Synchronization of Switched Complex Bipartite Neural Networks with Infinite Distributed Delays and Derivative Coupling

Abstract

A new model of switched complex bipartite neural network (SCBNN) with infinite distributed delays and derivative coupling is established. Using linear matrix inequality (LMI) approach, some synchronization criteria are proposed to ensure the synchronization between two SCBNNs by constructing effective controllers. Some numerical simulations are provided to illustrate the effectiveness of the theoretical results obtained in this paper.

1. Introduction

In recent years, neural networks have been intensively studied due to their potential applications in many different areas such as signal and image processing, content-addressable memory, optimization, and parallel computation [1–3]. Bidirectional associative memory (BAM) neural networks were first proposed by Kosko in [4, 5]. This class of networks has good applications in pattern recognition, solving optimization problems, and automatic control engineering. A large number of results on the dynamical behavior of BAM neural networks have been reported [6–9].

Switched systems, as an important kind of hybrid systems, have drawn considerable attention of researchers because of their theoretical significance and practical applications [10–12]. Switched systems are composed of a family of continuous-time or discrete-time subsystems and a rule that specifies the switching among them [13, 14]. Recently, the switched neural networks, whose individual subsystems are a set of neural networks, have found applications in the field of high speed signal processing, artificial intelligence, and biology, so there are many theoretical results about the switched neural networks [15–17].

Complex networks, which are a set of interconnected nodes with specific dynamics, have sparked the interest of many researchers from various fields of science and engineering such as the World Wide Web, electrical power grids, global economic markets, sensor networks; for example, see [18–20] and references therein. Bipartite networks are an important kind of complex networks, whose nodes can be divided into two disjoint nonempty sets such that every edge only connects a pair of nodes, which belong to different sets. Many real-world networks are naturally bipartite networks, such as the papers-scientists networks [21] and producer-consumer networks [22]. Recently, authors [23] have introduced a bipartite-graph complex dynamical network model that is only linearly coupled and has no delays. It is well known that time delays exist commonly in real-world systems. Therefore, many models of coupled networks with coupling delays are proposed, for example, constant single time delay [24], time-varying delays [25], and mix-time delays [26]. On the other hand, the coupled network often occurs in other forms, for example, nonlinearly coupled networks [27] and linearly derivative coupled networks [28]. In [29], a general model of bipartite dynamical network (BDN) with distributed delays and nonlinear derivative coupling was introduced. Synchronization of complex networks has been intensively investigated since they can be applied in power system control, secure communication, automatic control, chemical reaction, and so on [30–32]. The study of synchronization of coupled neural networks is an important step for both understanding brain science and designing coupled neural networks for practical use. Yu et al. [33] consider the synchronization of switched linearly coupled neural networks with constant delays, but the controllers are complex and changed with the switched rule. Synchronization of two coupled BDNs was investigated by adaptive method [29], but the controllers are complicated and the model does not include infinite distributed delays coupling and switching. Extending BAM neural networks to complex networks, we get complex bipartite dynamical networks (CBDNs). The dynamics of individual node in CBDNs is switched system and the switched coupling is considered; switched complex bipartite neural network (SCBNN) can be obtained. To the best of our knowledge, up to now, there is not any work that discusses the synchronization problem in SCBNN.

Motivated by the previous discussion, we first proposed a model of SCBNN, and then investigated the synchronization between two SCBNNs with infinite distributed delays and derivative coupling. Using adaptive controllers and linear matrix inequality (LMI) approach, some synchronization criteria are proposed to ensure the synchronization between two coupled SCBNNs. In our paper, the proposed controllers are simpler and do not change with the switched rule, which can be realize more easily.

The paper is organized as follows. In Section 2, a model of SCBNN with infinite distributed delays and derivative coupling is presented, and some hypotheses and lemmas are given too. In Section 3, several synchronization criteria on the SCBNNs are deduced. In Section 4, numerical examples are given to demonstrate the effectiveness of the proposed controller design methods in Section 3. Finally, conclusions are given in Section 5.

Notations. Throughout this paper, 𝜌_max(⋅) and 𝜌_min(⋅) denote the maximum eigenvalue and minimum eigenvalue of a real symmetric matrix, respectively. The notation ∗ denotes the symmetric block.

2. Model Description, Assumptions, and Lemmas

Consider a complex bipartite dynamical network (CBDN) consisting of two disjoint nonempty node sets 𝑉₁ and 𝑉₂. Suppose that 𝑉₁={𝜇₁,𝜇₂,…,𝜇_𝑙} and 𝑉₂={𝜈₁,𝜈₂,…,𝜈_𝑚}, 𝑙, 𝑚 are integer. The coupled network is described as follows: 𝑥̇_𝑖(𝑡)=−𝐷𝑥_𝑖+𝑅₁𝑓₁𝑥_𝑖(𝑡)+𝑅₂𝑓₂𝑥_𝑖((𝑡+𝑡−𝜏))+𝐼_𝑚𝑗=1𝑎_𝑖𝑗𝑦_𝑗𝑡−𝜏₁(𝑡)+^𝑚_𝑗=1𝑏_𝑖𝑗𝑔𝑦̇_𝑗𝑡−𝜏₂(𝑡)+^𝑚_𝑗=1𝑐_𝑖𝑗^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑦_𝑗(𝑠)𝑦̇d𝑠,𝑖=1,2,…,𝑙,_𝑗(𝑡)=−𝐷𝑦_𝑗+𝑅₁𝑓₁𝑦_𝑗(𝑡)+𝑅₂𝑓₂𝑦_𝑗((𝑡+𝑡−𝜎))+𝐽_𝑙𝑖=1𝑎_𝑗𝑖𝑥_𝑖𝑡−𝜎₁(𝑡)+^𝑙_𝑖=1𝑏_𝑗𝑖𝑔𝑥̇_𝑖𝑡−𝜎₂(𝑡)+^𝑙_𝑖=1𝑐_𝑗𝑖^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑥_𝑖(𝑠)d𝑠,𝑗=1,2,…,𝑚, (1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇, 𝑦_𝑗(𝑡)=(𝑦_𝑗1(𝑡),𝑦_𝑗2(𝑡),…,𝑦_𝑗𝑛(𝑡))^𝑇∈R^𝑛 denotes the state variables of nodes 𝜇_𝑖 and 𝜈_𝑗, respectively. 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) and 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) are diagonal matrices with 𝑑_𝑖,𝑑_𝑖>0. 𝑅₁,𝑅₁∈R^𝑛×𝑛 are weight matrices, 𝑅₂,𝑅₂∈R^𝑛×𝑛 are delayed weight matrices, 𝑓_𝑘(𝑥_𝑖)=(𝑓_𝑘1(𝑥_𝑖1),𝑓_𝑘2(𝑥_𝑖2),…,𝑓_𝑘𝑛(𝑥_𝑖𝑛))^𝑇, 𝑓_𝑘(𝑦_𝑗)=(𝑓_𝑘1(𝑦_𝑗1),𝑓_𝑘2(𝑦_𝑗2),…,𝑓_𝑘𝑛(𝑦_𝑗𝑛))^𝑇∈R^𝑛, 𝑘=1,2, 𝑔(𝑦̇_𝑗)=(𝑔₁(𝑦̇_𝑗1),𝑔₂(𝑦̇_𝑗2),…,𝑔_𝑛(𝑦̇_𝑗𝑛))^𝑇, 𝑔(𝑥̇_𝑖)=(𝑔₁(𝑥̇_𝑖1),𝑔₂(𝑥̇_𝑖2),…,𝑔_𝑛(𝑥̇_𝑖𝑛))^𝑇, 𝑘(𝑦_𝑗)=(𝑘₁(𝑦_𝑗1),𝑘₂(𝑦_𝑗2),…,𝑘_𝑛(𝑦_𝑗𝑛))^𝑇, 𝑘(𝑥_𝑖)=(𝑘₁(𝑥_𝑖1),𝑘₂(𝑥_𝑖2),…,𝑘_𝑛(𝑥_𝑖𝑛))^𝑇∈R^𝑛 corresponds to the boundedness activation functions of neurons. ℎ(𝑡)=diag(ℎ₁(𝑡),ℎ₂(𝑡),…,ℎ_𝑛(𝑡)), ℎ(𝑡)=diag(ℎ₁(𝑡),ℎ₂(𝑡),…,ℎ_𝑛(𝑡))∈R^𝑛×𝑛 are the delay kernel functions. 𝜏(𝑡),𝜏₁(𝑡),𝜏₂(𝑡),𝜎(𝑡),𝜎₁(𝑡), and 𝜎₂(𝑡)>0 are time delays. ∫^𝑡_−∞ℎ(𝑡−𝑠)𝑘(𝑦_𝑗(𝑠))d𝑠 and ∫^𝑡_−∞ℎ(𝑡−𝑠)𝑘(𝑥_𝑖(𝑠))d𝑠 express infinite distributed delays. 𝐼=(𝐼¹,𝐼²,…,𝐼^𝑛)^𝑇 and 𝐽=(𝐽¹,𝐽²,…,𝐽^𝑛)^𝑇∈R^𝑛 are the constant external input vectors. The matrix 𝐴=(𝑎_𝑖𝑗)_𝑙×𝑚 is the delayed weight coupling matrix denoting coupling strength between nodes. If there is a connection from node 𝜇_𝑖 to 𝜈_𝑗, then 𝑎_𝑖𝑗≠0; otherwise, 𝑎_𝑖𝑗=0 and the matrix 𝐴 satisfies the sum of every row being zero. The definitions of the other coupling matrixes 𝐵=(𝑏_𝑖𝑗)_𝑙×𝑚, 𝐶=(𝑐_𝑖𝑗)_𝑙×𝑚, 𝐴=(𝑎_𝑗𝑖)_𝑚×𝑙,𝐵=(𝑏_𝑗𝑖)_𝑚×𝑙, and 𝐶=(𝑐_𝑗𝑖)_𝑚×𝑙 are similar to that of matrix 𝐴; hence, they are omitted here.

