Bipartite synchronization for inertia memristor-based neural networks on coopetition networks

Journal Pre-proof Bipartite synchronization for inertia memristor-based neural networks on coopetition networks Ning Li, Wei Xing Zheng

PII: DOI: Reference:

S0893-6080(19)30349-1 https://doi.org/10.1016/j.neunet.2019.11.010 NN 4318

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Neural Networks

Received date : 27 April 2019 Revised date : 10 November 2019 Accepted date : 12 November 2019 Please cite this article as: N. Li and W.X. Zheng, Bipartite synchronization for inertia memristor-based neural networks on coopetition networks. Neural Networks (2019), doi: https://doi.org/10.1016/j.neunet.2019.11.010. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Elsevier Ltd. All rights reserved.

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Neural Networks

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Neural Networks 00 (2019) 1–18

Bipartite synchronization for inertia memristor-based neural networks on coopetition networks✩ Ning Lia,b, Wei Xing Zhengb,∗ a College

of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, Henan, 450046, China of Computing, Engineering and Mathematics, Western Sydney University, Sydney, NSW 2751, Australia

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b School

Abstract

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This paper addresses the bipartite synchronization problem of coupled inertia memristor-based neural networks with both cooperative and competitive interactions. Generally, coopetition interaction networks are modeled by a signed graph, and the corresponding Laplacian matrix is different from the nonnegative graph. The coopetition networks with structural balance can reach a final state with identical magnitude but opposite sign, which is called bipartite synchronization. Additionally, an inertia system is a secondorder differential system. In this paper, firstly, by using suitable variable substitutions, the inertia memristor-based neural networks (IMNNs) are transformed into the first-order differential equations. Secondly, by designing suitable discontinuous controllers, the bipartite synchronization criteria for IMNNs with or without a leader node on coopetition networks are obtained. Finally, two illustrative examples with simulations are provided to validate the effectiveness of the proposed discontinuous control strategies for achieving bipartite synchronization.

1. Introduction

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Keywords: Memristive neural networks; bipartite synchronization; discontinuous control; inertia term.

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Memristor was firstly proposed by Chua in 1971 (Chua, 1971) as the fourth basic circuit element, which revealed the relationship between magnetic flux and electric charge. Because of its good properties of memory and nanoscale, memristor can store and process the information very well (Strukov et al., 2008; Tour & He, 2008). So far, memristor has great potential applications in many fields, such as artificial neural network, secure communications and so on. Different from traditional neural networks, memristor-based neural networks can better emulate the human brain and compute complex data. From the viewpoint of mathematical model, memristor-based neural networks can be seen as a switched discontinuous system (Zhu & Zheng, 2019). Dynamical behaviors of memristor-based neural networks become more complex, and it is easy to produce chaotic behaviors. Up to now, many results concerning the dynamics analysis of memristor-based neural networks have been obtained. For example, in Di Marco et al. (2018), ✩ This work was supported in part by the National Natural Science Foundation of China under Grant 61603125 and 11872175, the Australian Research Council under Grant DP120104986, the Xinhe Huang Tingfang Young Scholars’ Fund of HUEL under Grant hncjzfdxxhhtf201913, the Scientific and Technological Innovative Talents of Henan Province under Grant 20HASTIT024, the Chinese Scholarship Council under Grant 201708410029, the Key Program of Henan Universities under Grant 18A110003, and the NSW Cyber Security Network in Australia under Grant P00025091. ∗ Corresponding author Email addresses: [email protected] (Ning Li), [email protected] (Wei Xing Zheng)

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nonsymmetric neuron interconnections were considered and the global asymptotic stability of the unique equilibrium point was investigated. Synchronization (Kaviarasan et al., 2016; Sakthivel et al., 2018; Zeng et al., 2015b,c, 2017), as a kind of important collective behaviors in coupled networks, has attracted considerable attention. In Yang et al. (2017), the interval matrix method was introduced to deal with the global exponential synchronization of drive-response memristive neural networks, and suitable controllers were designed to overcome the memristor mismatched parameters. By designing a fixed-time controller, the fixed-time synchronization criteria for delayed memristor-based neural networks were attained in Cao & Li (2017). Generally speaking, synchronization can be achieved by either external forcing or coupling relationship through interactions among nodes. It should be noted that there are two kinds of synchronization problems. One problem is called leaderless synchronization (Bao et al., 2016; Li & Cao, 2016; Zhang et al., 2015), that is to say, networks systems can reach synchronization by using its neighbors’ coupling relationship. The other problem is called leader-following synchronization (Liu et al., 2016; Hu et al., 2015), in which a leader node is introduced and all other network nodes can track a prescribed leader node. On the other hand, many systems cannot reach synchronization without an external controller. Various control strategies (state feedback control, sampled-data control, event-trigger control and so on) (Lv et al., 2018) have been proposed to deal with synchronization issues. In He et al. (2015), by designing a distributed impulsive pinning controller, the leader-following synchronization criteria for heterogeneous dynamic networks were derived. Moreover, it was reported that a large coupling strength will destroy synchrony, in contrast to continuous pinning control. The distributed event-triggered controller with periodic sampling scheme was proposed in Lv et al. (2018), by introducing a weighted average state as a virtual leader, and the leaderless synchronization problem was converted to the stability problem of the error system. Most of the existing literature concerning the synchronization problem mainly focus on cooperative networks, where all network nodes achieve synchronization (Wu et al., 2017; Wei et al., 2018; Zhang et al., 2013; Wang et al., 2017; Rakkiyappan et al., 2017; Zhu & Cao, 2012, 2011; Zhu et al., 2019) or consensus through collaboration. However, the cooperative and competitive relationships (Zhai & Li, 2016; Qin et al., 2017) coexist in our daily life. For instance, in biological systems, interactions between genes may be cooperative or antagonistic in the form of activators or inhibitors; in social networks, relationships between people may be friendly or hostile. Hence, it is very important to consider the synchronization on coopetition networks, when there coexist both collaborative and antagonistic (competitive) relationships in a network. And then bipartite synchronization instead of synchronization will be reached. For a signed network with structurally balanced topology, Altafini (2013) first introduced the concept of bipartite consensus/synchroniztion (i.e., all states of nodes in coopetition networks will converge to the same value in modulus, but not in sign). Herein, the bipartite consensus/synchronization means that all agents will be split into two subgroups: agents in the same subgroup will converge to a unique decision while the decision of agents in different subgroup are opposite, which can be seen as the generalization of synchronization and anti-synchronization. Based on the work in Altafini (2013), Hu & Zhu (2015) extended the bipartite consensus results to second-order multi-agents, and under the decentralized adaptive law the bipartite consensus criteria were obtained. In recent years, many nonlinear systems have been studied over coopetition networks. For example, by using the signed graph theory and designing the suitable controller, Wu et al. (2018) investigated the bipartite synchronization problem of reactiondiffusion neural networks with multilayer coopetition networks. To the best of our knowledge, there is no literature that has been found for inertia memristor-based neural networks (IMNNs) on coopetition networks. Therefore, it is very meaningful and challenging to study this type of bipartite synchronization problems of IMNNs with competition relationship. Motivated by the above observations, the objective of this paper is to extend the results reported in Li & Zheng (2018) to the bipartite synchronization of IMNNs on coopetition networks. This is the open and important topic that deserves much more attentions. The main contributions of this paper are threefold: 1) The competitive coupling relationship between network nodes and inertia term are considered for the first time, and a signed graph is utilized to describe the coopetition networks. Hence, the established mathematical model of IMNNs on coopetition networks is more practical and challenging. 2) Through introducing a special matrix M and an isolated virtual leader, the concepts of bipartite synchronization with or without a leader node are defined respectively, and the bipartite synchronization problem is transformed to the stability problem of the synchronization error system.

