# Asymptotic Stability and Asymptotic Synchronization of Memristive Regulatory-Type Networks

Authors: Jin-E Zhang
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

Memristive regulatory-type networks are recently emerging as a potential successor to traditional complementary resistive switch models. Qualitative analysis is useful in designing and synthesizing memristive regulatory-type networks. In this paper, we propose several succinct criteria to ensure global asymptotic stability and global asymptotic synchronization for a general class of memristive regulatory-type networks. The experimental simulations also show the performance of theoretical results.


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## Body

## 1. Introduction

Using memristive devices as synapses is a focus in memristive networks. To extract the benefits of high-efficiency memristive memory, various memristive networks have been reported to date [1–18]. Unlike conventional two-terminal devices, memristive networks exhibit pinched memristor hysteresis loop characteristics, making them particularly suitable for linear-drift devices [10]. Moreover, as the modular compact model for memristors, memristive regulatory-type networks are further broadened to memristive systems that exhibit the phenomenon of closed-form sneak paths, which enable nanoscale geometries with short access latencies. A memristive regulatory-type network contains multiply-threshold synapses, which has been heralded as a new paradigm in large-scale circuits. Compared with some memristive systems, a memristive regulatory-type network has the following advantages: (1) it is more biomimetic in behaviors with simple system structure; (2) it simplifies the structure and complication of circuits and is easy to realize. With these coveted properties, memristive regulatory-type networks have the potential of realizations in module-based nanoscale neuromorphic computing systems.The underlying physics mechanism of memristor models is extremely complex. In order to explore the characteristics and applications of memristive networks, several attempts in [1–4, 6–18] have been made, using nonlinear system theory, to develop behavioral models of memristors. An ideal dynamic property is a critical requirement for the development and validation of memristive networks. Evolutional characteristics of memristive networks are an interesting and prosperous research area. However, deploying nonlinear analysis technology in memristive networks is challenging because a memristive network is basically a switched network cluster [13, 15]. Such switched network cluster thus possesses the synaptic action, in which the synaptic weight can be incrementally ameliorated by adjusting the charge or flux through it. There are two major obstacles to analyze and control the memristive networks, namely, high complexity and switched hybridity [12, 13]. On the other hand, dynamical analysis for memristive networks can explain carrier dynamics and associated transients. Once the electronic properties of memristive networks are revealed, then the circuit models can be implemented based upon the underlying dynamic nature. By tweaking physical structures and bias conditions, system designer can optimize the circuit performance, and then, numerous potential applications of the memristors have been exploited, such as neuromorphic, digital, and quantum computation.In spite of having significant progress in the area of nonlinear control systems [19–35], memristive regulatory-type networks constituting switched network cluster have received less attention. It has been reasoned that much like neuroevolutionary systems, memristive regulatory-type networks could be responsible for different neuromorphic architectures [36, 37]. To this end, we focus on the evolution of memristive regulatory-type networks. In this paper, we study global asymptotic stability and global asymptotic synchronization of a class of memristive regulatory-type networks. Based on M-matrix theory, we develop less conservative global asymptotic stability results and global asymptotic synchronization results for memristive regulatory-type networks. Such theoretical analysis can significantly help understand and identify system performance, especially in neuromorphic computing era where stability or synchronization is crucial. In fact, dynamic analysis of memristive regulatory-type networks can provide an overview for optimizing the circuit device and enhancing circuit performances.The rest of this paper is organized as follows. Section2 introduces model description and preliminaries. Section 3 gives main results. Section 4 discusses two numerical examples to demonstrate the effectiveness of theoretical results. Finally, Section 5 concludes the paper with some remarks.