In this paper, we consider a class of switched complex bipartite neural network with infinite distributed delays and derivative coupling, which is described as follows: 𝑥̇_𝑖(𝑡)=−𝐷_𝜆𝑥_𝑖+𝑅_𝜆1𝑓₁𝑥_𝑖(𝑡)+𝑅_𝜆2𝑓₂𝑥_𝑖(𝑡−𝜏(𝑡))+𝐼_𝜆+^𝑚_𝑗=1𝑎_𝜆𝑖𝑗𝑦_𝑗𝑡−𝜏₁(𝑡)+^𝑚_𝑗=1𝑏_𝜆𝑖𝑗𝑔𝑦̇_𝑗𝑡−𝜏₂(𝑡)+^𝑚_𝑗=1𝑐_𝜆𝑖𝑗^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑦_𝑗(𝑠)𝑦̇d𝑠,𝑖=1,2,…,𝑙,_𝑗(𝑡)=−𝐷_𝜆𝑦_𝑗+𝑅_𝜆1𝑓₁𝑦_𝑗(𝑡)+𝑅_𝜆2𝑓₂𝑦_𝑗(𝑡−𝜎(𝑡))+𝐽_𝜆+^𝑙_𝑖=1𝑎_𝜆𝑗𝑖𝑥_𝑖𝑡−𝜎₁(𝑡)+^𝑙_𝑖=1𝑏_𝜆𝑗𝑖𝑔𝑥̇_𝑖𝑡−𝜎₂(𝑡)+^𝑙_𝑖=1𝑐_𝜆𝑗𝑖^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑥_𝑖(𝑠)d𝑠,𝑗=1,2,…,𝑚, (2) where switching signal 𝜆 is piecewise constant functions, which is a value in the finite set ℵ={1,2,…,𝑁}. This means that the matrices {𝐷_𝜆,𝑅_𝜆1,𝑅_𝜆2,𝐴_𝜆=(𝑎_𝜆𝑖𝑗),𝐵_𝜆=(𝑏_𝜆𝑖𝑗),𝐶_𝜆=(𝑐_𝜆𝑖𝑗),𝐼_𝜆,𝐷_𝜆,𝑅_𝜆1,𝑅_𝜆2,𝐴_𝜆=(𝑎_𝜆𝑗𝑖),𝐵_𝜆=(𝑏_𝜆𝑗𝑖),and𝐶_𝜆=(𝑐_𝜆𝑗𝑖),𝐽_𝜆} are allowed to take values at particular time, in a finite set {(𝐷_𝑟,𝑅_𝑟1,𝑅_𝑟2,𝐴_𝑟,𝐵_𝑟,𝐶_𝑟,𝐼_𝑟,𝐷_𝑟,𝑅_𝑟1,𝑅_𝑟2,𝐴_𝑟,𝐵_𝑟,𝐶_𝑟,𝐽_𝑟)∣𝑟=1,2,…,𝑁}. We define the function as follows: 𝜉_𝑟()=𝑡,𝜆1,whentheswitchedsystemisdescribedbythe𝑟thmode,thatis,𝜆=𝑟,0,others. (3) It follows that under any switching rules ∑^𝑁_𝑟=1𝜉_𝑟(𝑡,𝜆)=1. Model (2) can be written as 𝑥̇_𝑖(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆−𝐷_𝑟𝑥_𝑖+𝑅_𝑟1𝑓₁𝑥_𝑖(𝑡)+𝑅_𝑟2𝑓₂𝑥_𝑖(𝑡−𝜏(𝑡))+𝐼_𝑟+^𝑚_𝑗=1𝑎_𝑟𝑖𝑗𝑦_𝑗𝑡−𝜏₁(𝑡)+^𝑚_𝑗=1𝑏_𝑟𝑖𝑗𝑔𝑦̇_𝑗𝑡−𝜏₂(𝑡)+^𝑚_𝑗=1𝑐_𝑟𝑖𝑗^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑦_𝑗(𝑠),𝑦̇d𝑠𝑖=1,2,…,𝑙,_𝑗(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×−𝑡,𝜆𝐷_𝑟𝑦_𝑗+𝑅_𝑟1𝑓₁𝑦_𝑗(𝑡)+𝑅_𝑟2𝑓₂𝑦_𝑗(𝑡−𝜎(𝑡))+𝐽_𝑟+^𝑙_𝑖=1𝑎_𝑟𝑗𝑖𝑥_𝑖𝑡−𝜎₁(𝑡)+^𝑙_𝑖=1𝑏_𝑟𝑗𝑖𝑔𝑥̇_𝑖𝑡−𝜎₂(𝑡)+^𝑙_𝑖=1𝑐_𝑟𝑗𝑖^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑥_𝑖(𝑠)d𝑠,𝑗=1,2,…,𝑚. (4) The response network of the drive network (4) is 𝑥̇_𝑖(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆−𝐷_𝑟𝑥_𝑖(𝑡)+𝑅_𝑟1𝑓₁𝑥_𝑖(𝑡)+𝑅_𝑟2𝑓₂𝑥_𝑖(𝑡−𝜏(𝑡))+𝐼_𝑟+^𝑚_𝑗=1𝑎_𝑟𝑖𝑗𝑦_𝑗𝑡−𝜏₁(𝑡)+^𝑚_𝑗=1𝑏_𝑟𝑖𝑗𝑔𝑦̇_𝑗𝑡−𝜏₂(𝑡)+^𝑚_𝑗=1𝑐_𝑟𝑖𝑗^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑦_𝑗(𝑠)d𝑠+𝑢_𝑖(𝑡),𝑦̇_𝑗(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×−𝑡,𝜆𝐷_𝑟𝑦_𝑗(𝑡)+𝑅_𝑟1𝑓₁𝑦_𝑗(𝑡)+𝑅_𝑟2𝑓₂𝑦_𝑗(𝑡−𝜎(𝑡))+𝐽_𝑟+^𝑙_𝑖=1𝑎_𝑟𝑗𝑖𝑥_𝑖𝑡−𝜎₁(𝑡)+^𝑙_𝑖=1𝑏_𝑟𝑗𝑖𝑔𝑥̇_𝑖𝑡−𝜎₂(𝑡)+^𝑙_𝑖=1𝑐_𝑟𝑗𝑖^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑥_𝑖(𝑠)d𝑠+𝑣_𝑗(𝑡), (5) where 𝑢_𝑖(𝑡) and 𝑣_𝑗(𝑡)∈𝑅^𝑛 are the control inputs.

Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡), 𝜀_𝑗(𝑡)=𝑦_𝑗(𝑡)−𝑦_𝑗(𝑡), 𝑖=1,2,…,𝑙, and 𝑗=1,2,…,𝑚. The error dynamical system of (4) and (5) is given by 𝑒_𝑖̇(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆−𝐷_𝑟𝑒_𝑖(𝑡)+𝑅_𝑟1𝐹₁𝑒_𝑖(𝑡)+𝑅_𝑟2𝐹₂𝑒_𝑖((𝑡+𝑡−𝜏))_𝑚𝑗=1𝑎_𝑟𝑖𝑗𝜀_𝑗𝑡−𝜏₁(𝑡)+^𝑚_𝑗=1𝑏_𝑟𝑖𝑗𝐺𝜀_𝑗̇𝑡−𝜏₂(𝑡)+^𝑚_𝑗=1𝑐_𝑟𝑖𝑗^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)d𝑠+𝑢_𝑖(𝑡),𝜀̇𝑖=1,2,…,𝑙,_𝑗(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×−𝑡,𝜆𝐷_𝑟𝜀_𝑗(𝑡)+𝑅_𝑟1𝐹₁𝜀_𝑗(𝑡)+𝑅_𝑟2𝐹₂𝜀_𝑗((𝑡+𝑡−𝜎))_𝑙𝑖=1𝑎_𝑟𝑗𝑖𝑒_𝑖𝑡−𝜎₁(𝑡)+^𝑙_𝑖=1𝑏_𝑟𝑗𝑖𝐺𝑒_𝑖̇𝑡−𝜎₂(𝑡)+^𝑙_𝑖=1𝑐_𝑟𝑗𝑖^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝑒_𝑖(𝑠)d𝑠+𝑣_𝑗(𝑡),𝑗=1,2,…,𝑚, (6) where 𝐹_𝑘𝑒_𝑖(𝑡)=𝑓_𝑘𝑥_𝑖(𝑡)−𝑓_𝑘𝑥_𝑖(𝑡),𝐹_𝑘𝜀_𝑗(𝑡)=𝑓_𝑘𝑦_𝑗(𝑡)−𝑓_𝑘𝑦_𝑗(𝑡)𝐺𝜀̇,𝑘=1,2,_𝑗(𝑡)=𝑔𝑦̇_𝑗(𝑡)−𝑔𝑦̇_𝑗(𝑡),𝐺𝑒_𝑖̇(𝑡)=𝑔𝑥̇_𝑖(𝑡)−𝑔𝑥̇_𝑖(𝑡),𝐾𝜀_𝑗(𝑠)=𝑘𝑦_𝑗(𝑠)−𝑘𝑦_𝑗(𝑠)=𝐾₁𝜀_𝑗1(𝑠),…,𝐾_𝑛𝜀_𝑗𝑛(𝑠)^𝑇,𝐾𝑒_𝑖(𝑠)=𝑘𝑥_𝑖(𝑠)−𝑘𝑥_𝑖(𝑠)=𝐾₁𝑒_𝑖1(𝑠),…,𝐾_𝑛𝑒_𝑖𝑛(𝑠)^𝑇. (7)

In this paper, the following assumptions and lemmas are needed.(𝑆₁) There exist diagonal matrices 𝐿⁻_𝑖=diag(𝑙⁻_𝑖1,𝑙⁻_𝑖2,…,𝑙⁻_𝑖𝑛) and 𝐿⁺_𝑖=diag(𝑙⁺_𝑖1,𝑙⁺_𝑖2,…,𝑙⁺_𝑖𝑛), such that 𝑙⁻_𝑘𝑗≤𝑓_𝑘𝑗(𝑥)−𝑓_𝑘𝑗𝑦𝑥−𝑦≤𝑙⁺_𝑘𝑗,𝑙⁻_3𝑗≤𝑘_𝑗(𝑥)−𝑘_𝑗𝑦𝑥−𝑦≤𝑙⁺_3𝑗,𝑙⁻_4𝑗≤𝑔_𝑗(𝑥)−𝑔_𝑗𝑦𝑥−𝑦≤𝑙⁺_4𝑗, (8) ∀𝑥,𝑦∈R and 𝑥≠𝑦, 𝑖=1,2,3,4,𝑗=1,2,…,𝑛, and 𝑘=1,2.(𝑆₂) There exist diagonal matrices 𝐿⁻_𝑖=diag(𝑙⁻_𝑖1,𝑙⁻_𝑖2,…,𝑙⁻_𝑖𝑛) and 𝐿⁺_𝑖=diag(𝑙⁺_𝑖1,𝑙⁺_𝑖2,…,𝑙⁺_𝑖𝑛), such that 𝑙⁻_𝑘𝑗≤𝑓_𝑘𝑗(𝑥)−𝑓_𝑘𝑗𝑦≤𝑥−𝑦𝑙⁺_𝑘𝑗,𝑙⁻_3𝑗≤𝑘_𝑗(𝑥)−𝑘_𝑗𝑦≤𝑥−𝑦𝑙⁺_3𝑗,𝑙⁻_4𝑗≤𝑔_𝑗(𝑥)−𝑔_𝑗𝑦≤𝑥−𝑦𝑙⁺_4𝑗, (9) ∀𝑥,𝑦∈R and 𝑥≠𝑦, 𝑖=1,2,3,4, 𝑗=1,2,…,𝑛, and  𝑘=1,2.(𝑆₃)𝜏(𝑡), 𝜏₁(𝑡), 𝜏₂(𝑡), 𝜎(𝑡), 𝜎₁(𝑡), and 𝜎₂(𝑡) are differential functions with 𝜏̇(𝑡)<𝜏<1, 𝜎̇(𝑡)<𝜎<1, 𝜏̇₁(𝑡)<𝜏₁<1, 𝜎̇₁(𝑡)<𝜎₁<1, 𝜏̇₂(𝑡)<𝜏₂<1, and 𝜎̇₂(𝑡)<𝜎₂<1.(𝑆₄)ℎ_𝑖(𝑡),ℎ_𝑖(𝑡) are real-value nonnegative continuous functions defined in [0,∞) satisfying ^∞₀ℎ_𝑖(𝑠)d𝑠<∞,^∞₀ℎ_𝑖(𝑠)d𝑠<∞,𝑖=1,2,…,𝑛. (10)

