Delay-Dependent Synchronization for Complex Dynamical Networks with Interval Time-Varying and Switched Coupling Delays

Abstract

We investigate the local exponential synchronization for complex dynamical networks with interval time-varying delays in the dynamical nodes and the switched coupling term simultaneously. The constraint on the derivative of the time-varying delay is not required which allows the time delay to be a fast time-varying function. By using common unitary matrix for different subnetworks, the problem of synchronization is transformed into the stability analysis of some linear switched delay systems. Then, when subnetworks are synchronizable and nonsynchronizable, a delay-dependent sufficient condition is derived and formulated in the form of linear matrix inequalities (LMIs) by average dwell time approach and piecewise Lyapunov-Krasovskii functionals which are constructed based on the descriptor model of the system and the method of decomposition. The new stability condition is less conservative and is more general than some existing results. A numerical example is also given to illustrate the effectiveness of the proposed method.

1. Introduction

Complex dynamical network, as an interesting subject, has been thoroughly investigated for decades. These networks show very complicated behavior and can be used to model and explain many complex systems in nature such as computer networks [1], the World Wide Web [2], food webs [3], cellular and metabolic networks [4], social networks [5], and electrical power grids [6]. In general, a complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. As an implicit assumption, these networks are described by the mathematical term graph. In such graphs, each vertex represents an individual element in the system, while edges represent the relationship between them. Two nodes are joined by an edge if and only if they interact.

In the last decade, the synchronization of complex dynamical networks has attracted much attention by several researchers in this field [7–12]. Synchronization of complex dynamical networks is one of the most important dynamical mechanisms for creating order in complex dynamical networks. Wang and Chen introduced a uniform dynamical network model and also investigated its synchronization [10, 11, 13]. They have shown that the synchronizability of a scale-free dynamical network is robust against random removal of nodes and yet is fragile to specific removal of the most highly connected nodes [11]. Li and Chen [7] considered the synchronization of complex dynamical network models with coupling delays (continuous and discrete-time) and derived delay-independent and delay-dependent synchronization conditions. By utilizing Lyapunov functional method, Yue and Li [12] studied the synchronization of continuous and discrete complex dynamical networks with interval time-varying delays in the dynamical nodes and the coupling term simultaneously in which delay-dependent synchronization conditions are derived in the form of linear matrix inequalities (LMIs). In [8], Li et al. studied the synchronization problem of general complex dynamical networks with time-varying delays in the network couplings and in the dynamical nodes, respectively. However, the time-varying delays are required to be differentiable.

It is well known that the existence of time delay in a system may cause instability and oscillations. Examples of time-delay systems are chemical engineering systems, biological modeling, electrical networks, physical networks, and many others, [14–20]. The stability criteria for systems with time delays can be classified into two categories: delay-independent and delay-dependent. Delay-independent criteria do not employ any information on the size of the delay, while delay-dependent criteria make use of such information at different levels. Delay-dependent stability conditions are generally less conservative than the delay-independent ones especially when the delay is small [20]. Recently, delay-dependent stability for interval time-varying delay was investigated in [12, 15–17, 19]. Interval time-varying delay is a time delay that varies in an interval in which the lower bound is not restricted to be 0. Jiang and Han [17] considered the problem of robust 𝐻∞ control for uncertain linear systems with interval time-varying delay based on the Lyapunov functional approach in which restriction on the differentiability of the interval time-varying delay was removed. Shao [19] presented a new delay-dependent stability criterion for linear systems with interval time-varying delay without introducing any free-weighting matrices. In order to reduce further the conservatism introduced by descriptor model transformation and bounding techniques, a free-weighting matrices method is proposed in [15, 21–24]. An advantage of this transformation is to transform the original system to an equivalent descriptor representation without introducing additional dynamics in the sense defined in [21]. In [22], based on the descriptor model transformation and decomposition technique of coefficient matrix, some new robust stability criteria are derived. In [12], the synchronization problem has been investigated for continuous/discrete complex dynamical networks with interval time-varying delays. Based on piecewise analysis method and the Lyapunov functional method, some new delay-dependent synchronization criteria are derived in the form of LMIs by introducing free-weighting matrices. It will be pointed out later that some existing results require more free-weighting matrix variables than our result.

