# Exponential Antisynchronization Control of Stochastic Memristive Neural Networks with Mixed Time-Varying Delays Based on Novel Delay-Dependent or Delay-Independent Adaptive Controller

Authors: Minghui Yu; Weiping Wang; Xiong Luo; Linlin Liu; Manman Yuan
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The global exponential antisynchronization in mean square of memristive neural networks with stochastic perturbation and mixed time-varying delays is studied in this paper. Then, two kinds of novel delay-dependent and delay-independent adaptive controllers are designed. With the ability of adapting to environment changes, the proposed controllers can modify their behaviors to achieve the best performance. In particular, on the basis of the differential inclusions theory, inequality theory, and stochastic analysis techniques, several sufficient conditions are obtained to guarantee the exponential antisynchronization between the drive system and response system. Furthermore, two numerical simulation examples are provided to the validity of the derived criteria.


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## Body

## 1. Introduction

Memristor is a nonlinear resistor with memory function. Because of the nonlinear nature, it has been found that memristor responds well to the variable strength of synapses in the human brain. Therefore, we use memristor instead of resistor to reform new neural networks model named memristive neural networks (MNNs). In recent decades, dynamic behaviors analysis of MNNs has been attracting increasing attentions [1–3].As a typical dynamic behavior, the stability and synchronization problems of MNNs have been widely discussed, including exponential synchronization [4, 5], lag synchronization [6], finite time synchronization [7, 8], and antisynchronization [9, 10]. In communication system, the digital signal is transmitted by switching back and forth continuously between synchronization and antisynchronization, which strengthen security and secrecy. Besides that, the antisynchronization analysis of MNNs can provide the designers with some amazing properties, extensive flexibility, and opportunities. Thus, there are a few articles dealing with the antisynchronization issues of MNNs [11–15]. In [11], they studied the exponential antisynchronization in mean square of MNNs with the uncertain terms which include nonmodeled dynamics with boundary and stochastic perturbations. By an intermittent sampled-data controller [12] and utilizing matrix measure approach and Halanay inequality [13], the antisynchronization control of MNNs with time-varying delays has been addressed. Wang et al. discussed the antisynchronization control of MNNs with multiple proportional delays [14] and mixed time-varying delays [15]. Recently, the antisynchronization control has been widely employed in many fields, for example, secure communication, harmonic oscillation generation, and information science [16–18].In practice, because of certain switching speeds of the amplifiers, time delays are often encountered in hardware implementation [19], which may affect the stability of the system, even result in oscillation, divergence, and instability phenomena. Consequently, a great deal of effort has been devoted to the stability analysis of MNNs with various types of time delays [20–22]. Moreover, since an amount of parallel pathways of multiple axon sizes and lengths reside in the MNNs, such a unique nature can be appropriately modeled by distributed delay which means the signal transmission is distributed over a specific period of time. Hence, the authors in [23–25] have concentrated on the mixed delays. Meng and Xiang [24] considered a class of recurrent MNNs with mixed time-varying delays, in which the passivity and exponential passivity were investigated.Furthermore, in [25], Yang et al. studied the pth moment exponential stochastic synchronization for MNNs with discrete and distributed time-varying delays. However, the discrete delay τ(t) in [25] must be differentiable. Clearly, such a constraint on the delay term τ(t) was relatively strong. So, we discuss two kinds of discrete time-varying delay in this paper. One is that its derivative is bounded; the other is that it is not differentiable, or the derivative is unknown or unrestricted.On the other hand, as a result of random fluctuations from the release of neurotransmitters or other probabilistic causes in the nervous system, synaptic transmission is indeed a process accompanied by noise. The stability analysis of MNNs with stochastic perturbation has aroused great interests of many researchers [26–28]. For example, Song and Wen [27] investigated the stochastic MNNs with mixed delays and proposed the exponential synchronization criteria in the pth moment. Subsequently, Bao et al. [28] discussed a class of coupled stochastic MNNs with probabilistic time-varying delay in order to achieve exponential synchronization. However, to the best of our knowledge, the research on the exponential antisynchronization analysis of stochastic MNNs with mixed delays is still an open problem.Motivated by the above discussion, we focus our minds on the exponential antisynchronization problem of the MNNs with stochastic perturbation and mixed time-varying delays. Compared with other existing articles [11, 28, 29], our model is more complex and closer to the actual system; the obtained results are less conservative. The main contributions of this paper can be summarized in the following:(1)
The presented MNNs model contains not only stochastic perturbation but also two types of time-varying delays, namely, discrete and distributed time-varying delays.(2)
It is known that the time delays have a great influence on the stability of the MNNs, so the time delays cannot be neglected. As far as we know, many articles are based on the assumption that the time delay is bounded and derivable. Actually, it may happen that the time delayτ(t) is not differentiable or its derivative is unknown or unrestricted. Therefore, we study two kinds of discrete time-varying delay.(3)
Under two new types of controllers with delay-dependent and delay-independent, the obtained criteria which need no excessive numerical calculation can be extended to other general MNNs models.The rest structure of this paper is outlined as follows. In Section2, the models of the stochastic MNNs with mixed time-varying delays and some preliminaries are introduced. In Section 3, the main results on exponential antisynchronization of the stochastic MNNs are derived. In Section 4, some numerical simulations are presented to demonstrate the efficiency of the theoretical results. In Section 5, the conclusion is showed.