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3) By adopting suitable variable substitution and designing discontinuous controllers, with the help of structural balance condition, two novel different methods are proposed to deal with the leaderless and leader-following bipartite synchronization issues for IMNNs, respectively, and the obtained bipartite synchronization criteria are easy to verify.

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The remainder of this paper is organized as follows. In Section 2, the signed graph theory and model descriptions are presented. Section 3 gives the main results of this paper. That is, by designing the discontinuous controller, two different bipartite synchronization criteria are derived. Two numerical examples are provided to demonstrate the effectiveness of the obtained results in Section 4. Some conclusions are drawn in Section 5. Notations. Throughout this paper, R, Rn , and Rm×n denote the set of real numbers, the set of n-dimensional vector, and the set of m × n real-valued matrices, respectively. 1TN = (1, 1, · · · , 1)T represents the N-dimensional vector with all entries being one. The notation A > 0 (A < 0) means that matrix A is positive definite (negative definite). For any matrices A ∈ Rm×n and B ∈ Rm×n , A  B (A  B) implies that each element of A and B satisfies the inequality ai j ≥ bi j (ai j ≤ bi j ). In stands for the identity matrix of order n. In addition, sign(·) represents the sign function. The operator ⊗ denotes the Kronecker product. C([t0 − τ, t0 ]; Rn ) represents a class of continuous mapping set from [t0 − τ, t0 ] to Rn . For a number x ∈ R, the notation |x| denotes the absolute value of number x. Matrices, if their dimensions not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

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2. Model Description and Preliminaries

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2.1. Signed Graph Theory For a signed graph G = {V, E, G} (Harary et al., 1953; Zaslavsky, 1981), V = {1, 2, · · · , N} is a set of nodes, E ⊆ V × V is a set of edges, and G = [Gi j ] is an adjacency matrix describing the interaction between the nodes. The edge weight Gi j , 0 if and only if there exists an edge connecting node i and node j. Throughout this paper, we define the Gii = 0 for all i ∈ V, that is to say, the graph G is a simple graph. A neighbor set of node i is defined by Ni = { j | (i, j) ∈ E}. To describe the competitive-cooperative interaction, the positive edge E+ = {(i, j) | Gi j > 0} denotes the cooperative interaction between nodes i and j, and in the same way, the negative edge E− = {(i, j) | Gi j < 0} denotes the competitive interaction between nodes i and j. A path with length l is a sequence of edges with the form (i0 , i1 ), (i1 , i2 ),· · · ,(il−1 , il ) for distinct nodes. A cycle is a path, which starts and ends at the same node. Generally, a cycle can include both positive and negative edges. A cycle is called a positive (negative) cycle if the product of the weights Gi j in the cycle is positive (negative), that is, it contains even (odd) number of negative edges. In addition, a signed graph is said to be connected if there is a path between any pair of distinct nodes. Different from the nonnegative graph, the signed Laplacian matrix (Hou et al., 2003; Kunegis et al., 2010) is defined P P
as L = −diag Nj=1 |G1 j |, · · · , Nj=1 |G N j | + G, that is,  PN  − j=1 |G1 j |  G21  L =  ..  .  −G N1

G P 12 − Nj=1 |G2 j | .. . −G N2

... G1N ... G2N .. .. . . PN . . . − j=1 |G N j |

     .  

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Remark 2.1. In real social systems or networked systems, the cooperation and competition interactions coexist (Hu & Zheng, 2014). Motivated by the competitive model of ecology systems, we can use x j − xi to express the cooperation relationship between nodes i and j, and use x j + xi to describe the competitive relationship between nodes i and j. The coopetition network interactions can be derived as follows: X G+i j (x j − xi ) + G−i j (x j + xi ) j∈Ni

=

X

j∈Ni

=

X

j∈Ni

Gi j x j − |Gi j |xi |Gi j |(sign(Gi j )x j − xi )

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=

N X

4

li j x j ,

j=1

where G+i j denotes the positive edge and G−i j denotes the negative edge. In the existing literature, the cooperation relationship is mainly considered. Obviously, when the set of all negative edges is E− = ∅, the signed graph reduces to a nonnegative graph. Hence, the nonnegative graph can be seen as a special case of the signed graph.

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Now, the definition of structural balance and some important properties are stated below.

Definition 2.1 ((Altafini, 2013)). A signed graph G is said to be structurally balanced if the node set V can be divided into two subgroups denoted by V1 and V2 , such that Gi j ≥ 0, ∀i, j ∈ Vl (l ∈ {1, 2}) and Gi j ≤ 0, ∀i ∈ Vl , j ∈ Vq , l , q (l, q ∈ {1, 2}). Otherwise, graph G is said to be structurally unbalanced. Remark 2.2. When the structural balance condition is satisfied, the coopetition network G has two competitive subgroups with the node subsets V1 and V2 . The interactions between two subgroups are competitive while the interactions within the same subgroup are cooperative. As pointed out in Altafini (2013), structural balance is an important property in the signed graph theory, which ensures the feasibility of bipartite synchronization on coopetition networks.

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Lemma 2.1 ((Altafini, 2013)). Given a connected signed graph G = {V, E, G}, let L be the Laplacian matrix. Then the following equivalent conditions hold: 1) G is structurally balanced; 2) All cycles of G are positive;

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3) There exists S ∈ RN×N such that S LS has all nonnegative entries;

4) All eigenvalues of L have nonnegative real parts and 0 is a simple eigenvalue. Here, the diagonal matrix S = diag(s1 , s2 , · · · , sN ) ∈ RN×N is called the gauge transformation, where the diagonal entry si = 1 for i ∈ V1 and si = −1 for i ∈ V2 . It is obvious that the matrix S satisfies S = S T = S −1 .