Lemma 1 (see [34]). Given any real matrices Σ₁, Σ₂, and Σ₃ of appropriate dimensions and a scalar 𝜀>0 such that 0<Σ₃=Σ^𝑇₃, then the following inequality holds: Σ^𝑇₁Σ₂+Σ^𝑇₂Σ₁≤𝜀Σ^𝑇₁Σ₃Σ₁+𝜀⁻¹Σ^𝑇₂Σ₃⁻¹Σ₂. (11)

Lemma 2 (see [35]). Given a positive definite matrix𝑃∈𝑅^𝑛×𝑛and a symmetric matrix 𝑄∈𝑅^𝑛×𝑛, then 𝜌_min𝑃⁻¹𝑄𝑥^𝑇𝑃𝑥≤𝑥^𝑇𝑄𝑥≤𝜌_max𝑃⁻¹𝑄𝑥^𝑇𝑃𝑥,∀𝑥∈𝑅^𝑛. (12)

Lemma 3 (Schur complement). Given constant symmetric matrices Σ₁, Σ₂, and Σ₃, where Σ₁=Σ^𝑇₁ and 0<Σ₂=Σ^𝑇₂, then Σ₁+Σ^𝑇₃Σ₂⁻¹Σ₃<0 if and only if Σ₁Σ^𝑇₃Σ₃−Σ₂<0𝑜𝑟−Σ₂Σ₃Σ^𝑇₃Σ₁<0. (13)
For convenience, let 𝐿_𝑖=diagmax𝑙⁻_𝑖1,𝑙⁺_𝑖1,max𝑙⁻_𝑖2,𝑙⁺_𝑖2,…,max𝑙⁻_𝑖𝑛,𝑙⁺_𝑖𝑛,𝑖=1,2,3,4,𝐿_𝑖=diagmax𝑙⁻_𝑖1,𝑙⁺_𝑖1,max𝑙⁻_𝑖2,𝑙⁺_𝑖2,…,max𝑙⁻_𝑖𝑛,𝑙⁺_𝑖𝑛,𝑖=1,2,3,4,𝐻=diag^∞₀ℎ₁(𝑣)d𝑣,^∞₀ℎ₂(𝑣)d𝑣,…,^∞₀ℎ_𝑛(𝑣)d𝑣,𝐻=diag^∞₀ℎ₁(𝑣)d𝑣,^∞₀ℎ₂(𝑣)d𝑣,…,^∞₀ℎ_𝑛(𝑣)d𝑣. (14)

3. Main Results

Theorem 4. Under assumptions (𝑆₁)--(𝑆₄), the two coupled SCBNNs (4) and (5) can be synchronized, if there exist positive constants, 𝛼,𝛽,𝑝,𝑝,𝛾_𝑖,𝜂_𝑗(𝑖=1,2,…,𝑙,𝑗=1,2,…,𝑚), 𝑛×𝑛 positive matrices 𝑃, 𝑄, 𝑈, 𝑃, 𝑄, 𝑈 and 𝑛×𝑛 diagonal positive matrices 𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛), 𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛), 𝑀_𝑖,𝑀_𝑖(𝑖=1,2,3) such that 𝑍_𝑟𝑖𝑃𝑅_𝑟1𝑃𝑅∑_𝑟2^𝑚_𝑗=1𝑎²_𝑟𝑖𝑗^1/2𝑃∑^𝑚_𝑗=1𝑏²_𝑟𝑖𝑗^1/2𝑃𝑙∑^𝑚_𝑗=1𝑐²_𝑟𝑖𝑗^1/2𝑃∗−𝑀₁0000∗∗−𝑀₂000∗∗∗−𝑀₃00∗∗∗∗−𝐼_𝑛0∗∗∗∗∗−𝑊<0,𝑍_𝑟𝑗𝑃𝑅_𝑟1𝑃𝑅∑_𝑟2^𝑙_𝑖=1𝑎²_𝑟𝑗𝑖^1/2𝑃∑^𝑙_𝑖=1𝑏²_𝑟𝑗𝑖^1/2𝑃𝑚∑^𝑙_𝑖=1𝑐²_𝑟𝑗𝑖^1/2𝑃∗−𝑀₁0000∗∗−𝑀₂000∗∗∗−𝑀₃00∗∗∗∗−𝐼_𝑛0∗∗∗∗∗−𝑊𝑚<0,1−𝜎₂−2𝛼𝑝≤0,𝑃≥𝑝𝐼_𝑛,𝑃≥𝑝𝐼_𝑛,𝑙1−𝜏₂−2𝛽𝐿𝑝≤0,₂𝑀₂𝐿₂−(1−𝜏)𝑄≤0,𝑀₃−1−𝜏₁𝑈≤0,𝐿₂𝑀₂𝐿₂−(1−𝜎)𝑄≤0,𝑀₃−1−𝜎₁𝑈≤0, (15)(16)(17)(18)(19) and the adaptive feedback controllers are designed as 𝑢_𝑖(𝑡)=−𝛾_𝑖+𝛼_𝑖(𝑡)𝑒_𝑖(𝑡),𝑣_𝑗(𝑡)=−𝜂_𝑗+𝛽_𝑗(𝑡)𝜀_𝑗(𝑡),𝛼_𝑖(𝑡)=𝐺𝑒_𝑖̇(𝑡)²𝑒_𝑖(𝑡)²𝛼,𝑒_𝑖(𝑡)²𝑒≠0,0,_𝑖(𝑡)²𝛽=0,_𝑗(𝑡)=𝐺𝜀_𝑗̇(𝑡)²𝜀_𝑗(𝑡)²𝛽,𝜀_𝑗(𝑡)²𝜀≠0,0,_𝑗(𝑡)²=0, (20) where 𝑍_𝑟𝑖=−2𝑃𝐷_𝑟+𝐿₃𝐻𝑊𝐻𝐿₃+𝐿₁𝑀₁𝐿₁+𝑚𝑈−2𝛾_𝑖𝑃+𝑄, 𝑍_𝑟𝑗=−2𝑃𝐷_𝑟+𝐿₃𝐻𝑊𝐻𝐿₃+𝐿₁𝑀₁𝐿₁+𝑙𝑈+𝑄−2𝜂_𝑗𝑃, 𝑟∈ℵ, 𝑖=1,2,…,𝑙, and 𝑗=1,2,…,𝑚.