## 2. Model Description and Preliminaries

On the basis of the above discussion, we propose a class of MNNs with discrete and distributed time-varying delays described by the following differential equations:(1)dxitdt=-dixit+∑j=1naijxitfjxjt+∑j=1nbijxit-τtgjxjt-τt+∑j=1ncijxit∫t-ρttfjxjsds+Ii,where n corresponds to the number of neurons in system (1). xi(t) denotes the voltage of the capacitor Ci at time t. The self-inhibition connection weight is di>0. The neuron activation functions fj(·) and gj(·) are continuous and bounded. τ(t) and ρ(t) are two time-varying delays satisfying 0≤τ(t)≤θ1, 0≤ρ(t)≤θ2, τ˙t≤δ<1 (θ1, θ2, and δ are positive constants). Ii(t) represents the continuous external inputs on the ith neuron at time t. aijxit, bijxit-τt, and cijxit are the memristive connection weights: (2)aijxit=MijCi×signij,cijxit=M¯ijCi×signij,bijxit-τt=M~ijCi×signij,signij=1,i≠j,-1,i=j,in which Mij, M~ij, and M¯ij denote the memductances of memristors Rij, R~ij, and R¯ij, respectively. In addition, Rij represents the memristor between the activation function fi(xi(t)) and xit, R~ij represents the memristor between the activation function gixit-τt and xi(t), and R¯ij represents the memristor between the activation function ∫t-ρttfixisds and xi(t). As far as we know, capacitor Ci is constant while memductances of memristors Mij, M~ij, and M¯ij respond to change in pinched hysteresis loops [30]. As a consequence, aij(xi(t)), bijxit-τt, and cijxit will change with time. Based on the current-voltage characteristic of the memristor, we let (3)aijxit=a^ij,xit>Φi,aˇij,xit≤Φi,bijxit-τt=b^ij,xit-τt>Φi,bˇij,xit-τt≤Φi,cijxit=c^ij,xit>Φi,cˇij,xit≤Φi,where the switching jumps Φi>0, for i,j=1,2,…,n. Then a^ij, aˇij, b^ij, bˇij, c^ij, and cˇij are known constants with respect to the memristance. Moreover, the initial condition of system (1) is given to be x(s)=ϕ(s), ϕ(s)∈C-τ,0,Rn.Remark 1.
According to the above explanation,aij(xi(t)), bijxit-τt, and cijxit in system (1) are changed as the state of the memristance is switching or the state has switched. Therefore, the MNNs is considered as a time-varying system with state-dependent switching. When aijxit, bijxit-τt, and cijxit are constants, system (1) becomes the general recurrent neural networks.Sinceaij(xi(t)), bij(xi(t-τ(t))), and cij(xi(t)) are discontinuous, in this paper, the solutions of all the following systems are illustrated in Filippov’s sense. [·,·] represents the interval. Set a¯ij=max⁡a^ij,aˇij, a_ij=min⁡a^ij,aˇij, b¯ij=max⁡b^ij,bˇij, b_ij=min⁡b^ij,bˇij, c¯ij=max⁡c^ij,cˇij, and c_ij=min⁡c^ij,cˇij for i,j=1,2,…,n. co⁡u,v denotes closure of the convex hull generated by real numbers u and v. Considering system (1), we define the following set-valued maps: (4)co⁡aijxit=a^ij,xit>Φi,co⁡a^ij,aˇij,xit=Φi,aˇij,xit<Φico⁡bijxit-τt=⁡b^ij,xit-τt>Φi,co⁡b^ij,bˇij,xit-τt=Φi,bˇij,xit-τt<Φico⁡cijxit=c^ij,xit>Φi,co⁡c^ij,cˇij,xit=Φi,cˇij,xit<Φi.Obviously,co⁡a^ij,aˇij=a_ij,a¯ij, co⁡b^ij,bˇij=b_ij,b¯ij, and co⁡c^ij,cˇij=c_ij,c¯ij, for i,j=1,2,…,n. By the theory of differential inclusions, system (1) can be written as follows: (5)dxitdt∈-dixit+∑j=1nco⁡aijxitfjxjt+∑j=1nco⁡bijxit-τtgjxjt-τt+∑j=1nco⁡cijxit∫t-ρttfjxjsds+Ii,or equivalently, for i,j=1,2,…,n, there exist a^ij(xi(t))∈co⁡aijxit, b^ijxit-τt∈co⁡bijxit-τt, and c^ijxit∈co⁡cijxit, such that (6)dxitdt=-dixit+∑j=1na^ijxitfjxjt+∑j=1nb^ijxit-τtgjxjt-τt+∑j=1nc^ijxit∫t-ρttfjxjsds+Ii.In this paper, consider system (5) or (6) as the drive system; then the corresponding response system is designed by the following form: (7)dyit∈-diyit+∑j=1nco⁡aijxitfjyjt+∑j=1nco⁡bijxit-τtgjyjt-τt+∑j=1nco⁡cijxit∫t-ρttfjyjsds+Ii+uitdt+βit,xit+yit,xit-τt+yit-τtdωit,or equivalently, for i,j=1,2,…,n, there exist a^ij(xi(t))∈co⁡(aij(xi(t))), b^ijxit-τt∈co⁡bijxit-τt, c^ijxit∈co⁡cijxit, such that (8)dyit=-diyit+∑j=1na^ijxitfjyjt+∑j=1nb^ijxit-τtgjyjt-τt+∑j=1nc^ijxit∫t-ρttfjyjsds+Ii+uitdt+βit,xit+yit,xit-τt+yit-τtdωit,where ui(t), i=1,2,…,n, are the appropriate adaptive controllers designed to achieve the exponential antisynchronization stability. βit,xit+yit,xit-τt+yit-τtdωit denotes the stochastic perturbation, where ωi(t) is an n-dimensional Wiener process. It is defined on a complete probability space with a natural filtration Ft≥0 (i.e., Ft=βω(s):0≤s≤t).Remark 2.