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Remark 2.3. Via gauge transformation S , the matrix S LS can be seen as a Laplacian matrix of nonnegative graph. Since the matrix S is an invertible matrix, L and S LS are isospectral. From the nonnegative graph theory, we can get that 0 is the a simple eigenvalue of Laplacian matrix L, and s = (s1 , s2 , · · · , sN )T ∈ RN is the eigenvector corresponding to eigenvalue 0. Therefore, in the present of the structural balance, all states in coopetition networks will reach bipartite synchronization. 2.2. Model Formulation and Basic Lemmas

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Consider the ith node of coupled IMNNs, which can be described by the following equations: dxi (t) d2 xi (t) − Cxi (t) + A(xi (t)) f (xi (t)) =−D 2 dt dt + B(xi (t)) f (xi (t − τ(t))) + J,

(1)

where i = 1, 2, . . . , N, xi (t) = (xi1 (t), · · · , xin (t))T ∈ Rn represents the state of the ith dynamical node at time t, and the second derivative of xi (t) is called the inertial term of the system. f (xi (t)) = ( f1 (xi1 (t)), · · · , fn (xin (t)))T and f (xi (t − τ(t))) = ( f1 (xi1 (t − τ(t))), · · · , fn (xin (t − τ(t))))T are the activation functions of the ith neuron unit at time t and t − τ(t), respectively. J = (J1 ), · · · , Jn )T ∈ Rn denotes the bounded input. The time-varying delay τ(t) satisfies the conditions 0 ≤ τ(t) ≤ τ and τ˙ (t) ≤ µ < 1. D = diag(d1 , d2 , · · · , dn ) and C = diag(c1 , c2 , · · · , cn ) are positive diagonal matrices, and A(xi (t)) = [ak j (xi j (t))]n×n and B(xi (t)) = [bk j (xi j (t))]n×n denote the feedback connection weight

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matrix and the delayed connection memristive weight matrix, respectively. According to the feature of memristor, the memristor-based weights ak j (xi j (t)) and bk j (xi j (t)) satisfy the following conditions: ( a´ k j , |xi j | ≤ T j , ak j (xi j (t)) = a` k j , |xi j | > T j , (2) ( b´ , |xi j | ≤ T j , bk j (xi j (t)) = ` k j bk j , |xi j | > T j , where T j > 0 is the switching jump, and a´ k j , a` k j , b´ k j and b` k j are all constants, k, j = 1, 2, · · · , n. Denote ak j = min{´ak j , a` k j }, ak j = max{´ak j , a` k j }, bk j = min{b´ k j , b` k j }, bk j = max{b´ k j , b` k j }. Let a+k j = max{|ak j |, |ak j |}, b+k j = max{|bk j |, |bk j |}. Obviously, A  A(xi (t))  A, B  B(xi (t))  B¯ with A = (ak j )n×n , B = (bk j )n×n , A = (ak j )n×n , B = (bk j )n×n . The initial values associated with IMNNs (1) are given by xi j (s) = ϕi j (s),

dxi j (s) = ψi j (s), ds

(3)

where ϕi j (s), ψi j (s) ∈ C([−τ, 0]; R), and i = 1, 2, . . . , N ; j = 1, 2, . . . , n. Let xi (t) be the ith node. Then the mathematical model of IMNNs on coopetition networks can be built as follows:

j∈Ni

+c

X

(4)

! dx j (t) dxi (t) , − |Gi j |Γ sign(Gi j ) dt dt

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j∈Ni

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d2 xi (t) dxi (t) − Cxi (t) + A(xi (t)) f (xi (t)) =−D dt dt2 + B(xi (t)) f (xi (t − τ(t))) + J + ui (t) X   |Gi j |Γ sign(Gi j )x j (t) − xi (t) +c

where c is a positive real number denoting the coupling strength, Γ = diag(γ1 , γ2 , . . . , γn ) ∈ Rn×n is the inner coupling matrix, ui (t) is the control input which will be designed, and G = (Gi j )N×N is the symmetric weighted adjacency matrix of the signed graph, representing the competitive-cooperation relationship between neurons. According to the definition of Laplacian matrix L of the signed graph, multiple IMNNs (4) can be rewritten as

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d2 xi (t) dxi (t) =−D − Cxi (t) + A(xi (t)) f (xi (t)) + B(xi (t)) f (xi (t − τ(t))) dt2 dt ! X dx j (t) + x j (t) . li j Γ + J + ui (t) + c dt j∈N

(5)

i

The initial values associated with multiple IMNNs (5) are given in the following forms: xi (s) = ϕi (s),

dxi (s) = ψi (s), ds

(6)

where ϕi (s), ψi (s) ∈ C([−τ, 0]; Rn ), and i = 1, 2, . . . , N.

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Remark 2.4. Different from the nonnegative graph, the elements of the adjacency matrix G = (Gi j )N×N are not P always positive, and the corresponding Laplacian matrix L will not satisfy Nj=1 li j = 0. Hence, the dynamical analysis for coopetition networks is a new and challenging topic. Throughout this paper, we make the following assumptions. (H1 ): For any two different x, y ∈ R, there exist positive scalars κi > 0 (i = 1, 2, · · · , n) and F , such that | fi (x) − s j fi (y)| ≤ κi |x − s j y|, | fi (x)| ≤ F

hold for j = 1, 2, · · · , N, where s j is the jth diagonal entry of the orthogonal transformation matrix S . In the sequel, denote F = diag(κ1 , κ2 , · · · , κn ) for brevity.

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(H2 ): The cooperative-competitive network G is connected and structurally balanced. By adopting the variable substitution ri (t) =

dxi (t) + xi (t), dt

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the IMNNs model (5) can be transformed to  dxi (t)    = −xi (t) + ri (t),    dt        dri (t) = −Θxi (t) − Λri (t) + A(xi (t)) f (xi (t)) + J + ui (t)   dt   N  X      +B(x (t)) f (x (t − τ(t))) + c li j Γr j (t), i i   

(7)

j=1

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where Θ = I + C − D, Λ = D − I. According to the definition of Kronecker product, the above coupled system (7) can be further put in the following compact form:  dx(t)    = −x(t) + r(t),    dt    dr(t) (8)   ˆ ˆ ˆ  = −Θx(t) − Λr(t) + A(x(t)) f˜(x(t))    dt    ˆ + B(x(t)) f˜(x(t − τ(t))) + cL ⊗ Γr(t) + Jˆ + u(t).