Proof. For the error dynamical system (6), we design the following Lyapunov-Krasovskii function: 𝑉(𝑡)=𝑉₁(𝐸(𝑡))+𝑉₂(𝐸(𝑡)), (21) where 𝐸(𝑡)=(𝑒^𝑇₁(𝑡),𝑒^𝑇₂(𝑡),…,𝑒^𝑇_𝑙(𝑡),𝜀^𝑇₁(𝑡),𝜀^𝑇₂(𝑡),…,𝜀^𝑇_𝑚(𝑡))^𝑇,𝑉₁=^𝑙_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡)+^𝑙_𝑖=1^𝑡_{(𝑡)𝑡−𝜏}𝑒^𝑇_𝑖(𝑠)𝑄𝑒_𝑖(𝑠)+d𝑠_{𝑚𝑛𝑗=1𝑖=1}𝑤_𝑖^∞₀ℎ_𝑖(𝑣)d𝑣^∞₀ℎ_𝑖(𝜃)×^𝑡_𝑡−𝜃𝐾²_𝑖𝜀_𝑗𝑖(𝑠)d𝑠d𝜃+𝑙^𝑚_𝑗=1^𝑡_{𝑡−𝜏1(𝑡)}𝜀^𝑇_𝑗(𝑠)𝑈𝜀_𝑗(𝑠)+𝑙d𝑠1−𝜏₂^𝑚_𝑗=1^𝑡_{𝑡−𝜏2(𝑡)}𝐺^𝑇𝜀_𝑗̇(𝑠)𝐺𝜀_𝑗̇(𝑠)𝑉d𝑠,₂=^𝑚_𝑗=1𝜀^𝑇_𝑗(𝑡)𝑃𝜀_𝑗(𝑡)+^𝑚_𝑗=1^𝑡_{𝑡−𝜇(𝑡)}𝜀^𝑇_𝑗(𝑠)𝑄𝜀_𝑗(𝑠)+d𝑠_{𝑙𝑛𝑖=1𝑗=1}𝑤_𝑗^∞₀ℎ_𝑗(𝑣)d𝑣^∞₀ℎ_𝑗(𝜃)×^𝑡_𝑡−𝜃𝐾_𝑗𝑒_𝑖𝑗(𝑠)d𝑠d𝜃+𝑚^𝑙_𝑖=1^𝑡_{𝑡−𝜇1(𝑡)}𝑒^𝑇_𝑖(𝑠)𝑈𝑒_𝑖(𝑠)+𝑚d𝑠1−𝜎₂^𝑙_𝑖=1^𝑡_{𝑡−𝜇2(𝑡)}𝐺^𝑇𝑒_𝑖̇(𝑠)𝐺𝑒_𝑖̇(𝑠)d𝑠. (22)(23) Calculating the derivative of (22) along the trajectories of (6), we have 𝑉̇₁=^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆_𝑙𝑖=12𝑒^𝑇_𝑖(𝑡)𝑃×−𝐷_𝑟𝑒_𝑖(𝑡)+𝑅_𝑟1𝐹₁𝑒_𝑖(𝑡)+𝑅_𝑟2𝐹₂𝑒_𝑖((𝑡+𝑡−𝜏))_𝑚𝑗=1𝑎_𝑟𝑖𝑗𝜀_𝑗𝑡−𝜏₁(𝑡)+^𝑚_𝑗=1𝑏_𝑟𝑖𝑗𝐺𝜀_𝑗̇𝑡−𝜏₂(𝑡)+^𝑚_𝑗=1𝑐_𝑟𝑖𝑗×^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)𝑑𝑠+𝑢_𝑖(𝑡+_𝑙𝑖=1)𝑒^𝑇_𝑖(𝑡)𝑄𝑒_𝑖(𝑡)−(𝜏̇(𝑡×1−))_𝑙𝑖=1𝑒^𝑇_𝑖(𝑡−𝜏(𝑡))𝑄𝑒_𝑖((𝑡+𝑡−𝜏))_{𝑚𝑛𝑗=1𝑖=1}𝑤_𝑖𝐾_𝑖𝜀_𝑗𝑖(𝑡)^∞₀ℎ_𝑖(𝑣)𝑑𝑣²−^𝑚𝑛_𝑗=1𝑖=1𝑤_𝑖^∞₀ℎ_𝑖(𝑣)𝑑𝑣×^∞₀ℎ_𝑖(𝜃)𝐾²_𝑖𝜀_𝑗𝑖(𝑡−𝜃)𝑑𝜃+𝑙^𝑚_𝑗=1𝜀^𝑇_𝑗(𝑠)𝑈𝜀_𝑗(𝑠)−1−𝜏̇₁(𝑡)𝑙×^𝑚_𝑗=1𝜀^𝑇_𝑗𝑡−𝜏₁(𝑡)𝑈𝜀_𝑗𝑡−𝜏₁(𝑡)+𝑙1−𝜏₂^𝑚_𝑗=1𝐺^𝑇𝜀_𝑗̇(𝑡)𝐺𝜀_𝑗̇(𝑡)−1−𝜏̇₂(𝑡)1−𝜏₂𝑙×^𝑚_𝑗=1𝐺^𝑇𝜀_𝑗̇𝑡−𝜏₂(𝑡)𝐺𝜀_𝑗̇𝑡−𝜏₂(𝑡). (24) By Lemma 1, we can get from (𝑆₁)2𝑒^𝑇_𝑖(𝑡)𝑃𝑅_𝑟1𝐹₁𝑒_𝑖(𝑡)≤𝑒^𝑇_𝑖(𝑡)𝑃𝑅_𝑟1𝑀₁⁻¹𝑅^𝑇_𝑟1𝑃𝑒_𝑖(𝑡)+𝐹^𝑇₁𝑒_𝑖(𝑡)𝑀₁𝐹₁𝑒_𝑖(𝑡)≤𝑒^𝑇_𝑖(𝑡)𝑃𝑅_𝑟1𝑀₁⁻¹𝑅^𝑇_𝑟1𝑃+𝐿₁𝑀₁𝐿₁𝑒_𝑖(𝑡),2𝑒^𝑇_𝑖(𝑡)𝑃𝑅_𝑟2𝐹₂𝑒_𝑖(𝑡−𝜏(𝑡))≤𝑒^𝑇_𝑖(𝑡)𝑃𝑅_𝑟2𝑀₂⁻¹𝑅^𝑇_𝑟2𝑃𝑒_𝑖(𝑡)+𝐹^𝑇₂𝑒_𝑖(𝑡−𝜏(𝑡))𝑀₂𝐹₂𝑒_𝑖(𝑡−𝜏(𝑡))≤𝑒^𝑇_𝑖(𝑡)𝑃𝑅_𝑟2𝑀₂⁻¹𝑅^𝑇_𝑟2𝑃𝑒_𝑖(𝑡)+𝑒^𝑇_𝑖(𝑡−𝜏(𝑡))𝐿₂𝑀₂𝐿₂𝑒_𝑖(𝑡−𝜏(𝑡)),2𝑎_𝑟𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑃𝜀_𝑗𝑡−𝜏₁(𝑡)≤𝑎²_𝑟𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑃𝑀₃⁻¹𝑃𝑒_𝑖(𝑡)+𝜀^𝑇_𝑗𝑡−𝜏₁(𝑡)𝑀₃𝜀^𝑇_𝑗𝑡−𝜏₁(𝑡). (25) By assumptions (𝑆₁) and (𝑆₄), it is obvious that ^𝑚𝑛_𝑗=1𝑖=1𝑤_𝑖𝐾_𝑖𝜀_𝑗𝑖(𝑡)^∞₀ℎ_𝑖(𝑣)d𝑣²=^𝑚_𝑗=1𝐾^𝑇𝜀_𝑗(𝑡)𝐻𝑊𝐻𝐾𝜀_𝑗(𝑡)≤^𝑚_𝑗=1𝜀^𝑇_𝑗(𝑡)𝐿₃𝐻𝑊𝐻𝐿₃𝜀_𝑗(𝑡),2^𝑙_𝑖=1𝑐_𝑟𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑃^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)d𝑠≤𝑙^𝑙_𝑖=1𝑐²_𝑟𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑃𝑊⁻¹𝑃𝑒_𝑖(𝑡)+1𝑙^𝑙_𝑖=1^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)d𝑠^𝑇×𝑊^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)d𝑠=𝑙^𝑙_𝑖=1𝑐²_𝑟𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑃𝑊⁻¹𝑃𝑒_𝑖(𝑡)+^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)d𝑠^𝑇×𝑊^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)d𝑠. (26)(27) Observe that −𝑙1−𝜏̇₂(𝑡)1−𝜏₂^𝑚_𝑗=1𝐺^𝑇𝜀_𝑗̇𝑡−𝜏₂(𝑡)𝐺𝜀_𝑗̇𝑡−𝜏₂(𝑡)+2^𝑙𝑚_𝑖=1𝑗=1𝑒^𝑇_𝑖(𝑡)𝑃𝑏_𝑟𝑖𝑗𝐺𝜀_𝑗̇𝑡−𝜏₂(𝑡)≤−^𝑙𝑚_𝑖=1𝑗=1𝐺^𝑇𝜀_𝑗̇𝑡−𝜏₂(𝑡)𝐺𝜀_𝑗̇𝑡−𝜏₂(𝑡)+2^𝑙𝑚_𝑖=1𝑗=1𝑒^𝑇_𝑖(𝑡)𝑃𝑏_𝑟𝑖𝑗𝐺𝜀_𝑗̇𝑡−𝜏₂(𝑡)=−^𝑙𝑚_𝑖=1𝑗=1𝑏_𝑟𝑖𝑗𝑃𝑒_𝑖(𝑡)−𝐺𝜀_𝑗̇𝑡−𝜏₂(𝑡)^𝑇×𝑏_𝑟𝑖𝑗𝑃𝑒_𝑖(𝑡)−𝐺𝜀_𝑗̇𝑡−𝜏₂(𝑡)+^𝑙𝑚_𝑖=1𝑗=1𝑏²_𝑟𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑃²𝑒_𝑖(𝑡),≤^𝑙𝑚_𝑖=1𝑗=1𝑏²_𝑟𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑃²𝑒_𝑖(𝑡). (28) Using inequality ^∞₀𝑓²(𝑠)d𝑠^∞₀𝑔²(𝑠)d𝑠≥^∞₀𝑓(𝑠)𝑔(𝑠)d𝑠², (29) we have ^𝑛_𝑖=1𝑤_𝑖^∞₀ℎ_𝑖(𝑣)d𝑣^∞₀ℎ_𝑖(𝜃)𝐾²_𝑖𝜀_𝑗𝑖(≥𝑡−𝜃)d𝜃_𝑛𝑖=1𝑤_𝑖^∞₀ℎ_𝑖(𝜃)𝐾_𝑖𝜀_𝑗𝑖(𝑡−𝜃)d𝜃²=^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)d𝑠^𝑇×𝑊^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)d𝑠. (30) Using Lemma 2 and condition (17), we get −𝑒^𝑇(𝑡)𝑃𝐺𝑒_𝑖̇(𝑡)²𝑒_𝑖(𝑡)²𝑒(𝑡)≤−𝑝𝐺^𝑇𝑒_𝑖̇(𝑡)𝐺𝑒_𝑖̇(𝑡). (31) Substituting (20) into (24) and combining (24)–(31), it can be derived by condition (18) that 𝑉̇₁≤^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆_𝑙𝑖=1𝑒^𝑇_𝑖(𝑡)Ω_𝑟𝑖𝑒_𝑖(𝑡)+^𝑙_𝑖=1𝑒^𝑇_𝑖(𝑡−𝜏(𝑡))𝐿₂𝑀₂𝐿₂−(1−𝜏)𝑄×𝑒_𝑖((𝑡+𝑡−𝜏))_𝑚𝑗=1𝜀^𝑇_𝑗(𝑠)𝑙𝑈+𝐿₃𝐻𝑊𝐻𝐿₃𝜀_𝑗(𝑠)+^𝑚_𝑗=1𝜀^𝑇_𝑗𝑡−𝜏₁(𝑡)𝑙𝑀₃−𝑙1−𝜏₁𝑈×𝜀_𝑗𝑡−𝜏₁(𝑡)+𝑙1−𝜏₂^𝑚_𝑗=1𝐺^𝑇𝜀_𝑗̇(𝑡)𝐺𝜀_𝑗̇(𝑡)−2𝛼𝑝^𝑙_𝑖=1𝐺^𝑇𝑒_𝑖̇(𝑡)𝐺𝑒_𝑖̇(𝑡)≤^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆_𝑙𝑖=1𝑒^𝑇_𝑖(𝑡)Ω_𝑟𝑖𝑒_𝑖(𝑡)+^𝑚_𝑗=1𝜀^𝑇_𝑗(𝑡)𝑙𝑈+𝐿₃𝐻𝑊𝐻𝐿₃𝜀_𝑗(𝑡)+𝑙1−𝜏₂^𝑚_𝑗=1𝐺^𝑇𝜀_𝑗̇(𝑡)𝐺𝜀_𝑗̇(𝑡)−2𝛼𝑝^𝑙_𝑖=1𝐺^𝑇𝑒_𝑖̇(𝑡)𝐺𝑒_𝑖̇(𝑡), (32) where Ω_𝑖=−2𝑃𝐷_𝑟+𝑃𝑅_𝑟1𝑀₁⁻¹𝑅^𝑇_𝑟1𝑃+𝐿₁𝑀₁𝐿₁+𝑃𝑅_𝑟2𝑀₂⁻¹𝑅^𝑇_𝑟2𝑃+∑^𝑚_𝑗=1𝑎²_𝑟𝑖𝑗𝑃𝑀₃⁻¹𝑃+∑^𝑚_𝑗=1𝑏²_𝑟𝑖𝑗𝑃² + 𝑙∑^𝑚_𝑗=1𝑐²_𝑟𝑖𝑗𝑃𝑊⁻¹𝑃−2𝛾_𝑖𝑃+𝑄.
Meanwhile, by a similar process, the following inequality can be true: 𝑉̇₂≤^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆_𝑚𝑗=1𝜀^𝑇_𝑗(𝑡)Ω_𝑟𝑗𝜀_𝑗(𝑡)+^𝑙_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑚𝑈+𝐿₃𝐻𝑊𝐻𝐿₃𝑒_𝑖(𝑡)+𝑚1−𝜎₂^𝑙_𝑖=1𝐺^𝑇𝑒_𝑖̇(𝑡)𝐺𝑒_𝑖̇(𝑡)−2𝛽𝑝^𝑚_𝑗=1𝐺^𝑇𝜀_𝑗̇(𝑡)𝐺𝜀_𝑗̇(𝑡), (33) where Ω_𝑗=−2𝑃𝐷_𝑟+𝑃𝑅_𝑟1𝑀₁⁻¹𝑅^𝑇_𝑟1𝑃+𝐿₁𝑀₁𝐿₁+𝑃𝑅_𝑟2𝑀₂⁻¹𝑅^𝑇_𝑟2𝑃+∑^𝑙_𝑖=1𝑎²_𝑟𝑖𝑗𝑃𝑀₃⁻¹𝑃+∑^𝑙_𝑖=1𝑏²_𝑟𝑗𝑖𝑃²+𝑚∑^𝑙_𝑖=1𝑐²_𝑟𝑗𝑖𝑃𝑊⁻¹𝑃−2𝜂_𝑗𝑃+𝑄.
By condition (17), we have 𝑉̇≤^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆_𝑙𝑖=1𝑒^𝑇_𝑖(𝑡)Ω_𝑟𝑖+𝑚𝑈+𝐿₃𝐻𝑊𝐻𝐿₃𝑒_𝑖(𝑡)+^𝑚_𝑗=1𝜀^𝑇_𝑗(𝑡)Ω_𝑟𝑗+𝑙𝑈+𝐿₃𝐻𝑊𝐻𝐿₃𝜀_𝑗(𝑡). (34) By (15)-(16) and Lemma 3 (Schur complement), it can be obtained that Ω_𝑟𝑖+𝑚𝑈+𝐿₃𝐻𝑊𝐻𝐿₃<0,Ω_𝑟𝑗+𝑙𝑈+𝐿₃𝐻𝑊𝐻𝐿₃<0. Set 𝜌=min{𝜌₁,𝜌₂}, where 𝜌₁=−min𝜌_minΩ_𝑟𝑖+𝑚𝑈+𝐿₃𝐻𝑊𝐻𝐿₃,,𝜌𝑟∈ℵ,1≤𝑖≤𝑙₂=−min𝜌_minΩ_𝑟𝑗+𝑙𝑈+𝐿₃𝐻𝑊𝐻𝐿₃,𝑟∈ℵ,1≤𝑗≤𝑚, (35) then 𝜌>0, and 𝑉̇≤−𝜌₁^𝑙_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−𝜌₂^𝑚_𝑗=1𝜀^𝑇_𝑗(𝑡)𝜀_𝑗(𝑡)≤−𝜌𝐸^𝑇(𝑡)𝐸(𝑡). (36) Therefore, 𝑉 is nonincreasing in 𝑡≥0. One has 𝑉 bounded since 0≤𝑉(𝑡,𝐸(𝑡))≤𝑉(0,𝐸(0)), so lim_𝑡→+∞𝑉(𝑡,𝐸(𝑡)) exists and lim_𝑡→+∞^𝑡₀𝐸^𝑇(𝑠)𝐸(𝑠)1d𝑠≤−𝜌lim_𝑡→+∞^𝑡₀d𝑉=1d𝑠d𝑠𝜌𝑉((0−10,𝐸))𝜌lim_𝑡→+∞𝑉(𝑡,𝐸(𝑡)). (37) From (22)-(23) and conditions 𝑃≥𝑝𝐼, 𝑃≥𝑝𝐼 and we have 0≤𝐸^𝑇(𝑡)𝐸(𝑡)≤max{1/𝑝,1/𝑝}𝑉(𝑡,𝐸(𝑡)), so 𝐸^𝑇(𝑡)𝐸(𝑡) is bounded. According to error system (6), (d/d𝑡)𝐸^𝑇(𝑡)𝐸(𝑡)=2𝐸^𝑇(𝑡)𝐸̇(𝑡) is bounded for 𝑡≥0 due to the boundedness of activation functions. From the above we can see that 𝐸(𝑡)∈𝐿²∩𝐿^∞ and (d/d𝑡)𝐸^𝑇(𝑡)𝐸(𝑡)∈𝐿^∞. By using Barbǎlat lemma (see [36]), one has lim_𝑡→+∞𝐸^𝑇(𝑡)𝐸(𝑡)=0, so the two SBNNs (4) and (5) can obtain synchronization under the controllers (20). This completes the proof.