## 2. Model Description and Preliminaries

Consider a general class of memristive regulatory-type networks described by the following delay differential equations: fori=1,2,…,N, j=1,2,…,M,(1)p˙it=-aipit+∑j=1Mbijqjtfjqjt-ρij,q˙jt=-cjqjt+∑i=1Ndjipit-ϱji,where pi(t) and qj(t) represent the concentration variations of memristive messenger gene i and affiliated organic compound j, respectively, ai>0 and cj>0 denote the degradation rates of memristive messenger gene i and affiliated organic compound j, respectively, dji≥0 represents the translating rate, nonlinear function fj(·) is bounded and fj(0)=0, 0≤ρij≤τ and 0≤ϱji≤τ (τ≥0 is a constant) denote the regulating delay and the translating delay, respectively, and bij(qj(t)) represents regulatory relationship of the network, which is defined as(2)bijqjt=b^ij,qjt<0,bˇij,qjt>0,where b^ij and bˇij are constants.The initial conditions of system (1) are assumed to be(3)pis=ϕis,qjs=φjs,s∈-τ,0,i=1,2,…,N,j=1,2,…,M,where ϕi(s) and φj(s) are both continuous functions defined on [-τ,0].In addition, we also assume that the nonlinear functionfj(·)(j=1,2,…,M) satisfies the Lipschitz condition with the Lipschitz constant lj>0; that is,(4)fjχ-fjχ^≤ljχ-χ^,j=1,2,…,M,∀χ,χ^∈R.In this paper, solutions of all the systems considered in the following are in Filippov’s sense.K(P) denotes closure of the convex hull of set P. coΠ~,Π^ denotes closure of the convex hull generated by real numbers Π~ and Π^. Let b¯ij=max⁡b^ij,bˇij, b_ij=min⁡b^ij,bˇij, and b~ij=max⁡b^ij,bˇij, for i=1,2,…,N, j=1,2,…,M.When considering memristive regulatory-type networks (1), throughout this paper, let us define the set-valued maps as follows:(5)Kbijqjt=b^ij,qjt<0,cob^ij,bˇij,qjt=0,bˇij,qjt>0.Obviously, fori=1,2,…,N, j=1,2,…,M,(6)cob^ij,bˇij=b_ij,b¯ij.By the theory of differential inclusions, from (1), then for i=1,2,…,N, j=1,2,…,M,(7)p˙it∈-aipit+∑j=1MKbijqjtfjqjt-ρij,q˙jt=-cjqjt+∑i=1Ndjipit-ϱji.A solutionp(t)=(p1(t),p2(t),…,pN(t))T, q(t)=(q1(t),q2(t),…,qM(t))T in the sense of Filippov of system (1) with initial conditions pi(s)=ϕi(s), qj(s)=φj(s), and s∈[-τ,0], i=1,2,…,N, j=1,2,…,M, is absolutely continuous on any compact interval of [0,+∞), and(8)p˙it∈-aipit+∑j=1MKbijqjtfjqjt-ρij,q˙jt=-cjqjt+∑i=1Ndjipit-ϱji.Definition 1.
The constant vectorsp∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T are called an equilibrium point of system (1), if for i=1,2,…,N, j=1,2,…,M,(9)0∈-aipi∗+∑j=1MKbijqj∗fjqj∗,0=-cjqj∗+∑i=1Ndjipi∗.Definition 2.
The equilibrium point of system (1) is said to be globally asymptotically stable if it is locally stable in sense of Lyapunov and globally attractive.According to Lyapunov direct method, from Definition2, as we know, if there exists an appropriate Lyapunov function V which is positive definite and radially unbounded, such that the time-derivative of V along the trajectory of system (1) is negative definite, then the equilibrium point of system (1) is globally asymptotically stable.Definition 3.
For drive systemX˙(t)=F(t,X), X∈Rn, response system Y˙(t)=G(t,Y,U), Y∈Rn, and U∈Rn, define the synchronization error signal E(t)=X(t)-Y(t), E∈Rn; then the error dynamics can be expressed by the following form:(10)E˙t=Ft,X-Gt,Y,U,and we say that the response system can be globally asymptotically synchronized with the drive system if the zero solution of error system is globally asymptotically stable.


## 3. Main Results

In this section, we will first give two lemmas, which play important role in the analysis and synthesis of memristive regulatory-type networks (1).Lemma 4.
In system (1) at least one equilibrium point exists: p∗=(p1∗,p2∗,…,pN∗)T; q∗=(q1∗,q2∗,…,qM∗)T.Lemma 5.
For system (1), we have(11)KbijXjfjXj-KbijYjfjYj≤b~ijljXj-Yj,∀i=1,2,…,N,j=1,2,…,M,∀Xj,Yj∈R,where K(bij(Xj)) and K(bij(Yj)) are defined as those in (5).Using standard arguments as Lemmas1 and 2 in [15], Lemmas 4 and 5 of this paper can be proved, respectively.