ˆ = IN ⊗ Θ, Λ ˆ = IN ⊗ Λ, where Θ

r(t) = (r1T (t), r2T (t), · · · , rNT (t))T , Jˆ = 1TN × J = (J T , J T , · · · , J T )T ,

x(t) = (xT1 (t), xT2 (t), · · · , xTN (t))T ,

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u(t) = (uT1 (t), uT2 (t), · · · , uTN (t))T , f˜(x(t)) = ( f T (x1 (t)), f T (x2 (t)), · · · , f T (xN (t))T , f˜(x(t − τ(t))) = ( f T (x1 (t − τ(t))), f T (x2 (t − τ(t))), · · · , f T (xN (t − τ(t))T , ˆ A(x(t)) = diag(A(x1 (t)), A(x2 (t)), · · · , A(xN (t))), ˆ B(x(t)) = diag(B(x1 (t)), B(x2(t)), · · · , B(xN (t))).

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The objective of this paper is to design a controller ui (t) for the coupled system (7) by using the cooperativecompetitive interaction and information such that all neural nodes reach bipartite synchronization (Hu & Zheng, 2013), that is, lim (si xi (t) − s j x j (t)) = 0, (9) t→∞

for i, j = 1, 2, · · · , N, where si = 1 (s j = 1) for i ∈ V1 ( j ∈ V1 ) and si = −1 (s j = −1) for i ∈ V2 ) ( j ∈ V2 ). To obtain the main results of this paper, the following lemmas are introduced. Lemma 2.2 ((Wu & Chua, 1995)). Let L be an N × N Laplacian matrix of the signed graph and H be the (N − 1) × (N − 1) matrix defined by H = MLV. Then L and H satisfy the following equality:

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ML = HM,

where M and V are given respectively by   s1 −s2  0 s2  0 M =  0  . .. .  . .  0 0

Hi j =

j X k=1

si sk lik − si+1 sk li+1,k ,

0 −s3 s3 .. .

0 0 −s4 .. .

... ... ... .. .

0 0 0 .. .

0 0 0 .. .

0

0

...

sN−1

−sN

       

(N−1)×N

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and s1 0 0 .. .

s1 s2 0 .. .

s1 s2 s3 .. .

... ... s3 .. .

0 0

0 0

0 0

0... 0...

s1 s2 ... ... 0 0

s1 s2 s3 .. . sN−1 0

         

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      V =     

with si being the ith diagonal entry of the gauge transformation matrix S .

N×(N−1)

Remark 2.5. Matrices M and V are very important in analyzing the leaderless synchronization problem of coupled networks. The bipartite synchronization error function can be defined as d(x) = xT MT Mx with M = M ⊗ In . From the definition of matrix M, it is easy to see that d(x) → 0 if and only if ksi xi − s j x j k → 0 for all i and j, i, j = 1, 2, · · · , N. Hence, the leaderless bipartite synchronization problem has been changed to the stability problem of the synchronization error system.

1) (aA) ⊗ B = A ⊗ (aB), where a is a constant; 2) (A + B) ⊗ C = A ⊗ C + B ⊗ C; 3) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD).

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Lemma 2.3 ((Gu et al., 2003)). For any two matrices A ∈ Rm×n ,B ∈ R p×q , the following properties are valid:

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Lemma 2.4 ((Sanchez & Perez, 1999)). Given any real matrices X, Y and Q > 0 with appropriate dimensions, the following matrix inequality holds: X T Y + Y T X ≤ X T QX + Y T Q−1 Y.

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Lemma 2.5 ((Schur Complement (Boyd et al., 1994))). Let X be a symmetric matrix partitioned into blocks: # " X11 X12 < 0, X= T X22 X12 T T . Then X is equivalent to one of the following conditions: , X22 = X22 where X11 = X11 −1 T 1) X11 < 0, X11 − X12 X22 X12 < 0; T −1 2) X22 < 0, X22 − X12 X11 X12 < 0.

3. Main Results

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3.1. Leaderless Bipartite Synchronization Criteria for IMNNs In this subsection, a discontinuous controller is designed for each neuron node i as follows: ui (t) = −K si

N−1 X j=i

sign(s j r j (t) − s j+1 r j+1 (t)), i = 1, 2, · · · , N − 1,

(10)

where the feedback matrix K = diag(k1 , k2 , · · · , kn ) is a positive definite matrix to be determined. Since there are the switching parameters in the memristor system, the discontinuous controller plays an important rule in reaching synchronization, which can restrict the influence of the memristor parameters.

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Theorem 1. Under Assumptions (H1 ) and (H2 ), the IMNNs (7) on coopetition networks can reach the bipartite synchronization if there exist a positive matrix P ∈ Rn×n , and positive diagonal matrices T , S 1 , S 2 and Q = diag{q1 , q2 , · · · , qn } ∈ Rn×n , such that the following linear matrix inequalities (LMIs) hold: Υ11 ∗ ∗ ∗ ∗

P − QΘ Υ22 ∗ ∗ ∗

0 0 FS 2 F − FT F ∗ ∗

QA 0 0 −S 1 ∗

QB 0 0 0 −S 2

    < 0,   

pro of

    Υ =  

ˆ + HT Qˆ < 0, QH where Υ11 = −QΛ − ΛT Q, Υ22 = −P − PT + FS 1 F +

1 1−µ FT F,

(11)

(12)

and H = H ⊗ Γ.

re-

Proof. Before proceeding, for illustration convenience, we denote Pˆ = IN−1 ⊗ P, Qˆ = IN−1 ⊗ Q, Tˆ = IN−1 ⊗ T , ˆ 1 = IN−1 ⊗ Θ, Λ ˆ 1 = IN−1 ⊗ Λ, A = IN ⊗ A, B = IN ⊗ B, A1 = IN−1 ⊗ A, B1 = IN−1 ⊗ B. Θ T Let ξ(t) = Mx(t) = [ξ1T (t), ξ2T (t), · · · , ξN−1 (t)] and v(t) = Mr(t) = [vT1 (t), vT2 (t), · · · , vTN−1 (t)]. By the structure of M, we can get ξi (t) = si xi (t) − si+1 xi+1 (t), vi (t) = si ri (t) − si+1 ri+1 (t), i = 1, 2, · · · , N − 1. Consider the following Lyapunov functional: Z t 1 ˆ ˆ + + rT (t)MT QMr(t) V(t) = xT (t)MT PMx(t) (13) f˜T (x(s))MT Tˆ M f˜(x(s))ds. 1 − µ t−τ(t) Calculating the derivative of V(t) along the trajectory of (8) yields

lP

˙ ≤ 2xT (t)MT PM ˆ x˙(t) + 2rT (t)MT QM˙ ˆ r(t) + V(t)