We take CBDN (1) as drive network. The response network of the drive network (1) is 𝑥̇_𝑖(𝑡)=−𝐷𝑥_𝑖(𝑡)+𝑅₁𝑓₁𝑥_𝑖(𝑡)+𝑅₂𝑓₂𝑥_𝑖((𝑡(𝑡)+𝑡−𝜏))+𝐼^𝑚_𝑗=1𝑎_𝑖𝑗𝑦_𝑗𝑡−𝜏₁(𝑡)+^𝑚_𝑗=1𝑏_𝑖𝑗𝑔𝑦̇_𝑗𝑡−𝜏₂(𝑡)+^𝑚_𝑗=1𝑐_𝑖𝑗^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑦_𝑗(𝑠)d𝑠+𝑢_𝑖(𝑡),𝑦̇_𝑗(𝑡)=−𝐷𝑦_𝑗(𝑡)+𝑅₁𝑓₁𝑦_𝑗(𝑡)+𝑅₂𝑓₂𝑦_𝑗((𝑡(𝑡)+𝑡−𝜎))+𝐽^𝑙_𝑖=1𝑎_𝑗𝑖𝑥_𝑖𝑡−𝜎₁(𝑡)+^𝑙_𝑖=1𝑏_𝑗𝑖𝑔𝑥̇_𝑖𝑡−𝜎₂(𝑡)+^𝑙_𝑖=1𝑐_𝑗𝑖^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑥_𝑖(𝑠)d𝑠+𝑣_𝑗(𝑡), (38) where 𝑢_𝑖(𝑡),𝑣_𝑗(𝑡)∈𝑅^𝑛 are the control inputs.

From Theorem 4, we can get the following corollary.

Corollary 5. Under assumptions (𝑆₁)--(𝑆₄), the two coupled CBDNs (1) and (38) can be synchronized, if there exist positive constants 𝛼,𝛽,𝑝,𝑝,𝛾_𝑖,𝜂_𝑗(𝑖=1,2,…,𝑙,𝑗=1,2,…,𝑚), 𝑛×𝑛 positive matrices 𝑃, 𝑄, 𝑈, 𝑃, 𝑄, 𝑈 and 𝑛×𝑛 diagonal positive matrices 𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛), 𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛), 𝑀_𝑖, 𝑀_𝑖(𝑖=1,2,3)such that 𝑍_𝑖𝑃𝑅₁𝑃𝑅∑₂^𝑚_𝑗=1𝑎²_𝑖𝑗^1/2𝑃∑^𝑚_𝑗=1𝑏²_𝑖𝑗^1/2𝑃𝑙∑^𝑚_𝑗=1𝑐²_𝑖𝑗^1/2𝑃∗−𝑀₁0000∗∗−𝑀₂000∗∗∗−𝑀₃00∗∗∗∗−𝐼_𝑛0∗∗∗∗∗−𝑊<0,𝑍_𝑗𝑃𝑅₁𝑃𝑅∑₂^𝑙_𝑖=1𝑎²_𝑗𝑖^1/2𝑃∑^𝑙_𝑖=1𝑏²_𝑗𝑖^1/2𝑃𝑚∑^𝑙_𝑖=1𝑐²_𝑗𝑖^1/2𝑃∗−𝑀₁0000∗∗−𝑀₂000∗∗∗−𝑀₃00∗∗∗∗−𝐼_𝑛0∗∗∗∗∗−𝑊𝑚<01−𝜎₂−2𝛼𝑝≤0,𝑃≥𝑝𝐼_𝑛,𝑃≥𝑝𝐼_𝑛,𝑙1−𝜏₂−2𝛽𝐿𝑝≤0,₂𝑀₂𝐿₂−(1−𝜏)𝑄≤0,𝑀₃−1−𝜏₁𝑈≤0,𝐿₂𝑀₂𝐿₂−(1−𝜎)𝑄≤0,𝑀₃−1−𝜎₁𝑈≤0, (39) and the adaptive feedback controllers are designed as 𝑢_𝑖(𝑡)=−𝛾_𝑖+𝛼_𝑖(𝑡)𝑒_𝑖(𝑡),𝑣_𝑗(𝑡)=−𝜂_𝑗+𝛽_𝑗(𝑡)𝜀_𝑗(𝑡),𝛼_𝑖(𝑡)=𝐺𝑒_𝑖̇(𝑡)²𝑒_𝑖(𝑡)²𝛼,𝑒_𝑖(𝑡)²𝑒≠0,0,_𝑖(𝑡)²𝛽=0,_𝑗(𝑡)=𝐺𝜀_𝑗̇(𝑡)²𝜀_𝑗(𝑡)²𝛽,𝜀_𝑗(𝑡)²𝜀≠0,0,_𝑗(𝑡)²=0, (40)where 𝑍_𝑖=−2𝑃𝐷+𝐿₃𝐻𝑊𝐻𝐿₃+𝐿₁𝑀₁𝐿₁+𝑚𝑈−2𝛾_𝑖𝑃+𝑄,  𝑍_𝑗=−2𝑃𝐷+𝐿₃𝐻𝑊𝐻𝐿₃+𝐿₁𝑀₁𝐿₁+𝑙𝑈−2𝜂_𝑗𝑃+𝑄,𝑖=1,2,…,𝑙,𝑗=1,2,…,𝑚.

Remark 6. From Corollary 5, we can easily get that the controllers in this paper are simpler than those of Theorem 1 in [29].

Remark 7. If the coupling matrix of the SCBNN is not a diffusive matrix satisfying the sum of every row being zero, we can still obtain the same result from the proof of Theorem 4.
Theorem 8 presents another sufficient condition to ascertain that the two networks (4) and (5) can be synchronized, using the following simple adaptive feedback controllers: 𝑢_𝑖(𝑡)=−𝛾_𝑖𝑒_𝑖(𝑡),𝑣_𝑗(𝑡)=−𝛾_𝑗𝜀_𝑗(𝑡), (41) where 𝑖=1,2,…,𝑙,𝑗=1,2,…,𝑚, 𝛾_𝑖, and 𝛾_𝑗 are positive constants.
Let 𝑒(𝑡)=𝑒^𝑇₁(𝑡),𝑒^𝑇₂(𝑡),…,𝑒^𝑇_𝑙(𝑡)^𝑇,𝜀(𝑡)=𝜀^𝑇₁(𝑡),𝜀^𝑇₂(𝑡),…,𝜀^𝑇_𝑚(𝑡)^𝑇,𝐹_𝑘(𝑒(𝑡))=𝐹^𝑇_𝑘𝑒₁(𝑡),𝐹^𝑇_𝑘𝑒₂(𝑡),…,𝐹^𝑇_𝑘𝑒_𝑙(𝑡)^𝑇,𝐹𝑘=1,2,_𝑘(𝑒(𝑡))=𝐹^𝑇_𝑘𝜀₁(𝑡),𝐹^𝑇_𝑘𝜀₂(𝑡),…,𝐹^𝑇_𝑘𝜀_𝑚(𝑡)^𝑇,𝐺(𝜀̇(𝑡=𝐺𝑘=1,2,))^𝑇𝜀₁̇(𝑡),𝐺^𝑇𝜀₂̇(𝑡),…,𝐺^𝑇𝜀_𝑚̇(𝑡)^𝑇,𝐺(𝑒̇(𝑡))=𝐺^𝑇𝑒₁̇(𝑡),𝐺^𝑇𝑒₂̇(𝑡),…,𝐺^𝑇𝑒_𝑙̇(𝑡)^𝑇,𝐾(𝜀(𝑡))=𝐾^𝑇𝜀₁(𝑡),𝐾^𝑇𝜀₂(𝑡),…,𝐾^𝑇𝜀_𝑚(𝑡)^𝑇,𝐾(𝑒(𝑡))=𝐾^𝑇𝑒₁(𝑡),𝐾^𝑇𝑒₂(𝑡),…,𝐾^𝑇𝑒_𝑙(𝑡)^𝑇,Γ=−diag𝛾₁,𝛾₂,…,𝛾_𝑙,Γ=−diag𝛾₁,𝛾₂,…,𝛾_𝑚, (42) then the error dynamical system of (6) becomes 𝑒̇(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×−𝐼𝑡,𝜆_𝑙⊗𝐷_𝑟𝑒(𝑡)+𝐼_𝑙⊗𝑅_𝑟1𝐹₁(𝑒(𝑡+𝐼))_𝑙⊗𝑅_𝑟2𝐹₂(𝑒((𝑡+𝐴𝑡−𝜏)))_𝑟⊗𝐼_𝑛𝜀𝑡−𝜏₁(𝑡)+𝐵_𝑟⊗𝐼_𝑛𝐺𝜀̇𝑡−𝜏₂(𝑡)+𝐶_𝑟⊗𝐼_𝑛^𝑡_−∞𝐼_𝑚()𝐾(𝜀(𝑠+⊗ℎ𝑡−𝑠))𝑑𝑠Γ⊗𝐼_𝑛𝑒(𝑡,𝜀̇(𝑡)=_𝑁𝑟=1)𝜉_𝑟()×−𝐼𝑡,𝜆_𝑚⊗𝐷𝜀(𝑡)+𝐼_𝑚⊗𝑅_𝑟1𝐹₁(𝜀(𝑡+𝐼))_𝑚⊗𝑅_𝑟2𝐹₂(𝜀((𝑡+𝑡−𝜎)))𝐴_𝑟⊗𝐼_𝑛𝑒𝑡−𝜎₁(𝑡)+𝐵_𝑟⊗𝐼_𝑛𝐺𝑒̇𝑡−𝜎₂(𝑡)+𝐶_𝑟⊗𝐼_𝑛^𝑡_−∞𝐼_𝑙()𝐾(𝑒(𝑠+⊗ℎ𝑡−𝑠))𝑑𝑠Γ⊗𝐼_𝑛𝜀(𝑡), (43)