Switched systems are special class of hybrid systems which consist of a family of continuous-time or discrete-time subsystems and a switching law that orchestrates the switching between them. Such systems have drawn considerable attention in control and computer communities in the last decade [25–31]. In [32], some stability properties of switched linear systems consisting of both stable and unstable subsystems have been derived by using an average dwell time approach and piecewise Lyapunov functions. It was shown that when the average dwell time is sufficiently large and the total activation time of the unstable subsystems is relatively small compared with that of stable subsystems, then global exponential stability is guaranteed. Liu and Zhao [9] investigated synchronization of complex delayed networks with switching topology via switched system stability theory. In [9], when all subnetworks are synchronizable, a delay-dependent synchronization condition is given in terms of LMIs which guarantees the solvability of the synchronization problem under an average dwell time scheme. Liu et al. [33] studied the local and global exponential synchronization of a complex dynamical network with switching topology and time-varying coupling delays. However, the time-varying delays in the dynamical nodes were not considered.

In this paper, we will investigate the local exponential synchronization of complex dynamical networks with interval time-varying delays in both dynamical nodes and in switched coupling terms simultaneously. The restriction on the differentiability of interval time-varying delays is removed, which means that a fast time-varying delay is allowed. We use a common unitary matrix for different subnetworks to transform the synchronization problem of the network into the stability analysis of 𝑁−1 pieces of linear switched systems. Then by using the average dwell time approach, the descriptor model transformation, and the decomposition technique of coefficient matrix, a new class of piecewise Lyapunov-Krasovskii functionals are constructed in order to get improved delay-dependent synchronization criteria which are derived in the form of LMIs. Finally, numerical examples are given to demonstrate that the derived conditions are less conservative than some existing results given in the literature.

The rest of the paper is organized as follows. In Section 2, a complex dynamical network model with interval time-varying and switched coupling delays as well as some useful lemmas is given. In Section 3, common unitary matrix and descriptor system transformation strategy are introduced to reformulate the network model. Then delay-dependent synchronization criteria are presented based on this novel model. Numerical examples that illustrated the obtained results are given in Section 4. The paper ends with conclusions in Section 5 and cited references.

2. Network Model and Mathematic Preliminaries

The following notation will be used in this paper: 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space; 𝜆_min(⋅) and 𝜆_max(⋅) denote the minimum and maximum eigenvalue of a real symmetric matrix, respectively; ‖⋅‖ denotes the Euclidean vector norm of 𝑥; 𝐴^𝑇 denotes the transpose of the vector/matrix 𝐴; 𝐴 is symmetric if 𝐴=𝐴^𝑇; 𝐼 denotes the identity matrix; ‖⋅‖_cl=sup_{−ℎ𝑀≤𝜃≤0}{‖𝑥(𝑡+𝜃)‖,‖𝑥̇(𝑡+𝜃)‖}. The notation diag{…} stands for a block-diagonal matrix.