Compared with the articles already published [28, 29], the proposed system contains not only discrete time-varying delay τ(t) but also distributed time-varying delay ρ(t), while the stochastic perturbation is also taken into account. Therefore, the obtained results have a high value of practical application in this paper.As a matter of convenience, we will take advantage of the following assumptions.Assumption 3.
For∀x, y∈RN, we assume the activation functions fj(·) and gj(·) are odd and satisfy the Lipschitz continuous condition with positive constants Lj and Qj, such that (9)fjx-fjy≤Ljx-y,gjx-gjy≤Qjx-y,j=1,2,…,n.Assumption 4.
β
i
≕
R
+
×
R
×
R
→
R is locally Lipschitz continuous and meets the linear growth condition with βit,0,0=0. There exist nonnegative constants Gi, Ki, such that, for i,j=1,2,…,n, (10)Trace⁡βit,eit,eit-τt2≤Gieit2+Kieit-τt2.Assumption 5.
Fori,j=1,…,n, (11)co⁡a_ij,a¯ijfjyjt+co⁡a_ij,a¯ijfjxjt⊆co⁡a_ij,a¯ijfjyjt+fjxjt,co⁡b_ij,b¯ijgjyjt-τt+co⁡b_ij,b¯ijgjxjt-τt⊆co⁡b_ij,b¯ijgjyjt-τt+gjxjt-τt,co⁡c_ij,c¯ijfjyjt+co⁡c_ij,c¯ijfjxjt⊆co⁡c_ij,c¯ijfjyjt+fjxjt.Remark 6.
In [4], however, the error system was defined based on the following assumption: (12)co⁡a_ij,a¯ijfjyjt-co⁡a_ij,a¯ijfjxjt⊆co⁡a_ij,a¯ijfjyjt-fjxjt.In fact, this assumption has been proved not always to be correct in [5]. Recently, many researchers have tried to explore a novel and appropriate way to solve this problem. So far, there are two kinds of convincing method. One is, in [31], according to the switching parameter Φi; the authors discussed the parameter in four cases and obtained several synchronization conditions of the chaotic MNNs with time delays. The other is, in [11, 15–17], by Assumption 5; the authors turned to study the antisynchronization issues of MNNs.Now, we define the following error system:(13)eit=yit+xit,i=1,2,…,n.According to the theories of set-valued maps, stochastic differential inclusions, and Assumption 5, we derive the error dynamical system as follows: (14)deit∈-dieit+∑j=1nco⁡aijxithjejt+∑j=1nco⁡bijxit-τtzjejt-τt+∑j=1nco⁡cijxit∫t-ρtthjejsds+2Ii+uitdt+βit,eit,eit-τtdωit,or equivalently, for i,j=1,2,…,n, there exist a~ij(xi(t))∈co⁡aijxit, b~ijxit-τt∈co⁡bijxit-τt, and c~ij(xi(t))∈co⁡cijxit, such that (15)deit=-dieit+∑j=1na~ijxithjejt+∑j=1nb~ijxit-τtzjejt-τt+∑j=1nc~ijxit∫t-ρtthjejsds+2Ii+uitdt+βit,eit,eit-τtdωit,where hj(ej(t))=fj(yj(t))+fj(xj(t)), zj(ej(t-τ(t)))=gj(yj(t-τ(t)))+gj(xj(t-τ(t))), and hjejs=fjyjs+fjxjs.Since the activation functionsfi(·) and gi(·) are odd and bounded, in the light of the Assumption 3, for ∀x,y∈R, it is so clear that (16)fjx+fjy≤Ljx+y,gjx+gjy≤Qjx+y.According to the definition of functions hj(·) and zj(·), we deduce (17)hjejt≤Ljejt,zjejt-τt≤Qjejt-τt.Next, we introduce a definition and some lemmas as preparation.Lemma 7.
For the stochastic system,(18)det=gt,et,et-τtdt+βt,et,et-τtdωt,where ω(t) is the Wiener process and it is obvious that Eω(t)=0. LVt,et is the operator defined as follows: (19)LVt,et=Vtt,et+Vet,etgt,et,et-τt+12Trace⁡βTt,et,et-τt×Veeβt,et,et-τt,where(20)Vtt,et=∂Vt,et∂t,Vet,et=∂Vt,et∂x1,∂Vt,et∂x2,…,∂Vt,et∂xn,Vee=∂2Vt,et∂xi∂xjn×n.Lemma 8 (see [32]).
