DPW¯S2-1W¯TPTDT,Ψ25=-DPA¯,Ψ28=-DPB¯,Ψ33=λR3+μ2-1Q2,Ψ44=μ3-1Q3,Ψ55=-H1+τ1Q6+τ2Q7,Ψ66=-H2+Q4,Ψ77=τ3Q8-H3+Q5,Ψ88=μ2-1Q4,Ψ99=μ3-1Q5.And the adaptive feedback controller is designed as(18)ut=-Ket, where the feedback strength K=diag⁡(k1,k2,…,kn) is updated by the following law:(19)k˙i=φiei2t-φidieiteit-τ1t, with arbitrary constant φi>0(i=1,2,…,n).Proof.
We consider the following Lyapunov-Krasovskii functions:(20)Vt,et=∑i=110Vit,eit, where (21)V1t,et=et-Det-τ1tTPet-Det-τ1t,V2t,et=∫t-τ1ttesTQ1esds,V3t,et=∫t-τ2ttesTQ2esds,V4t,et=∫t-τ3ttesTQ3esds,V5t,et=∫t-τ2ttgTesQ4gesds,V6t,et=∫t-τ3tthTesQ5hesds,V7t,et=∫-τ1t0∫t+rtfTesQ6fesdsdr,V8t,et=∫-τ2t0∫t+rtfTesQ7fesdsdr,V9t,et=∫-τ3t0∫t+rthTesQ8hesdsdr,V10t,et=∑i=1npiki+α2φi. By Itô formula, it follows that(22)LVt,et=Vtt,et+Vet,et-Cet+A~tfet+B~tget-τ2t+W~t∫t-τ3tthesds+ut+12trσTt,et,et-τ1t,et-τ2t,et-τ3tVeeσt,et,et-τ1t,et-τ2t,et-τ3t,where Vt(t,e(t))=∂V(t,e(t))/∂t, Vet,et=∂Vt,et/∂e1,…,∂Vt,et/∂en, Vee(t,e(t))=∂2V(t,e(t))/∂ei∂ejn×n, and (23)LV1t,et=2et-Det-τ1tTP-Cet+A~tfet+B~tget-τ2t+W~t∫t-τ3tthesds-Ket+trσTt,et,et-τ1t,et-τ2t,et-τ3tP×σt,et,et-τ1t,et-τ2t,et-τ3t=eTt-2PCet+eTt2PA~tfet+eTt2PB~tget-τ2t+eTt2PW~t∫t-τ3tthesds+eTt-τ1t2DPCet+eTt-τ1t-2DPA~tfet+eTt-τ1t-2DPB~tget-τ2t+eTt-τ1t-2DPW~t∫t-τ3tthesds-2∑i=1npikiei2t+2∑i=1ndipikieiteit-τ1t+trσTt,et,et-τ1t,et-τ2t,et-τ3tPσt,et,et-τ1t,et-τ2t,et-τ3t. From Lemma 8, we get(24)eTt2PW~t∫t-τ3tthesds≤eTtPW¯S1-1W¯TPTet+∫t-τ3tthesdsTS1∫t-τ3tthesds,eTt-τ1t-2DPW~t∫t-τ3tthesds≤eTt-τ1tDPW¯S2-1W¯TPTDTet-τ1t+∫t-τ3tthesdsTS2∫t-τ3tthesds. Utilizing Lemma 9 yields(25)∫t-τ3tthesdsTS1+S2∫t-τ3tthesds≤τ3t∫t-τ3tthTesS1+S2hesds≤∫t-τ3tthTesτ3S1+S2hesds. It follows from Assumption 5 and (14) that(26)trσTt,et,et-τ1t,et-τ2t,et-τ3tP×σt,et,et-τ1t,et-τ2t,et-τ3t≤λmaxPtrσTt,et,et-τ1t,et-τ2t,et-τ3t×σt,et,et-τ1t,et-τ2t,et-τ3t≤λeTtR1tet+eTt-τ1tR2tet-τ1t+eTt-τ2tR3tet-τ2t+eTt-τ3tR4tet-τ3t. By Itô formula, we have(27)LV2t,et=eTtQ1et-1-τ˙1teTt-τ1tQ1et-τ1t≤eTtQ1et+μ1-1eTt-τ1tQ1et-τ1t.LV3t,et=eTtQ2et-1-τ˙2teTt-τ2tQ2et-τ2t≤eTtQ2et+μ2-1eTt-τ2tQ2et-τ2t,LV4t,et=eTtQ3et-1-τ˙3teTt-τ3tQ3et-τ3t≤eTtQ3et+μ3-1eTt-τ3tQ3et-τ3t,LV5t,et=gTetQ4get-1-τ˙2tgTet-τ2tQ4get-τ2t≤gTetQ4get+gTet-τ2tμ2-1Q4get-τ2t,LV6t,et=hTetQ5het-1-τ˙3thTet-τ2tQ5het-τ3t≤hTetQ5het+hTet-τ3tμ3-1Q5het-τ3t,LV7t,et=τ1tfTetQ6fet-∫t-τ1ttfTesQ6fesds≤fTetτ1Q6fet,LV8t,et=τ2tfTetQ7fet-∫t-τ2ttfTesQ7fesds≤fTetτ2Q7fet, From Assumption 1, it follows that(28)fTetH1fet≤eTtL1H1L1et,gTetH2get≤eTtL2H2L2et,hTetH3het≤eTtL3H3L3et, where H1,H2,H3 are positive diagonal matrices and Lj=diag⁡(lj1,lj2,…,ljn),lji=max⁡{lji-,lji+}(j=1,2,3) for i=1,2,…,n.