Consider a complex dynamical switched network consisting of 𝑁 identical coupled nodes, with each node being an 𝑛-dimensional dynamical system as 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−ℎ(𝑡))+𝑐^𝑁_{𝜎(𝑡)𝑗=1}𝑔^𝜎(𝑡)_𝑖𝑗𝐴_𝜎(𝑡)𝑥_𝑗((𝑡,𝑥(𝑡)(𝑡)𝑡𝑡−ℎ))=𝜙,𝑡∈₀−ℎ_𝑀,𝑡₀,ℎ_𝑀≥0, (1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈𝑅^𝑛 is the state vector of 𝑖th node, for all 𝑖∈{1,2,…,𝑁} are the state of node 𝑖th; 𝑓(⋅)∈𝑅^𝑛 is the continuously differentiable function; 𝜎(𝑡)∶[0,+∞)→𝑀={1,2,…,𝑚} is piecewise constant and left continuous and is called a switching signal; the constant 𝑐_𝜎(𝑡)>0 is the coupling strength; 𝐴_𝜎(𝑡)=(𝑎_𝑖𝑗)^𝜎(𝑡)_𝑛×𝑛∈𝑅^𝑛×𝑛 is a constant inner-coupling matrix, if some pairs (𝑖,𝑗), 1≤𝑖, 𝑗≤𝑛, with 𝑎^𝜎(𝑡)_𝑖𝑗≠0 which means two coupled nodes are linked through their 𝑖th and 𝑗th state variables, otherwise 𝑎^𝜎(𝑡)_𝑖𝑗=0; 𝐺_𝜎(𝑡)=(𝑔_𝑖𝑗)^𝜎(𝑡)_𝑁×𝑁∈𝑅^𝑁×𝑁 is the outer-coupling matrix of the network, in which 𝑔^𝜎(𝑡)_𝑖𝑗 is defined as follows: if there is a connection between node 𝑖 and node 𝑗(𝑗≠𝑖), then 𝑔^𝜎(𝑡)_𝑖𝑗=𝑔^𝜎(𝑡)_𝑗𝑖=1; otherwise, 𝑔^𝜎(𝑡)_𝑖𝑗=𝑔^𝜎(𝑡)_𝑗𝑖=0(𝑗≠𝑖), and the diagonal elements of matrix 𝐺 are defined by 𝑔^𝜎(𝑡)_𝑖𝑖=−^𝑁_{𝑗=1,𝑖≠𝑗}𝑔^𝜎(𝑡)_𝑖𝑗=−^𝑁_{𝑗=1,𝑖≠𝑗}𝑔^𝜎(𝑡)_𝑗𝑖,𝑖=1,2,…,𝑁. (2) It is assumed that switched network (1) is connected in the sense that there are no isolated clusters; that is namely, 𝐺_𝜎(𝑡) is an irreducible matrix. The time delay ℎ(𝑡) is a time-varying continuous function satisfying 0≤ℎ_𝑚≤ℎ(𝑡)≤ℎ_𝑀. (3) The initial condition function 𝜙(𝑡) is a continuous vector-valued function for 𝑡∈[𝑡₀−ℎ_𝑀,𝑡₀].

Corresponding to the switching signal 𝜎(𝑡), we have the switching sequence Σ={𝑥_𝑡0; (𝑖₀,𝑡₀),(𝑖₁,𝑡₁),…, (𝑖_𝑘,𝑡_𝑘),…,∣𝑖_𝑘∈𝑀}, and when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), the 𝑖_𝑘-th subnetwork is activated. Assume that there are no jumps in the state at the switching instants, and every network in (1) is connected; that is, 𝑘∈𝑀, 𝐺_𝑘 is irreducible.

Remark 1. For fixed 𝜎(𝑡), the system model (1) turns into the delayed complex dynamical network proposed by Yue and Li [12] as 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−ℎ(𝑡))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗𝐴𝑥_𝑗(𝑡−ℎ(𝑡)),𝑖=1,2,…,𝑁. (4) Therefore, (1) is a general complex networks model, with (4) as a special case.

Definition 2 (see [26]). For any 𝑇₂>𝑇₁≥0, let 𝑁_𝜎(𝑡)(𝑇₁,𝑇₂) denote the number of switching of 𝜎(𝑡) over (𝑇₁,𝑇₂). If 𝑁_𝜎(𝑡)(𝑇₁,𝑇₂)≤𝑁₀+((𝑇₂−𝑇₁)/𝑇_𝑎) holds for 𝑇_𝑎>0, 𝑁₀≥0, then 𝑇_𝑎 is called average dwell time and 𝑁₀ is called a chatter bound.

Definition 3 (see [12]). The delayed dynamical switched network (1) is said to achieve asymptotic synchronization if 𝑥₁(𝑡)=𝑥₂(𝑡)=⋯=𝑠(𝑡)as𝑡⟶∞, (5) where 𝑠(𝑡) is a solution of an isolated node satisfying 𝑠̇(𝑡)=𝑓(𝑠(𝑡),𝑠(𝑡−ℎ(𝑡))). (6)

Definition 4 (see [34]). If the matrix 𝐴∈𝑀_𝑛×𝑛 is similar to a diagonal matrix, then 𝐴 is said to be diagonalizable.

In this paper, we assume that 𝑠(𝑡) is an orbitally stable solution of the above system. Clearly, the stability of the synchronized states (5) of network (1) is determined by the dynamics of the isolated node, the coupling strength 𝑐_𝜎(𝑡), the inner-coupling matrix 𝐴_𝜎(𝑡), the outer-coupling matrix 𝐺_𝜎(𝑡), and the fast time-varying delays ℎ(𝑡).

The following lemmas will be used in the proof of the main result.