IfX and Y are real matrices which have appropriate dimensions, then there exists a constant ε>0, such that (21)XTY+YTX≤εXTX+1εYTY.Lemma 9 (see [33], Jensen’s inequality).
For any constant matrixM>0, a scalar a≤b, and a vector function e(s):a,b→Rn, then the following inequality holds:(22)∫abesdsTM∫abesds≤b-a∫abeTsMesds.Definition 10.
The error systeme(t) with mixed delays and stochastic perturbations can exponentially converge to zero in mean square if there exist positive constants μ>0 and ν>0, such that Eet2≤μe-νt, for t≥0, where ν is called the exponential convergence rate. Systems (6) and (8) are said to be exponential antisynchronization in mean square sense.


## 3. Main Results

In this section, we get some sufficient criteria to achieve the exponential antisynchronization of the stochastic MNNs with mixed time-varying delays. Then two corollaries are also derived for the stochastic MNNs. The novel delay-dependent adaptive controllerui(t) is designed as follows: (23)uit=-kiteit-miteit-τt-2Ii,k˙it=pieit2eνt,ki0=0,m˙it=qieit-τteiteνt,mi0=0,where the constants pi>0, qi>0, for i=1,2,…,n.For convenience, some notations are given. Leta˚ij=max⁡a^ij,aˇij, b˚ij=max⁡b^ij,bˇij, and c˚ij=max⁡c^ij,cˇij, for i,j=1,2,…,n.Theorem 11.
Under Assumptions3, 4, 5, and 13, systems (6) and (8) achieve exponential antisynchronization in mean square sense under the delay-dependent adaptive controller (23). For a given constant ν>0, if there exist constants ri>0, si>0 such that the conditions hold (24)ri=12ν-2di+n+Gi+neνθ11-δ+nθ2eνθ2-1ν+∑j=1na∘ij2Lj2+b∘ij2Qj2+c∘ij2Lj2,si=Ki,i=1,2,…,n,then error system (15) can be exponentially converged to zero.Proof.
We choose the following Lyapunov functional:(25)Vt,et=∑k=14Vkt,et,where(26)V1t,et=∑i=1neνtei2t,V2t,et=∑i=1nkit-ri2pi+∑i=1nmit-si2qi,V3t,et=∑i=1n11-δ∑j=1n∫t-τttej2seνs+θ1ds,V4t,et=θ2∑i=1n∑j=1n∫-θ20∫t+stej2zeνz-sdzds.Then we have (27)Vt,et≥eνtet2.By Ito^ differential formula, we obtain the stochastic derivative of V(t) in the form of (28)dVt,et=∑i=14LVit,etdt+2eνt×∑i=1nβit,eit,eit-τtdωit.From Lemma 7, it is clear that (29)LV1t,et=∑i=1nνeνtei2t+2∑i=1neνteit-dieit+∑j=1na~ijxithjejt+∑j=1nb~ijxi-τtzjejt-τt+∑j=1nc~ijxit∫t-ρtthjejsds-kiteit-miteit-τt+eνt∑i=1nTrace⁡βit,eit,eit-τtT×βit,eit,eit-τt,LV2t,et=2eνt∑i=1nkit-riei2t+2eνt∑i=1nmit-sieit-τteit,LV3t,et=∑i=1n11-δeνt+θ1∑j=1nej2t-∑i=1n11-δ1-τ˙t×∑j=1nej2t-τteνt+θ1-τt≤eνt∑i=1neνθ11-δ∑j=1nej2t-eνt∑i=1n∑j=1nej2t-τt=eνtneνθ11-δ∑i=1nei2t-neνt∑i=1nei2t-τt,LV4t,et=θ2eνt∑i=1n∑j=1n∫-θ20ej2te-νs-ej2t+sds=θ2∑i=1n∑j=1neνtej2te-νs-ν-θ20-θ2∑i=1n∑j=1neνt∫-θ20ej2t+sds≤neνtθ2∑i=1neνθ2-1νei2t-nθ2∑i=1neνt∫t-ρttei2sds.
Moreover, from mean value inequality and Assumption3, we have (30)2∑i=1neit∑j=1na~ijxithjejt≤2∑i=1n∑j=1neita~ijxitLjejt=∑i=1n∑j=1na˚ij2Lj2ei2t+n∑i=1nei2t,2∑i=1neit∑j=1nb~ijxit-τtzjejt-τt≤2∑i=1n∑j=1neitb~ijxit-τtQjejt-τt=∑i=1n∑j=1nb˚ij2Qj2ei2t+n∑i=1nei2t-τt,2∑i=1nsieiteit-τt≤∑i=1nei2t+∑i=1nsi2ei2t-τt,2∑i=1neit∑j=1nc~ijxit∫t-ρtthjejsds≤2∑i=1n∑j=1neitc~ijxitLj∫t-ρttejsds≤∑i=1n∑j=1nc˚ij2Lj2ei2t+∫t-ρttejsds2,and together with Lemma 9, it follows that (31)∫t-ρttejsds2≤θ2∫t-ρttej2sds.Then we get(32)2∑i=1neit∑j=1nc~ijxit∫t-ρtthjejsds≤∑i=1n∑j=1nc˚ij2Lj2ei2t+nθ2∑i=1n∫t-ρttei2sds.