### 3.1. Global Asymptotic Stability

According to Lemma4, memristive regulatory-type networks (1) have the equilibrium points p∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T; we shift the equilibrium points p∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T to the origin by the translation xi(t)=pi(t)-pi∗ and yj(t)=qj(t)-qj∗ in the differential inclusion (7), which results in(12)x˙it∈-aixit+∑j=1MKbijyjtfjyjt-ρij,y˙jt=-cjyjt+∑i=1Ndjixit-ϱji,where(13)Kbijyjtfjyjt-ρij=Kbijyjt+qj∗fjyjt-ρij+qj∗-Kbijqj∗fjqj∗.According to Lemma5,(14)Kbijyjtfjyjt-ρij≤b~ijljyjt-ρij.From (12)–(14), for i=1,2,…,N, j=1,2,…,M,(15)x˙it≤-aixit+∑j=1Mb~ijljyjt-ρij,y˙jt=-cjyjt+∑i=1Ndjixit-ϱji.Theorem 6.
The equilibrium pointsp∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T of system (1) is globally asymptotically stable, if the following matrix (16)W=A-B-DCN+M×N+Mis a nonsingular M-matrix, where A=diag(a1,a2,…,aN), B=(b~ijlj)N×M, C=diag(c1,c2,…,cM), and D=(dji)M×N.Proof.
Since matrixW is a nonsingular M-matrix, by the M-matrix theory, it follows that WT is a nonsingular M-matrix. Based on the fact that WT is a nonsingular M-matrix, then there exists an (N+M)-dimensional vector η>0 such that WTη>0; that is,(17)aiηi-∑j=1MdjiηN+j>0,fori=1,2,…,N,cjηN+j-∑i=1Nb~ijljηi>0,forj=1,2,…,M.
Choose(18)ϑ1=min1≤i≤N⁡aiηi-∑j=1MdjiηN+j,ϑ2=min1≤j≤M⁡cjηN+j-∑i=1Nb~ijljηi,and then we get (19)ϑ=min⁡ϑ1,ϑ2>0.
Consider the following positive definite and radially unbounded Lyapunov function:(20)Vxt,yt=∑i=1Nηixit+∑j=1MηN+jyjt+∑j=1MηN+j∑i=1Ndji∫t-ϱjitxisds+∑i=1Nηi∑j=1Mb~ijlj∫t-ρijtyjsds.
Calculating the upper right Dini derivative ofV(x(t),y(t)) along the trajectory of system (12) yields(21)D+Vxt,yt≤∑i=1Nηi-aixit+∑j=1Mb~ijljyjt-ρij+∑j=1MηN+j-cjyjt+∑i=1Ndjixit-ϱji+∑j=1MηN+j∑i=1Ndjixit-xit-ϱji+∑i=1Nηi∑j=1Mb~ijljyjt-yjt-ρij=-∑i=1Nηiaixit-∑j=1MηN+jcjyjt+∑j=1MηN+j∑i=1Ndjixit+∑i=1Nηi∑j=1Mb~ijljyjt=∑i=1N-ηiai+∑j=1MηN+jdjixit+∑j=1M-ηN+jcj+∑i=1Nηib~ijljyjt≤-ϑ∑i=1Nxit+∑j=1Myjt.
By Lyapunov global asymptotic stability theory, we can conclude system (12) is globally asymptotically stable. Thus, the equilibrium points p∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T of system (1) are globally asymptotically stable. The proof is completed.Next we extend Theorem6 to other possible cases.Corollary 7.
The equilibrium pointsp∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T of system (1) are globally asymptotically stable, if(22)ai>∑j=1Mdji,fori=1,2,…,N,cj>∑i=1Nb~ijlj,forj=1,2,…,M.Proof.
Select the(N+M)-dimensional unit vector as η in the proof of Theorem 6, from (22); it follows that (17) hold. Therefore, the conclusion of Corollary 7 is obvious.Corollary 8.
WhenM=N, the equilibrium points p∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T of system (1) are globally asymptotically stable, if the matrix (23)W=AC-DBis a nonsingular M-matrix, where A=diag(a1,a2,…,aN), B=(b~ijlj)N×M, C=diag(c1,c2,…,cM), and D=(dji)M×N.Proof.
The proof is a direct result of Theorem6.