(29)LV9t,et=τ3thTetQ8het-∫t-τ3tthTesQ8hesds≤hTetτ3Q8het-∫t-τ3tthTesQ8hesds,LV10t,et=2∑i=1npiφiki+αk˙i=2∑i=1npiki+αei2t-dieiteit-τ1t.Condition (15) yields(30)∫t-τ3tthTesτ3S1+S2hesds-∫t-τ3tthTesQ8hesds≤0. Substituting inequalities (23)–(30) into (22), we obtain(31)LVt,et=eTt-2PC+2Pα+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q1+Q2+Q3+λR1+R4et+eTt2PA¯fet+eTt2PB¯get-τ2t+eTt-τ1t2DPC-2αPDet+eTt-τ1t-2DPA¯fet+eTt-τ1t-2DPB¯get-τ2t+eTt-τ1tλR2+μ1-1Q1+DPW¯S2-1W¯TPTDTet-τ1t+eTt-τ2tλR3+μ2-1Q2et-τ2t+eTt-τ3tμ3-1Q3et-τ3t+hTetτ3Q8-H3+Q5het+fTetτ1Q6-H1+τ2Q7fet+gTet-H2+Q4get+hTet-τ3tμ3-1Q5het-τ3t+gTet-τ2tμ2-1Q4get-τ2t-eTtλR4et+eTt-τ3tλR4et-τ3t=ΦTtΨΦt-eTtλR4et+eTt-τ3tλR4et-τ3t, where (32)Φt=et,et-τ1t,et-τ2t,et-τ3t,fet,get,het,get-τ2t,het-τ2tT,Ψ=Ψ11Ψ1200Ψ1500Ψ180∗Ψ2200Ψ2500Ψ280∗∗Ψ33000000∗∗∗Ψ4400000∗∗∗∗Ψ550000∗∗∗∗∗Ψ66000∗∗∗∗∗∗Ψ7700∗∗∗∗∗∗∗Ψ880∗∗∗∗∗∗∗∗Ψ99, with(33)Ψ11=-2PC+2Pα+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q1+Q2+Q3+λR1+R4,Ψ12=DPC-αPD,Ψ15=PA¯,Ψ18=PB¯,Ψ22=λR2+μ1-1Q1+DPW¯S2-1W¯TPTDT,Ψ25=-DPA¯,Ψ28=-DPB¯,Ψ33=λR3+μ2-1Q2,Ψ44=μ3-1Q3,Ψ55=-H1+τ1Q6+τ2Q7,Ψ66=-H2+Q4,Ψ77=τ3Q8-H3+Q5,Ψ88=μ2-1Q4,Ψ99=μ3-1Q5.Using Lemma 12, if Ψ<0, let ζ=λmin(-Ψ), and, clearly, the constant ζ>0. This fact together with (31) gives(34)LVt,et≤-eTtλR4+ζIet+eTt-τ3tλR4-ζIet-τ3t=-ω1et+ω2et-τ3t, where ω1(e(t))=eT(t)(λR4+ζI)e(t) and ω2(e(t))=eT(t)(λR4-ζI)e(t).
It is obvious thatω1(e(t))>ω2(e(t)) for any e(t)≠0. Therefore, applying LaSalle-type invariance principle (see [27, 36]) for the stochastic differential delay equations, we can conclude that the coupled memristive neural networks (6) and (7) can be synchronized for almost every initial data. This completes the proof.WhenD=0, from Theorem 13, we obtain the following corollary.Corollary 14.
Under Assumptions1–5, the two coupled memristive neural networks (6) and (7) with D=0 can be synchronized for almost every initial data, if there exist positive diagonal matrices H1,H2,H3,P=diag⁡(p1,p2,…,pn), positive definite matrices Q2,Q3,Q4,Q5,Q7,Q8,S1, and a positive scalar λ such that the LMIs hold:(35)P≤λI,τ3S1≤Q8,Θ=Θ1100Ψ1500Ψ180∗Ψ33000000∗∗Ψ4400000∗∗∗Θ550000∗∗∗∗Ψ66000∗∗∗∗∗Ψ7700∗∗∗∗∗∗Ψ880∗∗∗∗∗∗∗Ψ99<0, where(36)Θ11=-2PC+2Pα+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q2+Q3+λR1+R4.Θ55=-H1+τ2Q7,And the adaptive feedback controller is designed as(37)ut=-Ket, where the feedback strength K=diag⁡(k1,k2,…,kn) is updated by the following law:(38)k˙i=φiei2t, with arbitrary constant φi>0(i=1,2,…,n).Remark 15.
When we remove the stochastic perturbations, our models become the model in [37], so our models are the extension of the model in [37]. Because the stochastic perturbations are unavoidable in practice, our models are more general and useful in practice.Remark 16.
IfA(xi(t))=aij(xi(t))n×n,B(xi(t))=bij(xi(t))n×n and W(xi(t))=wij(xi(t))n×n are constants, system (1) will reduce to a general network. What is more, when we remove the neutral terms, our models become the model in [10, 27], so our models are the extension of the model in [10, 27]. Because the neutral terms are important and complicated, our models are more general and useful in practice.