Under Assumption4, one has(33)LVt,et≤eνt∑i=1nν-2di+n+Gi-2ri+neνθ11-δ+nθ2eνθ2-1ν+∑j=1na˚ij2Lj2+b˚ij2Qj2+c˚ij2Lj2ei2t+eνt∑i=1nKiei2t-τt+neνt∑i=1nθ2∫t-ρttei2sds-eνt∑i=1n2sieiteit-τt-nθ2∑i=1neνt∫t-ρttei2sds;then we obtain the following estimation:(34)LVt,et≤eνt∑i=1nν-2di+n+Gi-2ri-1+neνθ11-δ+nθ2eνθ2-1ν+∑j=1na˚ij2Lj2+b˚ij2Qj2+c˚ij2Lj2ei2t+eνt∑i=1nKi-si2ei2t-τt.Let(35)ri=12ν-2di+n+Gi+neνθ11-δ+nθ2eνθ2-1ν+∑j=1na˚ij2Lj2+b˚ij2Qj2+c˚ij2Lj2,si=Ki,i=1,2,…,n;considering the condition of Theorem 11, we get (36)LVt,et≤-eνtet2≤0.It is easy to derive that (37)EVt,et≤EV0,e0,∀t≥0.Combining with (27), we finally have that (38)Eet2≤EVt,ete-νt≤EV0,e0e-νt.Therefore, by Definition 10, we see that systems (6) and (8) can be exponentially synchronized in mean square sense with the exponential convergence rate ν. The proof of Theorem 11 is completed.Remark 12.
In Theorem11, we assumed that the delay τ(t) satisfies 0≤τ˙(t)≤δ. Actually, when τ(t) is not differentiable or its derivatives are unknown or no bounded, Theorem 11 cannot be applicable any more. If the value of δ is unknown or not restriction, we will obtain the following corollary.Assumption 13.
The activation functionsfi(·) and gi(·) are bounded; namely, there exist constants κi, ϖi such that (39)fix≤κi,gix≤ϖi,∀x∈R,i=1,2,…,n.It is obvious that(40)hjejt≤2κj,∫t-ρtthjejsds≤2ϖjθ2,j=1,2,…,n.Corollary 14.
If the time-varying delayτ(t) is nondifferentiable, or the value of τ˙(t) is unknown or even the derivative τ˙(t) is unrestricted. According to Assumptions 3, 4, and 13, system (15) will converge to zero. For a given constant 0<ν<2di-Gi, if there exist constants ri>0, si>0 and the conditions hold (41)ri=∑j=1n2κja˚ij+b˚ij+∑j=1n2ϖjθ2c˚ij,si=Ki,i=1,2,…,n,systems (6) and (8) are exponentially antisynchronized in the mean square sense under the following controller: (42)uit=-kit-miteit-τt-2Ii,k˙it=pieiteνt,ki0=0,m˙it=qieit-τteiteνt,mi0=0.Proof.
We consider the following Lyapunov function:(43)Vt,et=∑i=1neνtei2t+∑i=1nkit-ri2pi+∑i=1nmit-si2qi.Combining with Assumption 13, the proof of Corollary 14 is analogous to Theorem 11, so we omit it here.Remark 15.
Based on the assumption that the delayτ(t) is bounded and differential, Theorem 11 provides a delay-dependent adaptive controller. It is worth mentioning that the criteria need no excessive numerical calculation such as solving LMIs [11] or computing complex algebraic conditions [28]. In Corollary 14, the delay τ(t) is not required to be differential but only bounded which makes our result more general. Further, there is an upper limit (2di-Gi) of the exponential convergence rate ν. When the exponent rate ν exceeds the upper limit, system (15) may not be able to converge or even cause unpredictable result.
Let(44)et=e1t,e2t,…,entT.The vector form of system (15) is given by (45)det=-Det+Axthet+Bxt-τt×zet-τt+Cxt∫t-ρthesds+2I+utdt+βt,et,e-τtdωt,where D=diag⁡d1,…,dn, Axt=a~ijxitn×n, Bxt-τt=b~ijxit-τtn×n, I=(I1,…,In), Cxt=c~ijxitn×n, and ω(t)=ω1t,…,ωnt.Moreover, we impose the following assumption and some lemmas.Assumption 16.
If there exist appropriate positive constant matricesT1>0 and T2>0, such that the inequality holds (46)Traceβt,et,et-τtTβt,et,et-τt≤eTtT1TT1et+eTt-τtT2TT2et-τt,for any e(t), et-τt∈Rn and t∈R+.Lemma 17 (see [24]).
For any given nonnegative matrixN∈Rn×n, if there exist a constant ν>0 and a vector function x:[a,b]→Rn, the following differential inequality holds: (47)∫baeνseTsNesds≥k∫baesdsTN∫baesds,where k=ν/e-νa-e-νb.If there is no time delay item, the delay-independent adaptive controller can be described as follows:(48)ut=-Mtet-2I,M˙t=μeνteTtet.Next, we will discuss the antisynchronization of systems (6) and (8) under the above delay-independent adaptive controller u(t).Theorem 18.