### 3.2. Global Asymptotic Synchronization

Let (1) be the drive memristive regulatory-type networks. The response memristive regulatory-type networks are described by the following: for i=1,2,…,N, j=1,2,…,M,(24)P˙it=-aiPit+∑j=1MbijQjtfjQjt-ρij+Uit,Q˙jt=-cjQjt+∑i=1NdjiPit-ϱji+Vjt,where Ui(t), Vj(t), i=1,2,…,N, j=1,2,…,M, denote the appropriate control inputs that will be designed in order to obtain a certain control objective.Next, the linear feedback scheme is used to achieve synchronization between drive memristive regulatory-type networks (1) and response memristive regulatory-type networks (24); that is, the controllers Ui(t), Vj(t), i=1,2,…,N, j=1,2,…,M, are designed as follows:(25)Uit=Kipit-Pit,Vjt=Hjqjt-Qjt,where Ki>0, Hj>0 denote the control gains.Let(26)eit=pit-Pit,Ejt=qjt-Qjt,for i=1,2,…,N, j=1,2,…,M. Then by drive memristive regulatory-type networks (1), response memristive regulatory-type networks (24), and the controllers (25), the error system can be described by(27)e˙it=-aieit+∑j=1Mbijqjtfjqjt-ρij-∑j=1MbijQjtfjQjt-ρij-Kieit,E˙jt=-cjEjt+∑i=1Ndjieit-ϱji-HjEjt,for i=1,2,…,N, j=1,2,…,M.To apply the theories of set-valued maps and differential inclusions, (27) is equivalent to(28)e˙it∈-aieit+∑j=1MKbijqjtfjqjt-ρij-∑j=1MKbijQjtfjQjt-ρij-Kieit,E˙jt=-cjEjt+∑i=1Ndjieit-ϱji-HjEjt.According to Lemma5,(29)Kbijqjtfjqjt-ρij-KbijQjtfjQjt-ρij≤b~ijljEjt-ρij.From (28) and (29), for i=1,2,…,N, j=1,2,…,M,(30)e˙it≤-ai+Kieit+∑j=1Mb~ijljEjt-ρij,E˙jt=-cj+HjEjt+∑i=1Ndjieit-ϱji.Theorem 9.
The zero solution of system (28) is globally asymptotically stable; that is, the response system (24) can be globally asymptotically synchronized with the drive system (1), if the following matrix (31)W=A-B-DCN+M×N+Mis a nonsingular M-matrix, where A=diag(a1+K1,a2+K2,…,aN+KN), B=(b~ijlj)N×M, C=diag(c1+H1,c2+H2,…,cM+HM), and D=(dji)M×N.Proof.
Since matrixW is a nonsingular M-matrix, by the M-matrix theory, it follows that WT is a nonsingular M-matrix. Based on the fact that WT is a nonsingular M-matrix, then there exists an (N+M)-dimensional vector η>0 such that WTη>0; that is,(32)ai+Kiηi-∑j=1MdjiηN+j>0,fori=1,2,…,N,cj+HjηN+j-∑i=1Nb~ijljηi>0,forj=1,2,…,M.
Choose(33)ϑ1=min1≤i≤N⁡ai+Kiηi-∑j=1MdjiηN+j,ϑ2=min1≤j≤M⁡cj+HjηN+j-∑i=1Nb~ijljηi,and then we get (34)ϑ=min⁡ϑ1,ϑ2>0.
Consider the following positive definite and radially unbounded Lyapunov function:(35)Vet,Et=∑i=1Nηieit+∑j=1MηN+jEjt+∑j=1MηN+j∑i=1Ndji∫t-ϱjiteisds+∑i=1Nηi∑j=1Mb~ijlj∫t-ρijtEjsds.
Calculating the upper right Dini derivative ofV(e(t),E(t)) along the trajectory of system (28) yields(36)D+Vet,Et≤∑i=1Nηi-ai+Kieit+∑j=1Mb~ijljEjt-ρij+∑j=1MηN+j-cj+HjEjt+∑i=1Ndjieit-ϱji+∑j=1MηN+j∑i=1Ndjieit-eit-ϱji+∑i=1Nηi∑j=1Mb~ijljEjt-Ejt-ρij=-∑i=1Nηiai+Kieit-∑j=1MηN+jcj+HjEjt+∑j=1MηN+j∑i=1Ndjieit+∑i=1Nηi∑j=1Mb~ijljEjt=∑i=1N-ηiai+Ki+∑j=1MηN+jdjieit+∑j=1M-ηN+jcj+Hj+∑i=1Nηib~ijljEjt≤-ϑ∑i=1Neit+∑j=1MEjt.
By Lyapunov global asymptotic stability theory, we can conclude that system (28) is globally asymptotically stable. Thus, the response system (24) can be globally asymptotically synchronized with the drive system (1). The proof is completed.Next we extend Theorem9 to other possible cases.Corollary 10.
The zero solution of system (28) is globally asymptotically stable; that is, the response system (24) can be globally asymptotically synchronized with the drive system (1), if(37)ai+Ki>∑j=1Mdji,fori=1,2,…,N,cj+Hj>∑i=1Nb~ijlj,forj=1,2,…,M.Proof.
Select the(N+M)-dimensional unit vector as η in the proof of Theorem 9, from (37), it follows that (32) hold. Therefore, the conclusion of Corollary 10 is obvious.Corollary 11.
WhenM=N, the zero solution of system (28) is globally asymptotically stable; that is, the response system (24) can be globally asymptotically synchronized with the drive system (1), if the matrix (38)W=AC-DBis a nonsingular M-matrix, where A=diag(a1+K1,a2+K2,…,aN+KN), B=(b~ijlj)N×M, C=diag(c1+H1,c2+H2,…,cM+HM), and D=(dji)M×N.Proof.
The proof is a direct result of Theorem9.Remark 12.
Theorem9 and Corollaries 10 and 11 show the feasibility of linear feedback scheme for designing a perfect control in memristive regulatory-type networks, and the sufficient conditions only depend on some system parameters, which are easy to be checked.Remark 13.
Compared with many other control strategies, linear feedback scheme is more suitable for implementation in memristive regulatory-type networks. For one thing, transient states are quite prevalent in memristive regulatory-type networks; that is, state-dependent jump abruptly spikes up or down with uncertainty. For another thing, linear feedback scheme itself is relatively cheaper and simpler to operate. It is more reasonable and implementable for linear feedback scheme only carried out at finite gain and bandwidth.Remark 14.
The asymptotic synchronization strategy contains more general synchronization behaviors. Through the node cluster, asymptotic synchronization in each group can achieve complete synchronization.