1 ˜T f (x(t))MT Tˆ M f˜(x(t)) 1−µ

− f˜T (x(t − τ(t)))MT Tˆ M f˜(x(t − τ(t))) ˆ ˆ ˆ ˆ Θx(t) − Λr(t) + r(t)] + 2rT (t)MT QM[− ≤ 2xT (t)MT PM[−x(t)

urn a

ˆ ˆ +A(x(t)) f˜(x(t)) + B(x(t)) f˜(x(t − τ(t))) + cL ⊗ Γr(t) + Jˆ + u(t)] 1 ˜T f (x(t))MT Tˆ M f˜(x(t)) − f˜T (x(t − τ(t)))MT Tˆ M f˜(x(t − τ(t))). + 1−µ

Obviously,

ˆ 1 Mx(t), ˆ ˆ Θx(t) = rT (t)MT Qˆ Θ rT (t)MT QM T T T T ˆ ˆ 1 Mr(t). ˆ = r (t)M Qˆ Λ r (t)M QMΛr(t)

(14)

With the help of Lemma 2.2 and Lemma 2.3, we know that

Jo

ˆ ˆ 2rT (t)MT QM(L ⊗ Γ)r(t) = 2rT (t)MT Q(M ⊗ In )(L ⊗ Γ)r(t) T T ˆ = 2r (t)M Q(ML ⊗ Γ)r(t)

ˆ = 2rT (t)MT Q(HM ⊗ Γ)r(t) T T ˆ = 2r (t)M Q(H ⊗ Γ)(M ⊗ In )r(t)

ˆ = 2rT (t)MT (QH)Mr(t).

Then we arrive at ˆ 1 Mr(t) ˆ 1 Mx(t) − rT (t)MT Qˆ Λ ˆ ˙ ≤ 2xT (t)MT PM[−x(t) + r(t)] − 2rT (t)MT Qˆ Θ V(t)

(15)

Journal Pre-proof

Ning Li and Wei Xing Zheng / Neural Networks 00 (2019) 1–18

9

ˆ B(x(t)) ˆ ˆ A(x(t)) ˆ f˜(x(t − τ(t))) f˜(x(t)) + 2rT (t)MT QM +2rT (t)MT QM 1 ˜T ˆ ˆ f (x(t))MT Tˆ M f˜(x(t)) +2crT (t)MT (QH)Mr(t) + 2rT (t)MT QMu(t) + 1−µ − f˜T (x(t − τ(t)))MT Tˆ M f˜(x(t − τ(t))). Since

pro of

ˆ ˆ − A) f˜(x(t)), MA(x(t)) f˜(x(t)) = MA f˜(x(t)) + M(A(x(t)) ˆ ˆ M B(x(t)) f˜(x(t − τ(t))) = MB f˜(x(t − τ(t))) + M( B(x(t)) − B) f˜(x(t − τ(t))), under Assumption (H1 ), we obtain ˆ ˆ A(x(t)) − A) f˜(x(t))] 2rT (t)MT QM[( i=1

=2

vTi (t)Q[si (A(xi (t)) − A) f (xi (t)) − si+1 (A(xi+1 (t)) − A) f (xi+1 (t))]

N−1 X n X

vi j q j

i=1 j=1

≤2

N−1 X n X i=1 j=1

n X s=1

[(a js(xis ) − a js )si f s (xis (t)) − (a js(xi+1,s ) − a js )si+1 f s (xi+1,s (t))]

re-

=2

N−1 X

(16)

q j α j |vi j (t)|.

Similarly, we can also get

(17)

=2

N−1 X i=1

=2

vTi (t)Q[si (B(xi(t)) − B) f (xi (t − τ(t))) − si+1 (B(xi+1 (t)) − B) f (xi+1 (t − τ(t)))]

N−1 X n X i=1 j=1

and

N−1 X n X

vi j q j

n X s=1

[(b js(xis ) − b js )si f s (xis (t − τ(t))) − (b js(xi+1,s ) − b js )si+1 f s (xi+1,s (t − τ(t)))]

q j γ j |vi j (t)|,

urn a

≤2

lP

ˆ B(x(t)) ˆ 2rT (t)MT QM( − B) f˜(x(t − τ(t)))

i=1 j=1

ˆ Jˆ = 2 2rT (t)MT QM

≤2

N−1 X i=1

(si ri − si+1 ri+1 )T Q(si J − si+1 J)

N−1 X n X i=1 j=1

q j % j |vi j (t)|,

Jo

P P where α j = 2 ns=1 |a js − a js |F , γ j = 2 ns=1 |b js − b js |F , and % j = 2J j . From Lemma 2.4 and Assumption (H1 ), it follows that ˆ 1 M f˜(x(t)) ˆ f˜(x(t)) = 2rT (t)MT QA 2rT (t)MT QMA

=2 ≤

N−1 X i=1

N−1 X i=1

(18)

(si ri − si+1 ri+1 )T QA(si f (xi (t)) − si+1 f (xi+1 (t)))

[(si ri − si+1 ri+1 )T QAS 1−1 AT Q(si ri − si+1 ri+1 )

(19)

Journal Pre-proof

Ning Li and Wei Xing Zheng / Neural Networks 00 (2019) 1–18

10

+(si xi − si+1 xi+1 )T FS 1 F(si xi − si+1 xi+1 )] ≤

N−1 X

[vTi (t)QAS 1−1 AT Qvi (t) + ξiT (t)FS 1 Fξi (t)],

(20)

i=1

and N−1 X i=1

[vTi (t)QBS 2−1 BT Qvi (t) + ξiT (t − τ(t))FS 2 Fξi (t − τ(t))].

Next, it is easy to verify that ˆ 2rT (t)MT QMu(t) = −2 = −2

pro of

ˆ f˜(x(t − τ(t))) ≤ 2rT (t)MT QMB

N−1 X n X

(21)

vi j (t)q j k j sign(vi j (t))

i=1 j=1

N−1 X n X i=1 j=1

k j q j |vi j (t)|.

(22)

˙ yields By selecting k j > α j + γ j + % j , substituting (16)–(22) into the derivative V(t) N−1 X {ξiT (t)[−P − PT + FS 1 F +

1 F T T F]ξi (t) 1−µ

re-

˙ ≤ V(t)

i=1

T

+2ξiT (t)[P − QΘ]vi (t) + vTi (t)[−QΛ − ΛT Q + QAS 1−1 A Q T

(23)

lP

+QBS 2−1 B Q]vi (t) + ξiT (t − τ(t))[FS 2 F − FT F]ξi (t − τ(t))} ˆ +2crT (t)MT QHMr(t). Using Schur Complement in Lemma 2.5, we can get ˙ ≤ V(t)

N−1 X

ˆ ζiT (t)Υζi (t) + 2crT (t)MT QHMr(t),

(24)

i=1

urn a

where ζi (t) = (vTi (t), ξiT (t), ξiT (t − τ(t)))T . ˙ From the conditions in Theorem 1, it follows that V(t) < 0, which implies that ξi (t) → 0. That is to say, limt→∞ (si xi (t) − si+1 xi+1 (t)) = 0. According to the definition of bipartite synchronization in (9), we can draw the conclusion that the IMNNs (7) on coopetition networks will reach the bipartite synchronization under the conditions (11) and (12). The proof of Theorem 1 is now complete.