Theorem 8. Under assumptions (𝑆₁)--(𝑆₄) and using the adaptive feedback controllers (41), the two coupled SCBNNs (4) and (5) can be synchronized, if there exist 𝑛×𝑛 positive matrices 𝑃,𝑈,𝑃,𝑈 and 𝑛×𝑛 diagonal positive matrices 𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛), 𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛), 𝑄, 𝑉, 𝑀, 𝑄, 𝑉, 𝑀 such that for 𝑟∈ℵ, the following matrix inequalities hold:Ω_𝑟=Ψ_𝑟1Ψ_𝑟2Ψ_𝑟3Ψ_𝑟4Ψ_𝑟5Ψ_𝑟6∗Ψ_𝑟7𝐼_𝑙⊗𝑅^𝑇_𝑟1𝑉𝑅_𝑟2𝐴_𝑟⊗𝑅^𝑇_𝑟1𝑉𝐵_𝑟⊗𝑅^𝑇_𝑟1𝑉𝐶_𝑟⊗𝑅^𝑇_𝑟1𝑉∗∗Ψ_𝑟8𝐴_𝑟⊗𝑅^𝑇_𝑟2𝑉𝐵_𝑟⊗𝑅^𝑇_𝑟2𝑉𝐶_𝑟⊗𝑅^𝑇_𝑟2𝑉∗∗∗Ψ_𝑟9𝐴^𝑇_𝑟𝐵_𝑟⊗𝑉𝐴^𝑇_𝑟𝐶_𝑟⊗𝑉∗∗∗∗Ψ_𝑟10𝐵^𝑇_𝑟𝐶_𝑟⊗𝑉∗∗∗∗∗Ψ_𝑟11<0,Ω_𝑟=Ψ_𝑟1Ψ_𝑟2Ψ_𝑟3Ψ_𝑟4Ψ_𝑟5Ψ_𝑟6∗Ψ_𝑟7𝐼_𝑙⊗𝑅^𝑇_𝑟1𝑉𝑅_𝑟2𝐴_𝑟⊗𝑅^𝑇_𝑟1𝑉𝐵_𝑟⊗𝑅^𝑇_𝑟1𝑉𝐶_𝑟⊗𝑅^𝑇_𝑟1𝑉∗∗Ψ_𝑟8𝐴_𝑟⊗𝑅^𝑇_𝑟2𝑉𝐵_𝑟⊗𝑅^𝑇_𝑟2𝑉𝐶_𝑟⊗𝑅^𝑇_𝑟2𝑉∗∗∗Ψ_𝑟9𝐵_𝑟⊗𝑉𝐴^𝑇_𝑟𝐶_𝑟⊗𝑉∗∗∗∗Ψ_𝑟10𝐵^𝑇_𝑟𝐶_𝑟⊗𝑉∗∗∗∗∗Ψ_𝑟11<0, (44)with Ψ_𝑟1=𝐼_𝑙⊗−𝑃𝐷_𝑟−𝐷^𝑇_𝑟𝑃+𝑄+𝑈+𝐿₃𝐻𝑊𝐻𝐿₃+𝐿₁𝑀𝐿₁+𝐷^𝑇_𝑟𝑉𝐷_𝑟+2Γ⊗𝑃+Γ²⊗𝑉−Γ⊗𝐷^𝑇_𝑟𝑉+𝑉𝐷_𝑟,Ψ_𝑟2=Γ⊗𝑉𝑅_𝑟1+𝐼_𝑙⊗𝑃𝑅_𝑟1−𝐷^𝑇_𝑟𝑉𝑅_𝑟1,Ψ_𝑟3=𝐼_𝑙⊗𝑃𝑅_𝑟2−𝐷^𝑇_𝑟𝑉𝑅_𝑟2+Γ⊗𝑉𝑅_𝑟2,Ψ_𝑟4=Γ𝐴_𝑟⊗𝑉+𝐴_𝑟⊗𝑃−𝐷^𝑇_𝑟𝑉,Ψ_𝑟5=Γ𝐵_𝑟⊗𝑉+𝐵_𝑟⊗𝑃−𝐷^𝑇_𝑟𝑉,Ψ_𝑟6=Γ𝐶_𝑟⊗𝑉+𝐶_𝑟⊗𝑃−𝐷^𝑇_𝑟𝑉,Ψ_𝑟7=𝐼_𝑙⊗𝑅^𝑇_𝑟1𝑉𝑅_𝑟1,Ψ−𝑀_𝑟8=𝐼_𝑙⊗𝑅^𝑇_𝑟2𝑉𝑅_𝑟2−(1−𝜏)𝐿₂⁻¹𝑄𝐿₂⁻¹,Ψ_𝑟9=𝐴^𝑇_𝑟𝐴_𝑟⊗𝑉−1−𝜏₁𝐼_𝑚,Ψ⊗𝑈_𝑟10=𝐵^𝑇_𝑟𝐵_𝑟⊗𝑉−1−𝜏₂𝐼_𝑚⊗𝐿₄⁻¹𝑉𝐿₄⁻¹,Ψ_𝑟11=𝐶^𝑇_𝑟𝐶_𝑟⊗𝑉−𝐼_𝑚⊗𝑊,Ψ_𝑟1=𝐼_𝑚⊗−𝑃𝐷_𝑟−𝐷^𝑇_𝑟𝑃+𝑄+𝑈+𝐿₃𝐻𝑊𝐻𝐿₃+𝐿₁𝑀𝐿₁+𝐷^𝑇_𝑟𝑉𝐷_𝑟+2Γ⊗𝑃+Γ²⊗𝑉−Γ⊗𝐷^𝑇_𝑟𝑉+𝑉𝐷_𝑟,Ψ_𝑟2=Γ⊗𝑉𝑅_𝑟1+𝐼_𝑚⊗𝑃𝑅_𝑟1−𝐷^𝑇_𝑟𝑉𝑅_𝑟1,Ψ₃=𝐼_𝑚⊗𝑃𝑅_𝑟2−𝐷^𝑇_𝑟𝑉𝑅_𝑟2+Γ⊗𝑉𝑅_𝑟2,Ψ_𝑟4=Γ𝐴_𝑟⊗𝑉+𝐴_𝑟⊗𝑃−𝐷^𝑇_𝑟𝑉,Ψ₅=Γ𝐵_𝑟⊗𝑉+𝐵_𝑟⊗𝑃−𝐷^𝑇_𝑟𝑉,Ψ_𝑟6=Γ𝐶_𝑟⊗𝑉+𝐶_𝑟⊗𝑃−𝐷^𝑇_𝑟𝑉,Ψ_𝑟7=𝐼_𝑚⊗𝑅^𝑇_𝑟1𝑉𝑅_𝑟1−𝑀,Ψ₈=𝐼_𝑚⊗𝑅^𝑇_𝑟2𝑉𝑅_𝑟2−(1−𝜎)𝐿₂⁻¹𝑄𝐿₂⁻¹,Ψ_𝑟9=𝐴^𝑇_𝑟𝐴_𝑟⊗𝑉−1−𝜎₁𝐼_𝑙⊗𝑈,Ψ_𝑟10=𝐵_𝑟^𝑇𝐵_𝑟⊗𝑉−1−𝜎₂𝐼_𝑙⊗𝐿₄⁻¹𝑉𝐿₄,Ψ_𝑟11=𝐶^𝑇_𝑟𝐶_𝑟⊗𝑉−𝐼_𝑙⊗𝑊. (45)