## 3.1. Global Asymptotic Stability

According to Lemma4, memristive regulatory-type networks (1) have the equilibrium points p∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T; we shift the equilibrium points p∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T to the origin by the translation xi(t)=pi(t)-pi∗ and yj(t)=qj(t)-qj∗ in the differential inclusion (7), which results in(12)x˙it∈-aixit+∑j=1MKbijyjtfjyjt-ρij,y˙jt=-cjyjt+∑i=1Ndjixit-ϱji,where(13)Kbijyjtfjyjt-ρij=Kbijyjt+qj∗fjyjt-ρij+qj∗-Kbijqj∗fjqj∗.According to Lemma5,(14)Kbijyjtfjyjt-ρij≤b~ijljyjt-ρij.From (12)–(14), for i=1,2,…,N, j=1,2,…,M,(15)x˙it≤-aixit+∑j=1Mb~ijljyjt-ρij,y˙jt=-cjyjt+∑i=1Ndjixit-ϱji.Theorem 6.
The equilibrium pointsp∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T of system (1) is globally asymptotically stable, if the following matrix (16)W=A-B-DCN+M×N+Mis a nonsingular M-matrix, where A=diag(a1,a2,…,aN), B=(b~ijlj)N×M, C=diag(c1,c2,…,cM), and D=(dji)M×N.Proof.
Since matrixW is a nonsingular M-matrix, by the M-matrix theory, it follows that WT is a nonsingular M-matrix. Based on the fact that WT is a nonsingular M-matrix, then there exists an (N+M)-dimensional vector η>0 such that WTη>0; that is,(17)aiηi-∑j=1MdjiηN+j>0,fori=1,2,…,N,cjηN+j-∑i=1Nb~ijljηi>0,forj=1,2,…,M.
Choose(18)ϑ1=min1≤i≤N⁡aiηi-∑j=1MdjiηN+j,ϑ2=min1≤j≤M⁡cjηN+j-∑i=1Nb~ijljηi,and then we get (19)ϑ=min⁡ϑ1,ϑ2>0.
Consider the following positive definite and radially unbounded Lyapunov function:(20)Vxt,yt=∑i=1Nηixit+∑j=1MηN+jyjt+∑j=1MηN+j∑i=1Ndji∫t-ϱjitxisds+∑i=1Nηi∑j=1Mb~ijlj∫t-ρijtyjsds.
Calculating the upper right Dini derivative ofV(x(t),y(t)) along the trajectory of system (12) yields(21)D+Vxt,yt≤∑i=1Nηi-aixit+∑j=1Mb~ijljyjt-ρij+∑j=1MηN+j-cjyjt+∑i=1Ndjixit-ϱji+∑j=1MηN+j∑i=1Ndjixit-xit-ϱji+∑i=1Nηi∑j=1Mb~ijljyjt-yjt-ρij=-∑i=1Nηiaixit-∑j=1MηN+jcjyjt+∑j=1MηN+j∑i=1Ndjixit+∑i=1Nηi∑j=1Mb~ijljyjt=∑i=1N-ηiai+∑j=1MηN+jdjixit+∑j=1M-ηN+jcj+∑i=1Nηib~ijljyjt≤-ϑ∑i=1Nxit+∑j=1Myjt.
By Lyapunov global asymptotic stability theory, we can conclude system (12) is globally asymptotically stable. Thus, the equilibrium points p∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T of system (1) are globally asymptotically stable. The proof is completed.Next we extend Theorem6 to other possible cases.Corollary 7.
The equilibrium pointsp∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T of system (1) are globally asymptotically stable, if(22)ai>∑j=1Mdji,fori=1,2,…,N,cj>∑i=1Nb~ijlj,forj=1,2,…,M.Proof.
Select the(N+M)-dimensional unit vector as η in the proof of Theorem 6, from (22); it follows that (17) hold. Therefore, the conclusion of Corollary 7 is obvious.Corollary 8.
WhenM=N, the equilibrium points p∗=(p1∗,p2∗,…,pN∗)T and q∗=(q1∗,q2∗,…,qM∗)T of system (1) are globally asymptotically stable, if the matrix (23)W=AC-DBis a nonsingular M-matrix, where A=diag(a1,a2,…,aN), B=(b~ijlj)N×M, C=diag(c1,c2,…,cM), and D=(dji)M×N.Proof.
The proof is a direct result of Theorem6.