## 3.2. Stochastic Adaptive Synchronization for the Two Coupled Memristive Neural Networks via the Linear Feedback Control

Theorem 17.
Under Assumptions1–6, two coupled memristive neural networks (6) and (7) can be synchronized for almost every initial data, if there exist positive diagonal matrices H1,H2,H3,P=diag⁡(p1,p2,…,pn), positive definite matrices Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8,S1,S2, and a positive scalar λ such that the LMIs hold:(39)P≤λI,τ3S1+S2≤Q8,X=X11X1200X1500X180∗X2200X2500X280∗∗X33000000∗∗∗X4400000∗∗∗∗X550000∗∗∗∗∗X66000∗∗∗∗∗∗X7700∗∗∗∗∗∗∗X880∗∗∗∗∗∗∗∗X99<0, where(40)X11=-2PC-2PK+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q1+Q2+Q3+λR1+R4,X12=DPC+KPD,X15=PA¯,X18=PB¯,X22=λR2+μ1-1Q1+DPW¯S2-1W¯TPTDT,X25=-DPA¯,X28=-DPB¯,X33=λR3+μ2-1Q1,X44=μ3-1Q3,X55=-H1+τ1Q6+τ2Q7,X66=-H2+Q4,X77=τ3Q8-H3+Q5,X88=μ2-1Q4,X99=μ3-1Q5.And the linear feedback controller is designed as(41)ut=-Ket, where K=diag⁡(k1,k2,…,kn),ki>0 is the feedback gain.Proof.
We consider the following Lyapunov-Krasovskii functions:(42)Vt,et=∑i=19Vit,eit, where (43)V1t,et=et-Det-τ1tTPet-Det-τ1t,V2t,et=∫t-τ1ttesTQ1esds,V3t,et=∫t-τ2ttesTQ2esds,V4t,et=∫t-τ3ttesTQ3esds,V5t,et=∫t-τ2ttgTesQ4gesds,V6t,et=∫t-τ3tthTesQ5hesds,V7t,et=∫-τ1t0∫t+rtfTesQ6fesdsdr,V8t,et=∫-τ2t0∫t+rtfTesQ7fesdsdr,V9t,et=∫-τ3t0∫t+rthTesQ8hesdsdr. Then, the proof of Theorem 17 is similar to Theorem 13, so the proof process is omitted here.Corollary 18.
Under Assumptions1–5, two coupled memristive neural networks (6) and (7) with D=0 can be synchronized for almost every initial data, if there exist positive diagonal matrices H1,H2,H3,P=diag⁡(p1,p2,…,pn), positive definite matrices Q2,Q3,Q4,Q5,Q7,Q8,S1, and a positive scalar λ such that the LMIs hold:(44)P≤λI,τ3S1≤Q8,Z=Z1100Ψ1500Ψ180∗Ψ33000000∗∗Ψ4400000∗∗∗Z550000∗∗∗∗Ψ66000∗∗∗∗∗Ψ7700∗∗∗∗∗∗Ψ880∗∗∗∗∗∗∗Ψ99<0, where(45)Z11=-2PC-2PK+PW¯S1-1W¯TPT+L1H1L1+L2H2L2+L3H3L3+Q2+Q3+λR1+R4,Z55=-H1+τ2Q7.And the linear feedback controller is designed as(46)ut=-Ket, where K=diag⁡(k1,k2,…,kn),ki>0 is the feedback gain.Remark 19.
WhenD=0, the systems are no longer neutral-type neural networks. We find that adaptive synchronization of other types of neural networks model has been researched (see [38, 39]). We can also get the synchronization results from Theorem 17 when D=0.Remark 20.
WhenW~(t)=0, the systems no longer have distributed time-varying delays. Our models become the model in [40]; we can also get the synchronization results, so our models are the extension of the model in [40] and they are more general than that.