Under Assumption13, for a given constant ν>0, if there exist constant m>0 and matrixes Π>0, Ξ>0, the conditions hold as follows: (49)Π=eνθ1Δ2-1ΘTΘ+T2TT2,Ξ=eνθ2-1νΔ3eνt,m=12ν-2D+Δ1A˚A˚T+Δ1-1ΛTΛ+Δ2B˚B˚T+Δ3C˚C˚T+Π1-δ+T1TT1+θ2ΞΛTΛ+1,where Λ=diag⁡L1,L2,…,Ln, Θ=diag⁡Q1,Q2,…,Qn, A˚=a˚ijn×n, B˚=b˚ijn×n, and C˚=c˚ijn×n. System (45) will converge to zero with the action of the above delay-independent adaptive controller.Proof.
We construct the following Lyapunov function:(50)Vt,et=∑k=14Vkt,et,where(51)V1t,et=eνteTtet,V2t,et=1μMt-m2,V3t,et=Π1-δ∫t-τtteνseTsesds,V4t,et=Ξ∫-θ20∫t+steνzhTezhezdzds.Then we have (52)Vt,et≥eνteTtet.According to Lemma 7, we get that (53)LV1t,et=νeνteTtet+2eνteTt-Det+Axthet+Bxt-τtzet-τt+Cxt∫t-ρtthesds-Mtet+eνtTrace⁡βTt,et,et-τt×βt,et,et-τt,LV2t,et=2μMt-mM˙t=2Mt-meνteTtet,LV3t,et=Πeνt1-δeTtet-Πeνt-τt1-δ1-τ˙teTt-τtet-τt≤Πeνt1-δeTtet-Πeνteνθ1eTt-τtet-τt,LV4t,et=Ξ∫-θ20eνthTethet-eνt+shTet+shet+sds≤θ2ΞeνthTethet-Ξ∫t-θ2teνt+shTet+shet+sdt+s≤θ2ΞeνteTtΛTΛet-Ξ∫t-ρtteνshTeshesds.
It follows from Lemma8 that (54)2eTtAxthet≤Δ1eTtAxtATxtet+Δ1-1hTethet≤Δ1eTA˚A˚Tet+Δ1-1eTtΛTΛet,2eTtBxt-τtzet-τt≤Δ2eTBxt-τtBTxt-τtet+Δ2-1zTet-τtzet-τt≤Δ2eTB˚B˚Tet+Δ2-1eTt-τtΘTΘet-τt,2eTtCxt∫t-ρtthesds≤Δ3eTC˚C˚Tet+Δ3-1∫t-ρtthetdsT∫t-ρtthetds.
Under Lemma17, we obtain that (55)-Ξ∫t-ρtteνshTeshesds≤-νΞe-νt-ρt-e-νt∫t-ρtthesdsT∫t-ρtthesds≤-νeνtΞeνρt-1∫t-ρtthesdsT∫t-ρtthesds≤-νeνtΞeνθ2-1∫t-ρtthesdsT∫t-ρtthes.
From Assumption16, it is clear that (56)LVt,et≤eνteTtν-2D+Δ1A˚A˚T+Δ1-1ΛTΛ+Δ2B˚B˚T+Δ3C˚C˚T-2m+Πeνt1-δ+θ2ΞΛTΛet+Δ2-1eνteTt-τtΘTΘet-τt+eνteTtT1TT1et+eTt-τtT2TT2et-τt-Πeνteνθ1eTt-τtet-τt-Δ3-1-νeνtΞeνθ2-1×∫t-ρtthesdsT∫t-ρtthesds.Then we have(57)LVt,et≤eνteTtν-D+Δ1A˚A˚T+Δ1-1ΛTΛ+Δ2B˚B˚T-2m+Δ3C˚C˚T+Π1-δ+T1TT1+θ2ΞΛTΛet+eνteTt-τtΔ2-1ΘTΘ+T2TT2-Πeνθ1×et-τt+Δ3-1-νeνtΞeνθ2-1×∫t-ρtthesdsT∫t-ρtthesds.
Let(58)Π=eνθ1Δ2-1ΘTΘ+T2TT2,Ξ=eνθ2-1νΔ3eνt,m=12ν-2D+Δ1A˚A˚T+Δ1-1ΛTΛ+Δ2B˚B˚T+Δ3C˚C˚T+Π1-δ+T1TT1+θ2ΞΛTΛ+1.Considering the condition of Corollary 14, we get (59)LVt≤-eνteTtet≤0.That is, (60)EVt,et≤EV0,e0,∀t≥0.Combining with (52), we obtain that (61)Eet2=EetTet≤e-νtEVt,et≤EV0,e0e-νt.According to Definition 10, this implies that the exponential antisynchronization of systems (6) and (8) is achieved. Thus, the proof is completed.When there is no distributed time-varying delay in system (45), the error system is introduced as follows: (62)det=-Det+Athet+Btzet-τt+2I+utdt+βt,et,et-τtdωt.Corollary 19.