Jo

Remark 3.1. The leaderless synchronization criteria for the IMNNs (7) have been derived in the above Theorem 1, which may be easily verified by using the Matlab LMI toolbox. In the leaderless case, under the suitably designed controller, the networks can reach synchronization or consensus by utilizing the network nodes’ coupling information. However, in practical applications, there may exist a leader in community, which can be seen as a tracking problem. Hence, leader-following synchronization is also a common and important phenomenon in nature. The following subsection will deal with the leader-following bipartite synchronization issue. 3.2. Leader-Following Bipartite Synchronization Criteria for IMNNs In this subsection, we will investigate the bipartite synchronization problem for the coupled IMNNs (7) with a virtual node, which can be described by the following equation: d2 x0 (t) dx0 (t) − Cx0 (t) + A(x0 (t)) f (x0 (t)) =− D dt dt2 + B(x0(t)) f (x0 (t − τ(t))) + J.

(25)

Journal Pre-proof

Ning Li and Wei Xing Zheng / Neural Networks 00 (2019) 1–18

11

Here, x0 (t) = (x01 (t), x02 (t), · · · , x0n (t))T ∈ Rn may be an equilibrium point, a periodic orbit, or even a chaotic attractor, which is the tracking state of the coupled IMNNs (7). The other parameters in (25) are similar to those with (1). The bipartite synchronization tracking problem is mainly devoted to designing the control input ui (t), such that all coupled IMNNs states xi (t) in two antagonistic subnetworks can reach synchronization regarding the leader state x0 (t), that is, lim (xi (t) − si x0 (t)) = 0, t→∞

ui = −cσi Γ(

pro of

where si = 1 if i ∈ V1 and si = −1 if i ∈ V2 . Equivalently, limt→∞ (xi (t) − x0 (t)) = 0 for i ∈ V1 and limt→∞ (xi (t) + x0 (t)) = 0 for i ∈ V2 . Let the bipartite synchronization error be ei (t) = xi (t) − si x0 (t). We design the following controller: de j (t) de j (t) + e j (t)) − βsign( + e j (t)), dt dt

where σi > 0, i = 1, 2, · · · , N and β = diag(β1 , β2 , · · · , βn ) > 0. Then, from (5) and (25), we get the following error dynamical system:

re-

d2 ei (t) dei (t) − Cei (t) + A(xi (t)) f (xi (t)) − si A(x0 (t)) f (x0 (t)) =−D 2 dt dt + B(xi (t)) f (xi (t − τ(t))) − si B(x0 (t)) f (x0 (t − τ(t))) ! X de j (t) de j (t) li j Γ +c + e j (t) − cσi Γ( + e j (t)) dt dt j∈N i

− βi sign(

(26)

de j (t) + e j (t)). dt

lP

By introducing the variable substitution

ωi (t) =

dei (t) + ei (t), dt

urn a

the IMNNs model (26) can be transformed to  de (t) i    = −ei (t) + ωi (t),    dt      dωi (t)    = −Θei (t) − Λωi (t) + A(xi (t)) f (xi (t)) − si A(x0 (t)) f (x0 (t))   dt  N  X      +B(x (t)) f (x (t − τ(t))) − s B(x (t)) f (x (t − τ(t))) + c li j Γω j (t) i i i 0 0      j=1     −cσi Γω j (t) − βsign(ω j (t)),

(27)

where Θ = I + C − D and Λ = D − I. Furthermore, the above error system (27) can be rewritten in a compact form as

Jo

 de(t)    = −e(t) + ω(t),    dt       dω(t) = −Θe(t) ˆ ˆ ˆ − Λω(t) + A(x(t)) f˜(x(t)) − S ⊗ A(x0 (t)) f˜(x0 (t))   dt     ˆ  + B(x(t)) f˜(x(t − τ(t))) − S ⊗ B(x0 (t)) f˜(x0 (t − τ(t))) + c(L ⊗ Γ)ω(t)      −c(Σ ⊗ Γ)ω(t) − (IN ⊗ β)sign(ω(t)),

where

e(t) = (eT1 (t), eT2 (t), · · · , eTN (t))T ,

ω(t) = (ωT1 (t), ωT2 (t), · · · , ωTN (t))T ,

(28)

Journal Pre-proof

Ning Li and Wei Xing Zheng / Neural Networks 00 (2019) 1–18

12

f˜(x0 (t)) = ( f T (x0 (t)), f T (x0 (t)), · · · , f T (x0 (t))T , Σ = diag(σ1 , σ2 , · · · , σN ), sign(ω(t)) = (sign(ω1 (t)), sign(ω2 (t)), · · · , sign(ωN (t))), and the other notations are the same as those defined in (8).

pro of

Theorem 2. Under Assumptions (H1 ) and (H2 ), the IMNNs (7) on coopetition networks will reach the bipartite synchronization if there exist positive definite matrices P, Q ∈ Rn×n , and a positive diagonal matrix Z = diag(z1 , z2 , · · · , zn ) ∈ Rn×n , such that the following LMI holds: " # −P + Q P − ZΘ Υ1 = < 0. (29) ∗ −ZΛ Proof. Consider the following Lyapunov-Krasovskii functional: Z t 1 1 ˆ ˆ ˆ eT (s)Qe(s) + ωT (t)Zω(t), + V(t) = eT (t)Pe(t) 2 2 t−τ(t) where Pˆ = IN ⊗ P, Qˆ = IN ⊗ Q, and Zˆ = IN ⊗ Z. Calculating the derivative of V(t) along the trajectories of system (28) yields

re-

˙ = eT (t)P˙ ˆ e(t) + eT (t)Qe(t) ˆ ˆ − τ(t)) + ωT (t)Zˆ ω(t) V(t) − (1 − µ)eT (t − τ(t))Qe(t ˙ T T T ˆ ˆ ˆ − τ(t)) ≤ e (t)P[−e(t) + ω(t)] + e (t)Qe(t) − (1 − µ)e (t − τ(t))Qe(t

ˆ ˆ ˆ Θe(t) ˆ +ωT (t)Z[− − Λω(t) + A(x(t)) f˜(x(t)) − S ⊗ A(x0 (t)) f˜(x0 (t)) ˆ + B(x(t)) f˜(x(t − τ(t))) − S ⊗ B(x0 (t)) f˜(x0 (t − τ(t))) + c(L − Σ) ⊗ Γω(t)

Under Assumption (H1 ), we obtain

lP

−(IN ⊗ β)sign(ω(t))].