## 3.2. Global Asymptotic Synchronization

Let (1) be the drive memristive regulatory-type networks. The response memristive regulatory-type networks are described by the following: for i=1,2,…,N, j=1,2,…,M,(24)P˙it=-aiPit+∑j=1MbijQjtfjQjt-ρij+Uit,Q˙jt=-cjQjt+∑i=1NdjiPit-ϱji+Vjt,where Ui(t), Vj(t), i=1,2,…,N, j=1,2,…,M, denote the appropriate control inputs that will be designed in order to obtain a certain control objective.Next, the linear feedback scheme is used to achieve synchronization between drive memristive regulatory-type networks (1) and response memristive regulatory-type networks (24); that is, the controllers Ui(t), Vj(t), i=1,2,…,N, j=1,2,…,M, are designed as follows:(25)Uit=Kipit-Pit,Vjt=Hjqjt-Qjt,where Ki>0, Hj>0 denote the control gains.Let(26)eit=pit-Pit,Ejt=qjt-Qjt,for i=1,2,…,N, j=1,2,…,M. Then by drive memristive regulatory-type networks (1), response memristive regulatory-type networks (24), and the controllers (25), the error system can be described by(27)e˙it=-aieit+∑j=1Mbijqjtfjqjt-ρij-∑j=1MbijQjtfjQjt-ρij-Kieit,E˙jt=-cjEjt+∑i=1Ndjieit-ϱji-HjEjt,for i=1,2,…,N, j=1,2,…,M.To apply the theories of set-valued maps and differential inclusions, (27) is equivalent to(28)e˙it∈-aieit+∑j=1MKbijqjtfjqjt-ρij-∑j=1MKbijQjtfjQjt-ρij-Kieit,E˙jt=-cjEjt+∑i=1Ndjieit-ϱji-HjEjt.According to Lemma5,(29)Kbijqjtfjqjt-ρij-KbijQjtfjQjt-ρij≤b~ijljEjt-ρij.From (28) and (29), for i=1,2,…,N, j=1,2,…,M,(30)e˙it≤-ai+Kieit+∑j=1Mb~ijljEjt-ρij,E˙jt=-cj+HjEjt+∑i=1Ndjieit-ϱji.Theorem 9.
The zero solution of system (28) is globally asymptotically stable; that is, the response system (24) can be globally asymptotically synchronized with the drive system (1), if the following matrix (31)W=A-B-DCN+M×N+Mis a nonsingular M-matrix, where A=diag(a1+K1,a2+K2,…,aN+KN), B=(b~ijlj)N×M, C=diag(c1+H1,c2+H2,…,cM+HM), and D=(dji)M×N.Proof.
Since matrixW is a nonsingular M-matrix, by the M-matrix theory, it follows that WT is a nonsingular M-matrix. Based on the fact that WT is a nonsingular M-matrix, then there exists an (N+M)-dimensional vector η>0 such that WTη>0; that is,(32)ai+Kiηi-∑j=1MdjiηN+j>0,fori=1,2,…,N,cj+HjηN+j-∑i=1Nb~ijljηi>0,forj=1,2,…,M.
Choose(33)ϑ1=min1≤i≤N⁡ai+Kiηi-∑j=1MdjiηN+j,ϑ2=min1≤j≤M⁡cj+HjηN+j-∑i=1Nb~ijljηi,and then we get (34)ϑ=min⁡ϑ1,ϑ2>0.
Consider the following positive definite and radially unbounded Lyapunov function:(35)Vet,Et=∑i=1Nηieit+∑j=1MηN+jEjt+∑j=1MηN+j∑i=1Ndji∫t-ϱjiteisds+∑i=1Nηi∑j=1Mb~ijlj∫t-ρijtEjsds.