Proof. For the error dynamical system (43), we define the following Lyapunov-Krasovskii function: 𝑉(𝑡,𝑒(𝑡),𝜀(𝑡))=𝑉₁(𝑡,𝑒(𝑡),𝜀(𝑡))+𝑉₂((𝑡)(𝑡,𝑉𝑡,𝑒,𝜀))₁=𝑒^𝑇(𝑡)𝐼_𝑙⊗𝑃𝑒(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑒^𝑇(𝑠)𝐼_𝑙𝑒(𝑠)+⊗𝑄d𝑠_{𝑡𝑡−𝜏1(𝑡)}𝜀^𝑇(𝑠)𝐼_𝑚𝜀(𝑠)+⊗𝑈d𝑠_{𝑚𝑛𝑗=1𝑖=1}𝑤_𝑖^∞₀ℎ_𝑖(𝑣)d𝑣^∞₀ℎ_𝑖(𝜃)×^𝑡_𝑡−𝜃𝐾²_𝑖𝜀_𝑗𝑖(𝑠)+d𝑠d𝜃_{𝑡𝑡−𝜏2(𝑡)}𝜀^𝑇̇(𝑠)𝐼_𝑚𝜀̇(𝑠)𝑉⊗𝑉d𝑠,₂=𝜀^𝑇(𝑡)𝐼_𝑚⊗𝑃𝜀(𝑡)+^𝑡_{𝑡−𝜎(𝑡)}𝜀^𝑇(𝑠)𝐼_𝑚⊗𝑄𝜀(𝑠)+d𝑠_{𝑡𝑡−𝜎1(𝑡)}𝑒^𝑇(𝑠)𝐼_𝑙⊗𝑈𝑒(𝑠)+d𝑠_{𝑙𝑛𝑖=1𝑗=1}𝑤_𝑗^∞₀ℎ_𝑗(𝑣)d𝑣^∞₀ℎ_𝑗(𝜃)×^𝑡_𝑡−𝜃𝐾_𝑗𝑒_𝑖𝑗(𝑠)+d𝑠d𝜃_{𝑡𝑡−𝜎2(𝑡)}𝑒^𝑇̇(𝑠)𝐼_𝑙⊗𝑉𝑒̇(𝑠)𝑉̇d𝑠,₁=𝑒^𝑇(𝑡)𝐼_𝑙⊗𝑃𝑒̇(𝑡)+𝑒^𝑇̇(𝑡)𝐼_𝑙⊗𝑃𝑒(𝑡)+𝑒^𝑇(𝑡)𝐼_𝑙⊗𝑄𝑒(𝑡)−(1−𝜏̇(𝑡))𝑒^𝑇((𝑡×𝐼𝑡−𝜏))_𝑙⊗𝑄𝑒(𝑡−𝜏(𝑡))+𝜀^𝑇(𝑡)𝐼_𝑚𝜀(𝑡)−𝜏̇⊗𝑈1−₁(𝑡)𝜀^𝑇𝑡−𝜏₁(𝑡)𝐼_𝑚⊗𝑈𝜀𝑡−𝜏₁(𝑡)+^𝑚𝑛_𝑗=1𝑖=1𝑤_𝑖𝐾_𝑖𝜀_𝑗𝑖(𝑡)^∞₀ℎ_𝑖(𝑣)d𝑣²−^𝑚𝑛_𝑗=1𝑖=1𝑤_𝑖^∞₀ℎ_𝑖(𝑣)d𝑣^∞₀ℎ_𝑖(𝜃)𝐾²_𝑖𝜀_𝑗𝑖()+𝜀̇𝑡−𝜃d𝜃_𝑇(𝑡)𝐼_𝑚𝜀̇(𝑡)−𝜏̇⊗𝑉1−₂(𝑡)𝜀^𝑇̇𝑡−𝜏₂(𝑡)𝐼_𝑚⊗𝑉𝜀̇𝑡−𝜏₂(𝑡),𝑉̇₂=𝜀^𝑇(𝑡)𝐼_𝑚⊗𝑃𝜀̇(𝑡)+𝜀^𝑇̇(𝑡)𝐼_𝑚⊗𝑃𝜀(𝑡)+𝜀(𝑡)𝐼_𝑚⊗𝑄𝜀(𝑡)−(𝜎̇(𝑡𝜀((𝑡×𝐼1−))𝑡−𝜎))_𝑚⊗𝑄𝜀(𝑡−𝜎(𝑡))+𝑒^𝑇(𝑡)𝐼_𝑙⊗𝑈𝑒(𝑡)−1−𝜎̇₁(𝑡)𝑒^𝑇𝑡−𝜎₁(𝑡)𝐼_𝑙⊗𝑈𝑒𝑡−𝜎₁(𝑡)+^𝑙𝑛_𝑖=1𝑗=1𝑤_𝑗𝐾_𝑗(𝑒_𝑖𝑗(𝑡))^∞₀ℎ_𝑗(𝑣)d𝑣²−^𝑙𝑛_𝑖=1𝑗=1𝑤_𝑗^∞₀ℎ_𝑗(𝑣)d𝑣^∞₀ℎ_𝑗(𝜃)𝐾²_𝑗𝑒_𝑖𝑗()+𝑒̇𝑡−𝜃d𝜃_𝑇(𝑡)𝐼_𝑙⊗𝑉𝑒̇(𝑡)−1−𝜎̇₂(𝑡)𝑒^𝑇̇𝑡−𝜎₂(𝑡)×𝐼_𝑙⊗𝑉𝑒̇𝑡−𝜎₂(𝑡). (46) By (26), we have ^𝑚𝑛_𝑗=1𝑖=1𝑤_𝑖𝐾_𝑖𝜀_𝑗𝑖(𝑡)^∞₀ℎ_𝑖(𝑣)d𝑣²≤^𝑚_𝑗=1𝜀^𝑇_𝑗(𝑡)𝐿₃𝐻𝑊𝐻𝐿₃𝜀_𝑗(𝑡)=𝜀^𝑇(𝑡)𝐼_𝑚⊗𝐿₃𝐻𝑊𝐻𝐿₃𝜀(𝑡). (47) Using (30), we get ^𝑚𝑛_𝑗=1𝑖=1𝑤_𝑖^∞₀ℎ_𝑖(𝑣)d𝑣^∞₀ℎ_𝑖(𝜃)𝐾²_𝑖𝜀_𝑗𝑖(𝑠≥)(𝑡−𝜃)d𝜃_{𝑚𝑗=1𝑡−∞}ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)d𝑠^𝑇×𝑊^𝑡_−∞ℎ(𝑡−𝑠)𝐾𝜀_𝑗(𝑠)=d𝑠_𝑡−∞𝐼_𝑚⊗ℎ(𝑡−𝑠)𝐾(𝜀(𝑠))d𝑠^𝑇𝐼_𝑚×⊗𝑊_𝑡−∞𝐼_𝑚⊗ℎ(𝑡−𝑠)𝐾(𝜀(𝑠))d𝑠. (48) From (𝑆₃) and (46)–(48), we have 𝑉̇₁≤𝑒^𝑇(𝑡)𝐼_𝑙⊗𝑃𝑒̇(𝑡)+𝑒^𝑇̇(𝑡)𝐼_𝑙⊗𝑃𝑒(𝑡)+𝑒^𝑇(𝑡)𝐼_𝑙⊗𝑄𝑒(𝑡)−(1−𝜏)𝑒^𝑇((𝑡×𝐼𝑡−𝜏))_𝑙⊗𝑄𝑒(𝑡−𝜏(𝑡))+𝜀^𝑇(𝑡)𝐼_𝑚⊗𝑈+𝐿₃𝐻𝑊𝐻𝐿₃𝜀(𝑡)−1−𝜏₁𝜀^𝑇𝑡−𝜏₁(𝑡)𝐼_𝑚⊗𝑈𝜀𝑡−𝜏₁(𝑡)−^𝑡_−∞𝐼_𝑚⊗ℎ(𝑡−𝑠)𝐾(𝜀(𝑠))d𝑠^𝑇𝐼_𝑚×⊗𝑊_𝑡−∞𝐼_𝑚(𝐾(𝜀(𝑠+𝜀̇⊗ℎ𝑡−𝑠)))d𝑠_𝑇(𝑡)𝐼_𝑚⊗𝑉𝜀̇(𝑡)−1−𝜏₂𝜀^𝑇̇𝑡−𝜏₂(𝑡)×𝐼_𝑚⊗𝑉𝜀̇𝑡−𝜏₂(𝑡). (49) In the same way, we have 𝑉̇₂≤𝜀^𝑇(𝑡)𝐼_𝑚⊗𝑃𝜀̇(𝑡)+𝜀^𝑇̇(𝑡)𝐼_𝑙⊗𝑃𝜀(𝑡)+𝜀^𝑇(𝑡)𝐼_𝑚⊗𝑄𝜀(𝑡)−(1−𝜎)𝜀^𝑇((𝑡×𝐼𝑡−𝜎))_𝑚⊗𝑄𝜀(𝑡−𝜎(𝑡))+𝑒^𝑇(𝑡)𝐼_𝑙⊗𝑈+𝐿₃𝐻𝑊𝐻𝐿₃𝑒(𝑡)−1−𝜎₁𝑒^𝑇𝑡−𝜎₁(𝑡)𝐼_𝑙⊗𝑈𝑒𝑡−𝜎₁(𝑡)−^𝑡_−∞𝐼_𝑙⊗ℎ(𝑡−𝑠)𝐾(𝑒(𝑠))d𝑠^𝑇𝐼_𝑙⊗𝑊×^𝑡_−∞𝐼_𝑙⊗ℎ()𝐾(𝑒(𝑠+𝑒̇𝑡−𝑠))d𝑠_𝑇(𝑡)𝐼_𝑙⊗𝑉𝑒̇(𝑡)−1−𝜎₂𝑒^𝑇̇𝑡−𝜎₂(𝑡×𝐼_𝑙)⊗𝑉𝑒̇𝑡−𝜎₂(𝑡). (50) From (𝑆₁) and (𝑆₂), 𝑒^𝑇(𝑡)𝐼_𝑙⊗𝐿₁𝑀𝐿₁𝑒(𝑡)−𝐹^𝑇₁(𝑒(𝑡))𝐼_𝑙⊗𝑀𝐹₁(𝑒(𝑡𝜀))≥0,_𝑇(𝑡)𝐼_𝑚⊗𝐿₁𝑀𝐿₁𝜀(𝑡)−𝐹^𝑇₁(𝜀(𝑡))𝐼_𝑚⊗𝑀𝐹₁(𝜀(𝑡𝑒))≥0,_𝑇(𝑡−𝜏(𝑡))𝐼_𝑙𝑒((𝑡≥𝐹⊗𝑄𝑡−𝜏))_𝑇2(𝑒(𝑡−𝜏(𝑡)))𝐼_𝑙⊗𝐿₂⁻¹𝑄𝐿₂⁻¹𝐹₂(𝑒((𝑡,𝜀𝑡−𝜏)))_𝑇(𝑡−𝜎(𝑡))𝐼_𝑚⊗𝑄𝜀((𝑡≥𝐹𝑡−𝜎))_𝑇2(𝜀(𝑡−𝜎(𝑡)))𝐼_𝑚⊗𝐿₂⁻¹𝑄𝐿₂⁻¹𝐹₂(𝜀((𝑡,𝜀̇𝑡−𝜎)))_𝑇𝑡−𝜏₂(𝑡)𝐼_𝑚⊗𝑉𝜀̇𝑡−𝜏₂(𝑡)≥𝐺^𝑇𝜀̇𝑡−𝜏₂(𝑡)𝐼_𝑚⊗𝐿₄⁻¹𝑉𝐿₄⁻¹𝐺𝜀̇𝑡−𝜏₂(𝑡,𝑒̇_𝑇)𝑡−𝜎₂(𝑡)𝐼_𝑙⊗𝑉𝑒̇𝑡−𝜎₂(𝑡)≥𝐺^𝑇𝑒̇𝑡−𝜎₂(𝑡)𝐼_𝑙⊗𝐿₄⁻¹𝑉𝐿₄⁻¹𝐺𝑒̇𝑡−𝜎₂(𝑡). (51) With the aid of (43) and (51), we have 𝑉̇≤^𝑁_𝑟=1𝜉_𝑟(𝑡,𝜆)𝜂^𝑇(𝑡)Ω_𝑟𝜂(𝑡)+𝜂^𝑇(𝑡)Ω_𝑟𝜂(𝑡), (52) where 𝜂(𝑡)=𝑒^𝑇(𝑡),𝐹^𝑇₁(𝑒(𝑡)),𝐹^𝑇₂(𝑒(𝑡−𝜏(𝑡))),𝜀^𝑇𝑡−𝜏₁(𝑡),𝐺^𝑇𝜀̇𝑡−𝜏₂(𝑡),_𝑡−∞𝐼_𝑚⊗ℎ(𝑡−𝑠)K(𝜀(𝑠))d𝑠^𝑇𝑇,𝜂(𝑡)=𝜀^𝑇(𝑡),𝐹^𝑇₁(𝜀(𝑡)),𝐹^𝑇₂(𝜀(𝑡−𝜎(𝑡))),𝑒^𝑇𝑡−𝜎₁(𝑡),𝐺^𝑇𝑒̇𝑡−𝜎₂(𝑡),_𝑡−∞𝐼_𝑙⊗ℎ(𝑡−𝑠)K(𝑒(𝑠))d𝑠^𝑇𝑇. (53) Let𝜌=min{𝜌₁,𝜌₂}, where 𝜌₁=−min{𝜌_min(Ω_𝑟),𝑟∈ℵ}, 𝜌₂=−min{𝜌_min(Ω_𝑟),𝑟∈ℵ}, then 𝜌>0 and 𝑉̇≤−𝜌₁^𝑙_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−𝜌₂^𝑚_𝑗=1𝜀^𝑇_𝑗(𝑡)𝜀_𝑗(𝑡)≤−𝜌^𝑙_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+^𝑚_𝑗=1𝜀^𝑇_𝑗(𝑡)𝜀_𝑗(𝑡). (54) The following proof is similar to that of Theorem 4 and is omitted here.