Assumptions3, 4, and 16 hold, for a given constant ν>0; if there exist matrix Π>0 and constant m>0 such that the conditions hold (63)Π=eνθ1T2TT2,m=12ν-2D+Δ1A˚A˚T+Δ1-1ΛTΛ+Δ2B˚B˚T+Π1-δ+T1TT1+1,system (62) will exponentially converge to zero with the action of the delay-independent controller (48).Proof.
We construct the following Lyapunov function:(64)Vt,et=eνteTtet+1μMt-m2+Π1-δ∫t-τtteνseTsesds.According to the proof of Theorem 18, Corollary 19 is not difficult to obtain. Hence, it is omitted here.Remark 20.
Theorem18 holds only when some special conditions are fulfilled: (1) all parameters meet some certain conditions; (2) the novel delay-independent adaptive controller which is much easier to implement in practice is provided. What makes that all the more remarkable is that there is no upper bound constraint on the exponent convergence speed ν in Theorem 18. Then we can improve the exponent convergence speed ν appropriately within the permissible range to achieve synchronization as soon as possible. So Theorem 18 is less conservative from a certain perspective.


## 4. Numerical Simulation

In this section, several numerical examples are offered to illustrate the effectiveness of the results obtained in the above section.Example 21.
Consider the following2-dimensional MNNs with mixed time-varying delays: (65)dxit=-dixit+∑j=12aijxitfjxjt+∑j=12bijxit-τtgjxjt-τt+∑j=12cijxit∫t-ρttfjxjsds+Iidt,i=1,2,where(66)a11x1t=0.4,x1t>Φ1,0.38,x1t≤Φ1,a12x1t=-1,x1t>Φ1,-1.1,x1t≤Φ1,a21x2t=-0.7,x2t>Φ2,-0.64,x2t≤Φ2,a22x2t=0.75,x2t>Φ2,0.71,x2t≤Φ2,b11x1t-τt=-0.76,x1t-τt>Φ1,-0.82,x1t-τt≤Φ1,b12x1t-τt=0.7,x1t-τt>Φ1,0.68,x1t-τt≤Φ1,b21x2t-τt=0.35,x2t-τt>Φ2,0.29,x2t-τt≤Φ2,b22x2t-τt=0.74,x2t-τt>Φ2,0.88,x2t-τt≤Φ2,c11x1t=0.3,x1t>Φ1,0.25,x1t≤Φ1,c12x1t=0.24,x1t>Φ1,0.29,x1t≤Φ1,c21x2t=-0.21,x2t>Φ2,-0.3,x2t≤Φ2,c22x2t=-0.98,x2t>Φ2,-0.89,x2t≤Φ2.For system (65), we obtain the following response system with stochastic perturbation:(67)dyit=-diyit+∑j=12aijyitfjyjt+∑j=12bijyit-τtgjyjt-τt+∑j=12cijyit∫t-ρttfjyjsds+Ii+uitdt+βit,xit+yit,xit-τt+yit-τtdωit,where the delay-dependent controller ui(t) is designed as follows: (68)uit=-kiteit-miteit-τt-2Ii.And the adaptive law is (69)k˙it=pieit2eνt,ki0=0,m˙it=qieit-τteiteνt,mi0=0,i=1,2.Hered1=2.12, d2=2.17, I1=-0.01, and I2=0.01. The activation functions are taken as fi(s)=gi(s)=tanh⁡s, i=1,2. Time delays (t)=ρ(t)=0.5cos⁡(t)+0.5. It is easy to calculate that θ1=θ2=1, δ=0.5. According to Assumption 3, it can be verified that L1=L2=L3=1 and Q1=Q2=Q3=1. The initial conditions of system (65) are chosen as x1(t)=1.75, x2(t)=0.75, t∈[-1,0]. We take ν=1.53, p1=1.61, p2=1.33, q1=1.23, and q2=1.03. And the stochastic perturbations are given such as (70)β1t,e1t,e1t-τt=0.5e1t+0.89e1t-τt,β2t,e2t,e2t-τt=-0.5e2t-0.56e2t-τt.WhenΦ1=Φ2=1, according to Theorem 11, it can be concluded that systems (65) and (67) with the above parameters achieve exponential antisynchronization in the mean square. Define error system ei(t)=yi(t)+xi(t); Figure 1 depicts the time response curves of state variables of systems (65) and (67) and the curves of synchronization errors e1(t) and e2(t) without or with the delay-dependent adaptive controller (68). It is clear that Figure 1 testifies that the synchronization errors under the effect of the delay-dependent controller (68) tend to zero as time goes on.Figure 1
(a) Time response curves of statesx1(t) and y1(t); (b) time response curves of states x2(t) and y2(t); (c) curves of error systems without controller; (d) curves of error systems with controller (68).
(a)
(b)
(c)
(d)When the time-varying delayτ(t) is not differentiable or the derivative τ˙(t) is unknown or not restriction, we design the following controller: (71)uit=-kit-miteit-τt-2Ii,k˙it=pieiteνt,ki0=0,m˙it=qieit-τteiteνt,mi0=0,i=1,2,where κi=ϖi=1, i=1,2 and all other parameter values are the same as those mentioned from above.According to Corollary14, systems (65) and (67) under the above controller implement exponential antisynchronization. Likewise, define error system ei(t)=yi(t)+xi(t); Figure 2 shows the curves of error systems without or with controller (71). The error system with controller converges significantly to zero which is a stable state.Figure 2
(a) Curves of error systems without controller; (b) curves of error systems with delay-dependent controller (71).