Lemma 5 (see [34]). Let G be a family of diagonalizable matrices. Then G is a commuting family (under multiplication) if and only if it is a simultaneously diagonalizable family.

Lemma 6 (see [14]). For any constant symmetric matrix 𝑀∈𝑅^𝑛×𝑛, 𝑀=𝑀^𝑇>0, 0≤ℎ_𝑚≤ℎ(𝑡)≤ℎ_𝑀, 𝑡≥0, and any differentiable vector function 𝑥(𝑡)∈𝑅^𝑛, we have (a)^𝑡_{𝑡−ℎ𝑚}𝑥̇(𝑠)𝑑𝑠^𝑇𝑀^𝑡_{𝑡−ℎ𝑚}𝑥̇(𝑠)𝑑𝑠≤ℎ_𝑚^𝑡_{𝑡−ℎ𝑚}𝑥̇^𝑇(𝑠)𝑀𝑥̇(𝑠)(b𝑑𝑠,)^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠^𝑇𝑀^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)≤ℎ(𝑡)𝑑𝑠−ℎ_𝑚^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑀𝑥̇(𝑠)≤ℎ𝑑𝑠_𝑀−ℎ_𝑚^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑀𝑥̇(𝑠)𝑑𝑠. (7)

3. Synchronization of Switched Network

In this section, we will obtain some delay-dependent synchronization criteria for general complex dynamical network with interval time-varying delay and switched coupling delays (1) by strict LMI approaches. Let 𝐴_𝑖=𝐽(𝑡), 𝐵_{𝑖𝜎(𝑡)}=𝐽_ℎ(𝑡−ℎ(𝑡))+𝑐_𝜎(𝑡)𝜆_{𝑖𝜎(𝑡)}𝐴_𝜎(𝑡), 𝐽(𝑡)=𝑓^′(𝑠(𝑡),and𝑠(𝑡−ℎ(𝑡)))∈𝑅^𝑛×𝑛 is the Jacobian of 𝑓(𝑥(𝑡),𝑥(𝑡−ℎ(𝑡))) at 𝑠(𝑡) with the derivative of 𝑓(𝑥(𝑡),𝑥(𝑡−ℎ(𝑡))) with respect to 𝑥(𝑡). 𝐽_ℎ(𝑡−ℎ(𝑡))=𝑓^′(𝑠(𝑡),𝑠(𝑡−ℎ(𝑡)))∈𝑅^𝑛×𝑛 is the Jacobian of 𝑓(𝑥(𝑡),𝑥(𝑡−ℎ(𝑡))) at 𝑠(𝑡−ℎ(𝑡)) with the derivative of 𝑓(𝑥(𝑡),𝑥(𝑡−ℎ(𝑡))) with respect to 𝑥(𝑡−ℎ(𝑡)).

Lemma 7. Consider the switched coupling delays dynamical network described by (1). Let 0=𝜆_1𝜎(𝑡)>𝜆_2𝜎(𝑡)≥𝜆_3𝜎(𝑡)≥⋯≥𝜆_{𝑁𝜎(𝑡)} be the eigenvalues of the outer-coupling matrix 𝐺_𝜎(𝑡). If the 𝑛-dimensional linear switched system 𝑧̇_𝑖(𝑡)=𝐴_𝑖𝑧_𝑖(𝑡)+𝐵_{𝑖𝜎(𝑡)}𝑧_𝑖(𝑡−ℎ(𝑡)),𝑖=2,3,…,𝑁 (8) is asymptotically (or exponentially) stable, then the synchronized state (5) is asymptotically (or exponentially) stable.