4. Simulations

In this section, numerical examples are provided to demonstrate the validity of the synchronization criteria obtained in the previous sections. Consider the following network as drive network: 𝑥̇_𝑖(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆−𝐷_𝑟𝑥_𝑖+𝑅_𝑟1𝑓₁𝑥_𝑖(𝑡)+𝑅_𝑟2𝑓₂𝑥_𝑖(𝑡−𝜏(𝑡))+𝐼_𝑟+^𝑚_𝑗=1𝑎_𝑟𝑖𝑗𝑦_𝑗𝑡−𝜏₁(𝑡)+^𝑚_𝑗=1𝑏_𝑟𝑖𝑗𝑔𝑦̇_𝑗𝑡−𝜏₂(𝑡)+^𝑚_𝑗=1𝑐_𝑟𝑖𝑗^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑦_𝑗(𝑠),𝑦̇d𝑠𝑖=12,…,𝑙,_𝑗(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×−𝑡,𝜆𝐷_𝑟𝑦_𝑗+𝑅_𝑟1𝑓₁𝑦_𝑗(𝑡)+𝑅_𝑟2𝑓₂𝑦_𝑗(𝑡−𝜎(𝑡))+𝐽_𝑟+^𝑙_𝑖=1𝑎_𝑟𝑗𝑖𝑥_𝑖𝑡−𝜎₁(𝑡)+^𝑙_𝑖=1𝑏_𝑟𝑗𝑖𝑔𝑥̇_𝑖𝑡−𝜎₂(𝑡)+^𝑙_𝑖=1𝑐_𝑟𝑗𝑖^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑥_𝑖(𝑠)d𝑠,𝑗=1,2,…,𝑚, (55) where 𝑥_𝑖(𝑡),𝑦_𝑗(𝑡)∈R², 𝑙=3, and 𝑚=3. 𝑓₁(𝑧(𝑡))=0.1(tanh(𝑧₁(𝑡)),tanh(𝑧₂(𝑡)))^𝑇, 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡))^𝑇, 𝑓₁=𝑔=𝑔=𝑘=𝑘=𝑓₂=𝑓₂=𝑓₁, and ℎ(𝑡)=ℎ(𝑡)=diag(𝑒^−𝑡,𝑒^−𝑡). Choose time delays 𝜏(𝑡)=1+0.4sin𝑡,𝜏₁(𝑡)=2+0.2arctan(𝑡), 𝜏₂(𝑡)=0.6+0.5cos𝑡, and 𝜎(𝑡)=1+0.8sin𝑡, 𝜎₁(𝑡)=0.7+0.1cos𝑡, 𝜎₂(𝑡)=0.5+(0.3𝑒^𝑡/(1+𝑒^𝑡)). We define a switching rule 𝜆∶𝑡∈[0,+∞)→{1,2},𝜆(𝑡)=int(𝑡)mod2+1. The other parameters are as follows: 𝐷₁=1.8004,𝑅₁₁=,𝑅−1100.2₁₂=10.50.6−1,𝐼₁=𝐽₁=(1,2)^𝑇,𝐴₁=𝑎_1𝑖𝑗=,𝐵−2−2002−211−2₁=𝑏_1𝑖𝑗=,𝐶−0.20−0.20.1−0.40.30.20.1−0.3₁=𝑐_1𝑖𝑗=,1−1011−2−101𝐷₁=,2001𝑅₁₁=,−0.310.20.3𝑅₁₂=,0.30.40.6−0.5𝐴₁=𝑎_1𝑗𝑖=,40−411−210−1𝐵₁=𝑏_1𝑗𝑖=,0.10−0.10.2−0.30.10.10.2−0.2𝐶₁=𝑐_1𝑗𝑖=,𝐷−31220−2−404₂=1.4001.4,𝑅₂₁=,𝑅1−1−53₂₂=−1.5−0.1−3−1,𝐴₂=𝑎_2𝑖𝑗=,𝐵−11002−21.21−2.2₂=𝑏_2𝑖𝑗=,𝐶0.1−0.100.1−0.50.40.20−0.2₂=𝑐_2𝑖𝑗=,2−203−1−20−11𝐷₂=,1.2001.2𝑅₂₁=,−0.31−41𝑅₂₂=0.30.4−2−1,𝐼₂=𝐽₂=(3,4)^𝑇,𝐴₂=𝑎_2𝑗𝑖=,1−1012−311−2𝐵₂=𝑏_2𝑗𝑖=,0.2−0.200.1−0.20.10.30.1−0.4𝐶₂=𝑐_2𝑗𝑖=.−51411−2−101 (56) The response network of drive network (55) is 𝑥̇_𝑖(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×𝑡,𝜆−𝐷_𝑟𝑥_𝑖(𝑡)+𝑅_𝑟1𝑓₁𝑥_𝑖(𝑡)+𝑅_𝑟2𝑓₂𝑥_𝑖(𝑡−𝜏(𝑡))+𝐼_𝑟+^𝑚_𝑗=1𝑎_𝑟𝑖𝑗𝑦_𝑗𝑡−𝜏₁(𝑡)+^𝑚_𝑗=1𝑏_𝑟𝑖𝑗𝑔𝑦̇_𝑗𝑡−𝜏₂(𝑡)+^𝑚_𝑗=1𝑐_𝑟𝑖𝑗^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑦_𝑗(𝑠)d𝑠+𝑢_𝑖(𝑡),𝑦̇_𝑗(𝑡)=^𝑁_𝑟=1𝜉_𝑟()×−𝑡,𝜆𝐷_𝑟𝑦_𝑗(𝑡)+𝑅_𝑟1𝑓₁𝑦_𝑗(𝑡)+𝑅_𝑟2𝑓₂𝑦_𝑗(𝑡−𝜎(𝑡))+𝐽_𝑟+^𝑙_𝑖=1𝑎_𝑟𝑗𝑖𝑥_𝑖𝑡−𝜎₁(𝑡)+^𝑙_𝑖=1𝑏_𝑟𝑗𝑖𝑔𝑥̇_𝑖𝑡−𝜎₂(𝑡)+^𝑙_𝑖=1𝑐_𝑟𝑗𝑖^𝑡_−∞ℎ(𝑡−𝑠)𝑘𝑥_𝑖(𝑠)d𝑠+𝑣_𝑗(𝑡), (57) where 𝑢_𝑖(𝑡),𝑣_𝑗(𝑡)∈R².

Let 𝛾₁=𝛾₂=𝛾₃=15, 𝜂₁=𝜂₂=𝜂₃=16, 𝛼=0.5, 𝛽=0.5, and the feasible solution of the matrix inequalities (15)–(19) by employing MATLAB LMI Toolbox be as follows: 𝑝=7.5267,,,,𝑀𝑝=7.8951,𝑃=12.95080.28170.281710.7871,𝑄=83.64053.66743.667468.8593𝑈=119.10800.22160.221655.7869𝑊=233.787000223.7801₁=,𝑀227.115300202.1669₂=,𝑀206.303400217.8341₃=,86.97230041.4980𝑃=,19.45480.03360.033611.6060𝑄=,26.19730.54110.54119.6902𝑈=,30.91260.98500.985022.6634𝑊=,316.694300287.9793𝑀₁=,183.041900162.4080𝑀₂=,250.278700409.0398𝑀₃=.13.87040010.1694 (58) The initial values are chosen as 𝑥_𝑖(𝑠)=(−5,9), 𝑦_𝑗(𝑠)=(−6,7)^𝑇, 𝑥_𝑖(𝑠)=2𝑖(2,5)^𝑇, 𝑦_𝑗(𝑠)=3𝑗(2,−1)^𝑇, and 𝑠∈[−2,0]. Clearly, the two coupled networks (55) and (57) satisfy the conditions of Theorem 4. Figure 1 presents the synchronization errors of the state variables between the two networks. The simulation result shows that the synchronization is achieved under the proposed controllers (20). Thus, the proposed synchronization control scheme in Theorem 4 is valid.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

Synchronization errors of BDN (55) and (57) with adaptive feedback controllers (20).

Let 𝛾₁=𝛾₂=𝛾₃=12, 𝛾₁=𝛾₂=𝛾₃=17, then the feasible solution of the matrix inequalities (44) in Theorem 8 by employing MATLAB LMI Toolbox is as follows: ,,,,𝑃=0.02510.00050.00050.0214,𝑈=3.7826−0.0043−0.00433.5381𝑄=4.2792003.7789𝑉=0.0116000.0079,𝑀=3.4521003.1482𝑊=3.7071003.3433𝑃=,0.0298000.0268𝑈=,3.3712−0.0132−0.01323.1782𝑄=,7.2228006.3479𝑉=,0.0077000.0052𝑀=,3.3924003.0142𝑊=.3.4120003.2474 (59) Using the controllers (41), the simulation result is given in Figure 2, which shows that the proposed synchronization control scheme in Theorem 8 is effective.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Synchronization errors with adaptive feedback controllers (41).

5. Conclusions

In this paper, we have proposed a general SCBNN with distributed delays and derivative coupling and investigated the synchronization problem in the two coupled SCBNNs. Using linear matrix inequality (LMI) approach and Barbǎlat lemma, we have deviated some useful synchronization criteria to ensure the synchronization of these two SCBNNs by constructing effective controllers. Compared with relative previous jobs, the controllers proposed by us are more simple and feasible. Some simulation results have been presented to demonstrate our theoretical results. In our future work, we will consider using pinning control to realize the synchronization of SCBNNs and identify the network topology of the unknown SCBNNs.

Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grants nos. 61272530 and 11072059, the Natural Science Foundation of Jiangsu Province of China under Grants no. BK2012741, Specialized Research Fund for the Doctoral Program of Higher Education under Grants no. 20110092110017, and the National Social Science Fund of China under Grant no. 11BGL039.