Calculating the upper right Dini derivative ofV(e(t),E(t)) along the trajectory of system (28) yields(36)D+Vet,Et≤∑i=1Nηi-ai+Kieit+∑j=1Mb~ijljEjt-ρij+∑j=1MηN+j-cj+HjEjt+∑i=1Ndjieit-ϱji+∑j=1MηN+j∑i=1Ndjieit-eit-ϱji+∑i=1Nηi∑j=1Mb~ijljEjt-Ejt-ρij=-∑i=1Nηiai+Kieit-∑j=1MηN+jcj+HjEjt+∑j=1MηN+j∑i=1Ndjieit+∑i=1Nηi∑j=1Mb~ijljEjt=∑i=1N-ηiai+Ki+∑j=1MηN+jdjieit+∑j=1M-ηN+jcj+Hj+∑i=1Nηib~ijljEjt≤-ϑ∑i=1Neit+∑j=1MEjt.
By Lyapunov global asymptotic stability theory, we can conclude that system (28) is globally asymptotically stable. Thus, the response system (24) can be globally asymptotically synchronized with the drive system (1). The proof is completed.Next we extend Theorem9 to other possible cases.Corollary 10.
The zero solution of system (28) is globally asymptotically stable; that is, the response system (24) can be globally asymptotically synchronized with the drive system (1), if(37)ai+Ki>∑j=1Mdji,fori=1,2,…,N,cj+Hj>∑i=1Nb~ijlj,forj=1,2,…,M.Proof.
Select the(N+M)-dimensional unit vector as η in the proof of Theorem 9, from (37), it follows that (32) hold. Therefore, the conclusion of Corollary 10 is obvious.Corollary 11.
WhenM=N, the zero solution of system (28) is globally asymptotically stable; that is, the response system (24) can be globally asymptotically synchronized with the drive system (1), if the matrix (38)W=AC-DBis a nonsingular M-matrix, where A=diag(a1+K1,a2+K2,…,aN+KN), B=(b~ijlj)N×M, C=diag(c1+H1,c2+H2,…,cM+HM), and D=(dji)M×N.Proof.
The proof is a direct result of Theorem9.Remark 12.
Theorem9 and Corollaries 10 and 11 show the feasibility of linear feedback scheme for designing a perfect control in memristive regulatory-type networks, and the sufficient conditions only depend on some system parameters, which are easy to be checked.Remark 13.
Compared with many other control strategies, linear feedback scheme is more suitable for implementation in memristive regulatory-type networks. For one thing, transient states are quite prevalent in memristive regulatory-type networks; that is, state-dependent jump abruptly spikes up or down with uncertainty. For another thing, linear feedback scheme itself is relatively cheaper and simpler to operate. It is more reasonable and implementable for linear feedback scheme only carried out at finite gain and bandwidth.Remark 14.
The asymptotic synchronization strategy contains more general synchronization behaviors. Through the node cluster, asymptotic synchronization in each group can achieve complete synchronization.