(30)

ˆ A(x(t)) ˆ ωT (t)Z[ f˜(x(t)) − S ⊗ A(x0 (t)) f˜(x0 (t))] =

N X i=1

ˆ i (t)) f (xi (t)) − si A(x0 (t)) f (x0 (t))] ωTi (t)Z[A(x n X

urn a

N X n X

=

ωi j (t)z j

s=1

i=1 j=1

=

N X n X i=1 j=1

P where π j = ns=1 2a+js F . Similarly, we have

[a js(xis (t) f (xis (t)) − si a js (x0s ) f (x0s (t)))]

(31)

z j π j |ωi j (t)|,

ˆ B(x(t)) ˆ ωT (t)Z[ f˜(x(t − τ(t))) − S ⊗ B(x0(t)) f˜(x0 (t − τ(t)))] N X

Jo =

i=1

≤

≤

ωTi (t)Z[B(xi (t)) f (xi (t − τ(t))) − si B(x0 (t)) f (x0 (t − τ(t)))]

N X n X i=1 j=1

N X n X i=1 j=1

ωi j (t)z j

n X s=1

[b js(xis (t) f (xis (t − τ(t))) − si b js (x0s ) f (x0s (t − τ(t))))]

z j ρ j |ωi j (t)|,

(32)

Journal Pre-proof

Ning Li and Wei Xing Zheng / Neural Networks 00 (2019) 1–18

13

P where ρ j = ns=1 2b+jsF . Next, it is easy to verify that

≤−

N X

ωTi (t)Zβsign(ωi (t))

i=1

N X n X i=1 j=1

(33) z j β j |ωi j (t)|.

pro of

ˆ N ⊗ β)sign(ω(t)) = − −ωT (t)Z(I

By choosing β j > π j + ρ j and substituting (31)–(33) into (30), we arrive at

ˆ ˆ ˙ ≤ eT (t)P[−e(t) ˆ ˆ ˆ Θe(t) V(t) + ω(t)] + eT (t)Qe(t) + ωT (t)Z[− − Λω(t)] +c(L − Σ) ⊗ Γω(t) ≤

N X i=1

eTi (t)[(−P + Q)ei (t) + eTi (t)(P − ZΘ)ωi (t) + ωTi (t)(−ZΛ)ωi (t)]

+c(L − Σ) ⊗ Γω(t).

(34)

re-

N ˙ ≤ Pi=1 According to Lemma 4 in Hu & Zhu (2015), the matrix L − Σ is negative definite. It is easy to get V(t) ζiT Υ1 ζi , T T T ˙ where ζi = (ei (t), ωi (t)) . From the conditions in Theorem 2, it follows that V(t) < 0. Hence, we can draw the conclusion that the bipartite synchronization error states ei (t) in (28) converge to zero. This completes the proof of Theorem 2.

lP

Remark 3.2. In the above Theorems 1–2, we have analyzed the bipartite synchronization problem for IMNNs on coopetition networks. Actually, the proposed results not only help the analysis of dynamics of engineering systems with signed graphs, but also benefit the study of social networks, opinion dynamics, economics, etc. For example, two adversary subcommunities in social community will reach diametric opinions. 4. Illustrative Examples

urn a

In this section, two numerical examples are given to illustrate the formation of the bipartite synchronization on coopetition network with 20 nodes. The connected coopetition network G is portrayed in Fig. 1. The blue solid edges and red dash edges represent the cooperative relationship and the competitive relationship, respectively. It is obvious the all the cycles in the coopetition network G are positive, so the coopetition network is structurally balanced. Network nodes 7 ∼ 14 and nodes 1 ∼ 6, 15 ∼ 20 constitute two antagonistic subgroups, respectively. Then si = 1, i = 1, 2, · · · , 6, 15, 16, · · · , 20, and si = −1, i = 7, 8, · · · , 14. By choosing the weighted edges and observing Fig. 1, it is not difficult to obtain the Laplacian matrix L of the signed graph G.

濆

濄

濄

Jo

濅

濄

濄

濄濃

濄

濄濊

濄濄

濄

濈

濄

濄濋

濄

濄

濄

激濅濁濈

濋

濄濉

濄濅

濄濌

激濅濁濈

濄

濄

濌

濇

濄 濉

濊

濄

濄濇

濄濆

濄

濄

濄

濄

濄

濄濈

激濅濁濈

濄

Figure 1. Coopetition network G.

濅濃

Journal Pre-proof

0.7

0.7

0.6

0.6

0.5

0.5

b 11 (x 11 (t))

a 11 (x 11 (t))

Ning Li and Wei Xing Zheng / Neural Networks 00 (2019) 1–18

0.4

0.3

0.3

0.2

0.1

0.1

0

0

0

2

4

6

8

10

12

14

16

18

0

t

2

4

6

8

10

12

14

16

18

t

Figure 2. Trajectory of connection weight coefficient a11 (x11 ) in Example 1.

Figure 3. Trajectory of connection weight coefficient b11 (x11 ) in Example 1.

3

3

2

2

x i2(t),i=1,2,3,...20

1

1

re-

x i1(t),i=1,2,3,...20

0.4

pro of

0.2

14

0

-1

-2

0

-1

-2

-3

0

2

4

6

8

10

12

t

14

lP

-3 16

18

Figure 4. Trajectories of states xi1 (t), i = 1, 2, 3, · · · , 20 in Example 1.

0

2

4

6

8

10

12

14

16

18

t

Figure 5. Trajectories of states xi2 (t), i = 1, 2, 3, · · · , 20 in Example 1.

urn a

Example 1. Consider the following second-order IMNNs on coopetition networks: d2 xi (t) dxi (t) − Cxi (t) + A(xi (t)) f (xi (t)) =−D dt dt2 + B(xi (t)) f (xi (t − τ(t))) + J + ui (t) ! X dx j (t) li j Γ +c + x j (t) . dt j∈N

(35)

i

i (t)−1| where the activation function f (xi (t)) = |xi (t)+1|−|x , the time delay τ(t) = ete+1 , the coupling strength c = 1, and the 2 T control input J = [1, 1] . Obviously, we can choose τ = 1 and µ = 0.5. Assumption (H)1 is satisfied with F = I2 and F = 1.The network parameters and configuration matrices are set as " # " # " # 2 0 2.5 0 1 0 D= , C= , Γ= , 0 2 0 2.5 0 1 " # " # a11 (xi1 ) 1.5 b11 (xi1 ) −0.5 A(xi (t)) = , B(xi(t)) = 0.3 0.7 0.5 0.5

Jo

with

t

a11 (xi1 ) =

(

0.4, |xi1 | ≤ 0.05, 0.5, |xi1 | > 0.05,

b11 (xi1 ) =

(

0.3, |xi1 | ≤ 0.05, 0.5, |xi1 | > 0.05.