Proof. To investigate the stability of the synchronized states (5), set 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡),𝑖=1,2,…,𝑁. (9) Substituting (9) into (1), for 1≤𝑖≤𝑁, we have 𝑒_𝑖̇(𝑡)=𝑥̇_𝑖(𝑡)−𝑠̇(𝑡)=𝑓𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−ℎ(𝑡))−𝑓(𝑠(𝑡),𝑠(𝑡−ℎ(𝑡)))+𝑐^𝑁_{𝜎(𝑡)𝑗=1}𝑔^𝜎(𝑡)_𝑖𝑗𝐴_𝜎(𝑡)𝑒_𝑗(𝑡−ℎ(𝑡)). (10) Since 𝑓(⋅) is continuous differentiable, it is easy to show that the origin of the nonlinear system (1) is an asymptotically stable equilibrium point if it is an asymptotically stable equilibrium point of the following linear switched time-varying delays systems: 𝑒_𝑖̇(𝑡)=𝐽(𝑡)𝑒_𝑖(𝑡)+𝐽_ℎ(𝑡)𝑒_𝑖(𝑡−ℎ(𝑡))+𝑐^𝑁_{𝜎(𝑡)𝑗=1}𝑔^𝜎(𝑡)_𝑖𝑗𝐴_𝜎(𝑡)𝑒_𝑗((𝑡(𝑡)𝑒𝑡−ℎ))=𝐽_𝑖(𝑡)+𝐽_ℎ(𝑡)𝑒_𝑖(𝑡−ℎ(𝑡))+𝑐_𝜎(𝑡)𝐴_𝜎(𝑡)𝑒₁(𝑡−ℎ(𝑡)),…,𝑒_𝑁((𝑡×𝑔𝑡−ℎ))_{𝜎(𝑡)𝑖1},…,𝑔^𝜎(𝑡)_𝑖𝑁^𝑇, (11) where 𝐽(𝑡)=𝑓^′(𝑠(𝑡),𝑠(𝑡−ℎ(𝑡)))∈𝑅^𝑛×𝑛 is the Jacobian of 𝑓(𝑥(𝑡),𝑥(𝑡−ℎ(𝑡))) at 𝑠(𝑡) and 𝐽_ℎ(𝑡−ℎ(𝑡))=𝑓^′(𝑠(𝑡),𝑠(𝑡−ℎ(𝑡)))∈𝑅^𝑛×𝑛 is the Jacobian of 𝑓(𝑥(𝑡),𝑥(𝑡−ℎ(𝑡))) at 𝑠(𝑡−ℎ(𝑡)).
Letting 𝑒(𝑡)=(𝑒₁(𝑡),…,𝑒_𝑁(𝑡))∈𝑅^𝑛×𝑁 and 𝑒(𝑡−ℎ(𝑡))=(𝑒₁(𝑡−ℎ(𝑡)),…,𝑒_𝑁(𝑡−ℎ(𝑡)))∈𝑅^𝑛×𝑁, we have 𝑒̇(𝑡)=𝐽(𝑡)𝑒(𝑡)+𝐽_ℎ(𝑡−ℎ(𝑡))𝑒(𝑡−ℎ(𝑡))+𝑐_𝜎(𝑡)𝐴_𝜎(𝑡)𝑒(𝑡−ℎ(𝑡))𝐺^𝑇_𝜎(𝑡). (12) Obviously, 𝐺_𝑘 is diagonalizable for fixed 𝜎(𝑡)=𝑘∈𝑀. If 𝐺_𝑘, 𝐺_𝑙, for all 𝑘, and 𝑙∈𝑀 commute pairwise;, that is, 𝐺_𝑘𝐺_𝑙=𝐺_𝑙𝐺_𝑘, then based on Lemma 5, one can get a common unitary matrix 𝑈∈𝑅^𝑁×𝑁 with 𝑢_𝑖∈𝑅^𝑁 such that 𝑈^𝑇𝐺_𝑘𝑈=Γ_𝑘, (13) where 𝑈^𝑇𝑈=𝐼, Γ_𝑘=diag{𝜆_1𝑘,…,𝜆_𝑁𝑘}, 𝜆_𝑖𝑘, 𝑖=1,2,…,𝑁 are eigenvalues of 𝐺_𝑘. In addition, with (2) and the irreducible feature of 𝐺_𝑘, we may choose 𝑢₁=(1/√𝑁)(1,1,…,1)^𝑇 such that 𝜆_1𝑘=0, for all 𝑘∈𝑀.
From (12), by using the nonsingular transform 𝑒(𝑡)𝑈=𝑧(𝑡)=(𝑧₁(𝑡),…,𝑧_𝑁(𝑡))∈𝑅^𝑁×𝑁, we have the following matrix equation: 𝑧̇(𝑡)=𝐽(𝑡)𝑧(𝑡)+𝐽_ℎ(𝑡−ℎ(𝑡))𝑧(𝑡−ℎ(𝑡))+𝑐_𝜎(𝑡)𝐴_𝜎(𝑡)𝑧(𝑡−ℎ(𝑡))Γ_𝜎(𝑡), (14) namely, 𝑧̇_𝑖(𝑡)=𝐴_𝑖𝑧_𝑖(𝑡)+𝐵_{𝑖𝜎(𝑡)}𝑧_𝑖(𝑡−ℎ(𝑡)),𝑖=1,2,…,𝑁, (15) where 𝐴_𝑖=𝐽(𝑡) and 𝐵_{𝑖𝜎(𝑡)}=𝐽_ℎ(𝑡−ℎ(𝑡))+𝑐_𝜎(𝑡)𝜆_{𝑖𝜎(𝑡)}𝐴_𝜎(𝑡). Thus, we have transformed the stability problem of the synchronized state (5) to the stability problem of the 𝑁 pieces of 𝑛-dimensional linear switched time-varying delays differential equations (15). Note that 𝜆_1𝑘=0 corresponding to the synchronization of the system states (5), where the state 𝑠(𝑡) is an orbitally stable solution of the isolated node as assumed above in (5). If the following 𝑁−1 pieces of 𝑛-dimensional linear switched time-varying delays systems 𝑧̇_𝑖(𝑡)=𝐴_𝑖𝑧_𝑖(𝑡)+𝐵_{𝑖𝜎(𝑡)}𝑧_𝑖(𝑡−ℎ(𝑡)),𝑖=2,3,…,𝑁, (16) are asymptotically (or exponentially) stable, then 𝑒(𝑡) will tend to the origin asymptotically (or exponentially), which is equivalent to the fact that the synchronized states (5) are asymptotically (or exponentially) stable. This completes the proof.