## 4. Illustrative Examples

In this section, we discuss two numerical examples to illustrate the theoretical results.Example 1.
Consider the following memristive regulatory-type networks:(39)p˙it=-pit+∑j=12bijqjtfjqjt-0.1,q˙jt=-qjt+∑i=130.2pit-0.3,where i=1,2,3, j=1,2, fj(ν)=(ν+1-ν-1)/2,(40)bijqjt=0.3,qjt<0,-0.3,qjt>0,i=1,2,3,j=1,2.
Obviously, we can calculate that(41)A=1000100013×3,-B=-0.3-0.3-0.3-0.3-0.3-0.33×2,C=10012×2,-D=-0.2-0.2-0.2-0.2-0.2-0.22×3,and then (42)W=100-0.3-0.3010-0.3-0.3001-0.3-0.3-0.2-0.2-0.210-0.2-0.2-0.2015×5,and the eigenvalues of matrix W are 0.4, 1, 1, 1, and 1.6; thus the matrix W is a nonsingular M-matrix. According to Theorem 6, system (39) is globally asymptotically stable.
The simulation results of system (39) with some initial values are depicted in Figures 1 and 2. Based on the dynamical evolutions in Figures 1 and 2, we can see that the experimental results agree with the theory very well.Figure 1
Transient behaviors of system (39).Figure 2
Phase portraits of system (39) in the three-dimensional space.Example 2.
Consider the following memristive regulatory-type networks:(43)p˙it=-0.2pit+∑j=12bijqjtfjqjt-0.1,q˙jt=-0.2qjt+∑i=130.2pit-0.3,where i=1,2,3, j=1,2, fj(ν)=ν+1-ν-1/2,(44)bijqjt=0.3,qjt<0,-0.3,qjt>0,i=1,2,3,j=1,2.
Let (43) be the drive system. The response system is described by(45)P˙it=-0.2Pit+∑j=12bijQjtfjQjt-0.1+Uit,Q˙jt=-0.2Qjt+∑i=130.2Pit-0.3+Vjt,where i=1,2,3, j=1,2, fj(ν)=(ν+1-ν-1)/2,(46)bijQjt=0.3,Qjt<0,-0.3,Qjt>0,i=1,2,3,j=1,2.
The controllersUi(t), Vj(t), i=1,2,3, j=1,2, are designed as follows:(47)Uit=0.8pit-Pit,Vjt=0.8qjt-Qjt,and to apply Theorem 9, then we can calculate that (48)A=1000100013×3,-B=-0.3-0.3-0.3-0.3-0.3-0.33×2,C=10012×2,-D=-0.2-0.2-0.2-0.2-0.2-0.22×3,and then(49)W=100-0.3-0.3010-0.3-0.3001-0.3-0.3-0.2-0.2-0.210-0.2-0.2-0.2015×5,and the eigenvalues of matrix W are 0.4, 1, 1, 1, and 1.6; thus the matrix W is a nonsingular M-matrix. According to Theorem 9, the response system (45) can be globally asymptotically synchronized with the drive system (43). The simulation result on the error dynamics e1(t)=p1(t)-P1(t), e2(t)=p2(t)-P2(t), e3(t)=p3(t)-P3(t), e4(t)=q1(t)-Q1(t), and e5(t)=q2(t)-Q2(t), with some initial values, is depicted in Figure 3. The dynamical evolutions in Figure 3 clearly indicate that the controller designed performs well.Figure 3
The error dynamicse1(t)=p1(t)-P1(t), e2(t)=p2(t)-P2(t), e3(t)=p3(t)-P3(t), e4(t)=q1(t)-Q1(t), and e5(t)=q2(t)-Q2(t).


## 5. Conclusion

Memristive network can achieve more expedient goal-finding behavior in spiking networks via memristive connections, which has aroused considerable interest by electronics researchers. The practical applications of memristive network popularizes real-time processing and recognition of natural signals. It is of great significance to investigate its nonlinear dynamics. In this paper, we study global asymptotic stability and global asymptotic synchronization for memristive regulatory-type networks, based on theM-matrix theory and Lyapunov stability theory. These criteria, which can be directly derived from the system parameters, are easily verified. The theoretical results developed in this paper may be applied to the synthesis of memristive regulatory-type networks.



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