Journal Pre-proof

Ning Li and Wei Xing Zheng / Neural Networks 00 (2019) 1–18

3

3

2

2

x 02 (t)

x 01 (t) 1

x i2(t),i=0,1,2,3,...20

0

-1

1

0

-1

pro of

x i1(t),i=0,1,2,3,...20

15

-2

-2

-3

-3

0

5

10

15

20

25

0

t

5

10

15

20

25

t

Figure 6. Trajectories of states xi1 (t), i = 0, 1, 2, · · · , 20 in Example 2.

Figure 7. Trajectories of states xi2 (t), i = 0, 1, 2, · · · , 20 in Example 2.

re-

According to the above parameters, solving the LMIs (11) and (12) in Theorem 1 by the Matlab LMI toolbox, the feasible positive definite matrices P, Q, T, S 1 , S 2 , are found to be " # " # " # 2.2910 −0.0789 1.2405 0 2.0271 0 P= , Q= , T= , −0.0789 3.3287 0 1.8132 0 2.2118 " # " # 1.2351 0 1.3882 0 S1 = , S2 = . 0 2.8875 0 1.3860

lP

For simulation, the feedback matrix is selected as K = diag(3, 3). The initial values are randomly chosen in the set [0, 1] with step h = 0.01. Fig. 2 and Fig. 3 depict the switched trajectories of the memristor parameters a11 (x11 ) and b11 (x11 ), respectively. Moreover, Fig. 4 and Fig. 5 plot the synchronization error trajectories of xi1 (t) and xi2 (t), respectively, which indicates that all states in different subgroups are the same in modulus, but not in sign. Hence, the coupled IMNNs reach bipartite synchronization as expected. Example 2. For the second-order IMNNs in (35), the virtual leader node can be given as

where

urn a

d2 x0 (t) dx0 (t) =− D − Cx0 (t) + A(x0 (t)) f (x0 (t)) dt2 dt + B(x0(t)) f (x0 (t − τ(t))) + J,

a11 (x01 ) =

(

0.4, |x01 | ≤ 0.05, 0.5, |x01 | > 0.05,

b11 (x01 ) =

(

(36)

0.3, |x01 | ≤ 0.05, 0.5, |x01 | > 0.05.

Jo

Assume that the other parameters are the same as those in Example 1. By computing the LMIs in (29), the suitable positive definite matrices P, Q, Z, can be obtained below: " # " # " # 81.8338 0 40.9169 0 55.9916 0 P= , Q= , Z= . 0 81.8338 0 40.9169 0 55.9916 In simulation, the control gain matrices are given by Σ = I5 and β = diag(3, 3). The initial values are chosen in the interval [0, 1]. The evolutions of the states xi (t), i = 1, 2, · · · , 20 are displayed in Fig. 6 and Fig. 7, which shows that the twenty nodes belong to two antagonistic subgroups. Specifically, nodes 1 ∼ 6, 15 ∼ 20 can follow node 0, while nodes 7 ∼ 14 can reach a reverse state −x0 (t). Furthermore, the switched trajectories of the connection weights a11 (x11 ) and b11 (x11 ) are illustrated in Fig. 8 and Fig. 9, respectively.

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0.7

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Figure 8. Trajectories of connection weight coefficient a11 (x11 ) in Example 2.

Figure 9. Trajectories of connection weight coefficient b11 (x11 ) in Example 2.

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Remark 4.1. In the IMNNs (35), if we remove the inertial term and the delay term, let matrices D = I and B = 0, and choose C and A as those in the example in Liu et al. (2018), it is easy to verify that the parameters in Lur’e neural networks model satisfy the proposed bipartite synchronization criteria in Theorem 1 and Theorem 2. Similarly, the proposed bipartite synchronization criteria of the IMNNs (35) can be applied to the multi-agent system in Hu & Zhu (2015). Hence, the proposed bipartite synchronization criteria have wider generality than the existing literature.

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Remark 4.2. Different from Li & Zheng (2018), the competitive relationship between network nodes has been considered in this paper, which is described by the negative edge in the signed graph. because of the existence of the negative edges, the traditional synchronization methods cannot be applied directly to achieve the bipartite synchronization. Due to the parameter switching and antagonistic interactions, the design of the desired controller is a more challenging task. By reconstructing the matrices M and V in Lemma 2.2 and redesigning the controller, the bipartite synchronization criteria are derived for IMNNs and are also easy to verify. This provides a new research method for bipartite synchronization. 5. Conclusion

References

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In this paper, we have studied the bipartite synchronization problem for IMNNs on coopetition networks, where the coopetition networks are described by signed graphs. By applying the signed graph theory and dynamical system theory and proposing the special M-matrix, the leaderless bipartite synchronization criteria for IMNNs have been obtained. Simultaneously, a virtual leader node has been introduced, which can be viewed as a reference signal. We have designed a discontinuous controller for each network node so as to drive all network nodes to belong to two competitive subcommunities and thus to reach bipartite synchronization on a reference signal. Finally, two illustrative examples have been presented to demonstrate the formation of bipartite synchronization. In real cases, exogenous disturbances and time delays are unavoidable. Thus, in the future we will consider the synchronization control problem of IMNNs in the presence of noise disturbances and time delays under the condition that time-varying delays are not required to be differentiable, so as to obtain less conservative synchronization criteria (Zeng et al., 2015a; Seuret & Gouaisbaut, 2017; Zeng et al., 2019; Long et al., 2019). Another interesting yet challenging topic for future work is to investigate event-triggered based synchronization control for delayed IMNNs with exogenous disturbances.

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Locked Bag 1797 Sydney, NSW 2751, Australia

27 April 2019

Professor DeLiang Wang Co-Editor-In-Chief, NEURAL NETWORKS Department of Computer Science and Engineering Ohio State University Columbus, Ohio 43210-1277 USA

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Dear Professor Wang,

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Professor Wei Xing Zheng, PhD, IEEE Fellow School of Computing, Engineering and Mathematics Tel: +61-2-4736 0608 Fax: +61-2-4736 0374 Email: [email protected]

I would appreciate it very much if you could consider our paper entitled

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Synchronization criteria for inertial memristor-based neural networks with linear coupling possible publication in Neural Networks. We do not have any conflict of interest. I also confirm that this manuscript is the authors' original work and has not been submitted simultaneously elsewhere.

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Yours sincerely,

Jo

Professor Wei Xing Zheng, IEEE Fellow School of Computing, Engineering and Mathematics Western Sydney University Sydney, NSW 2751 Australia