(a)
(b)Example 22.
We consider the following MNNs without stochastic perturbation as drive system; the vector form is given by(72)dxt=-Dxt+Axtfxt+Bxt-τtgxt-τt+Cxt∫t-ρtfxsds+Idt,where(73)Axt=a11x1ta12x1ta21x2ta22x2t,Bxt-τt=b11x1t-τtb12x1t-τtb21x2t-τtb22x2t-τt,Cxt=c11x1tc12x1tc21x2tc22x2t,a11x1t=0.5,x1t>Φ1,0.49,x1t≤Φ1,a12x1t=-0.6,x1t>Φ1,-0.67,x1t≤Φ1,a21x2t=0.3,x2t>Φ2,0.35,x2t≤Φ2,a22x2t=0.4,x2t>Φ2,0.45,x2t≤Φ2,b11x1t-τt=0.31,x1t-τt>Φ1,0.29,x1t-τt≤Φ1,b12x1t-τt=-0.15,x1t-τt>Φ1,-0.18,x1t-τt≤Φ1,b21x2t-τt=0.83,x2t-τt>Φ2,0.85,x2t-τt≤Φ2,b22x2t-τt=0.56,x2t-τt>Φ2,0.58,x2t-τt≤Φ2,c11x1t=-0.4,x1t>Φ1,-0.49,x1t≤Φ1,c12x1t=0.2,x1t>Φ1,0.29,x1t≤Φ1,c21x2t=0.13,x2t>Φ2,0.1,x2t≤Φ2,c22x2t=0.3,x2t>Φ2,0.33,x2t≤Φ2.HereD=diag⁡1.35,1.05T, I1=I2=0. The activation functions are taken as f(s)=g(s)=tanh⁡(s). The time delays are defined as τ(t)=ρ(t)=0.5sin⁡(t)+0.5; then it can be verified that θ1=θ2=1 and δ=0.5. From Assumption 3, we select Λ=Θ=diag⁡(1,1). The initial conditions of system (72) are chosen as x(t)=1.75,-1.75T, t∈-1,0.The corresponding response system under the delay-independent controlleru(t) is given as follows: (74)dyt=-Dyt+Axtfyt+Bxt-τtgyt-τt+Cxt∫t-ρtfysds+I+utdt+βt,et,et-τtdωt.We choose the initial conditions as y(t)=0.75,-0.75T, t∈[-1,0] and the stochastic perturbation is given as follows: (75)β1t,e1t,e1t-τt=0.04e1t+0.24e1t-τt,β2t,e2t,e2t-τt=0.04e2t+0.24e2t-τt,where ω(t)=ω1t,ω2tT is a 2-dimensional Wiener process satisfying Edωt=0. From Assumption 3, it is easy to get that T1=T2=diag⁡0.08,0.48T, Δ1=Δ2=Δ3=diag⁡(1,1). The remaining parameters of system (74) are the same as system (72).The delay-independent controller is defined as follows:(76)uit=-Mitet-2Ii,M˙it=μeνteTtet.In the simulation, we take ν=1.45 and μ=1. We define the error system e(t)=y(t)+x(t). When Φ1=Φ2=1, Figure 3 depicts the time response curves of state variables of system (72) and (74) and the curves of synchronization errors e1(t) and e2(t) without or with the delay-independent adaptive controller u(t). According to Theorem 18, systems (72) and (74) attain exponential antisynchronization in the mean square sense, which is verified in Figure 3.Figure 3
(a) Time response curves of statesx1(t) and y1(t); (b) time response curves of states x2(t) and y2(t); (c) curves of error systems without controller; (d) curves of error systems with delay-independent controller (76).
(a)
(b)
(c)
(d)When we ignore the impact of the distributed delay, namely,ρ(t)≡0, system (72) will be rewritten as (77)dxtdt=-Dxt+Axtfxt+Bxt-τtgxt-τt+I.Then system (74) also becomes (78)dyt=-Dyt+Axthyt+Bxt-τtzyt-τt+I+utdt+βt,et,et-τtdωt.Similarly, all the parameters are the same as those from above. According to Corollary19, systems (77) and (78) under controller (76) achieve the exponential antisynchronization in the mean square sense. Define error system e(t)=y(t)+x(t); Figure 4 depicts the curves of error systems without or with controller (76). It is obvious that Figure 4 proves the error system converges quickly to zero.Figure 4
(a) Curves of error systems without controller; (b) curves of error systems with delay-independent controller (76).
(a)
(b)


## 5. Conclusion

In this paper, we addressed the antisynchronization issues for a class of MNNs with stochastic perturbation and mixed delays, including discrete and distributed time-varying delays. Two novel adaptive controllers were designed on the basis of delay-dependent and delay-independent to ensure that the drive system can achieve the exponential antisynchronization with the response system in spite of their initial conditions. Through the use of the Lyapunov stability method, inequality analysis technique, and the differential inclusion theory, we proposed two types of exponential antisynchronization criteria for these systems. Moreover, the obtained criteria need no excessive numerical calculation. Some numerical examples have been shown to validate the effectiveness of our theoretical results.



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