## 4. Numerical Simulation

In this section, a numerical example is given to illustrate the effectiveness of Theorem13. Consider a two-dimensional synchronization error system (8) with u(t)=-Ke(t) such that k˙i=φiei2(t)-φidiei(t)ei(t-τ1(t)).Takef(e(t))=g(e(t))=h(e(t))=tanh⁡(e1(t)),tanh⁡(e1(t))T,τ1(t)=0.6,τ2(t)=0.1,τ3(t)=0.2,L1=L2=L3=I, and σ(t,e(t),e(t-τ1(t)),e(t-τ2(t)),e(t-τ3(t)))=(47)0.5e1t+0.2e1t-τ1t000.5e2t-τ2t+0.2e2t-τ3t,A~xt=a11x1ta12x1ta21x2ta22x2t,B~xt=b11x1tb12x1tb21x2tb22x2t,W~xt=w11x1tw12x1tw21x2tw22x2t,a11x1t=0.3x1t≤1,-0.3x1t>1,a12x1t=0.2x1t≤1,-0.2x1t>1,a21x2t=0.2x2t≤1,-0.2x2t>1,a22x2t=1x2t≤1,-1x2t>1,b11x1t=0.4x1t≤1,-0.4x1t>1,b12x1t=0.3x1t≤1,-0.3x1t>1,b21x2t=0.5x2t≤1,-0.5x2t>1,b22x2t=0.2x2t≤1,-0.2x2t>1,w11x1t=0.3x1t≤1,0.2x1t>1,w12x1t=0.5x1t≤1,0.3x1t>1,w21x2t=0.5x2t≤1,0.3x2t>1,w22x2t=0.3x2t≤1,0.2x2t>1,and R1=0.3I,R2=0.2I,R3=0.3I,R4=0.1I,μ1=μ2=μ3=0.2. Then (48)A¯=0.30.20.21,B¯=0.40.30.50.2,C=0.6000.7,D=0.1000.4,W¯=0.30.50.50.3.Letting α=20, using LMI toolbox in MATLAB, we obtain the following feasible solutions to LMIs in Theorem 13: (49)P=0.9183-0.3941-0.39410.4683,H1=0.3760-0.1907-0.19070.1717,H2=0.3823-0.1868-0.18680.1367,H3=0.7747-0.4052-0.40520.2813,Q1=0.3465-0.1538-0.15380.4013,Q2=0.2118-0.0897-0.08970.1025,Q3=0.1834-0.0959-0.09590.0666,Q4=0.2613-0.1224-0.12240.0927,Q5=0.2087-0.1090-0.10900.0760,Q6=0.2036-0.1067-0.10670.0716,Q7=0.1283-0.0672-0.06720.0453,Q8=0.2313-0.1210-0.12100.0839,S1=1.04021.50851.50852.8776,S2=1.04021.50851.50852.8776,λ=1.2143.So the conditions of Theorem13 are satisfied, and we conclude that two coupled memristive neural networks (6) and (7) can be synchronized for almost every initial data.Now by taking the initial date ase(0)=0.4,0.5T,K(0)=10,15T and φ1=0.1,φ2=0.2, we can draw the dynamic curves of the error system, the evolution of adaptive coupling strength k1,k2, and the Brownian motion ω(t), respectively, as Figures 1–3. Figure 1 shows that two coupled memristive neural networks (6) and (7) are synchronized.Figure 1
The curve of the synchronization errorse1,e2.Figure 2
The evolution graph of the adaptive coupling strengthsk1,k2.Figure 3
The evolution graph of the Brownian motionsω1,ω2.


## 5. Conclusions and Discussion

In this paper, by applying LaSalle-type invariance principle for stochastic differential delays equations, the stochastic differential inclusions theory, Lyapunov functional, and linear matrix inequalities method, linear feedback control and adaptive feedback control are proposed to achieve the synchronization of stochastic neutral-type memristive neural networks with mixed time-varying delays. Even though the synchronization problem of stochastic MNNs has been studied, there are few studies on the synchronization problem of stochastic neutral-type MNNs. Neutral terms are taken into account in this paper, which make the model have wider application and make research more meaningful. So we generalized the synchronization problem of MNNs. The effectiveness of our results has been illustrated by a numerical example. Furthermore, exponential synchronization and passivity of this model can be discussed in the near future.



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