Next, we consider the nonswitched system 𝑧̇_𝑖(𝑡)=𝐴_𝑖𝑧_𝑖(𝑡)+𝐵_𝑖𝑧_𝑖(𝑡−ℎ(𝑡)),𝑖=2,3,…,𝑁. (17) We rewrite system (17) in the following equivalent descriptor system form: 𝑧̇_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑦_𝑖(𝑡)=𝐴_𝑖𝑧_𝑖(𝑡)+𝐵_𝑖𝑧_𝑖(𝑡−ℎ(𝑡)),𝑖=2,3,…,𝑁. (18)(19) To derive delay-dependent stability conditions, which include the information of the time-varying delay ℎ(𝑡), one usually uses the fact 𝑧(𝑡−ℎ(𝑡))=𝑧(𝑡)−^𝑡_{(𝑡)𝑡−ℎ}𝑧̇(𝑠)(𝑡)−𝑑𝑠=𝑧^𝑡_{(𝑡)𝑡−ℎ}𝑦(𝑠)𝑑𝑠, (20) to transform the original system to a system with distributed delays. In order to improve the bound of time-varying delay ℎ(𝑡), let us decompose the matrix 𝐵_𝑖=𝐵_𝑖1+𝐵_𝑖2, where 𝐵_𝑖1 are 𝐵_𝑖2 are appropriate constant matrices to be determined later. Then the original system (17) can be represented in the form of a descriptor system with time-varying and distributed delays as 𝑧̇_𝑖(𝑡)=𝑦_𝑖(𝑡)0=−𝑦_𝑖(𝑡)+𝐴_𝑖+𝐵_𝑖1𝑧_𝑖(𝑡)+𝐵_𝑖2𝑧_𝑖((𝑡−𝐵𝑡−ℎ))_{𝑖1𝑡(𝑡)𝑡−ℎ}𝑦_𝑖(𝑠)𝑑𝑠,𝑖=2,3,…,𝑁. (21) Let 𝛿=ℎ_𝑀−ℎ_𝑚. Let 𝑉_𝑖(𝑡) be defined by 𝑉_𝑖(𝑡)=𝑉_𝑖1(𝑡)+𝑉_𝑖2(𝑡)+𝑉_𝑖3(𝑡)+𝑉_𝑖4(𝑡)+𝑉_𝑖5(𝑡)+𝑉_𝑖6(𝑡), (22) where 𝑉_𝑖1(𝑡)=𝜉^𝑇_𝑖(𝑡)𝐸𝑃_𝑖𝜉_𝑖(𝑡)=𝑧^𝑇_𝑖(𝑡)𝑃_𝑖1𝑧_𝑖(𝑡),𝑉_𝑖2(𝑡)=^𝑡_{𝑡−ℎ𝑚}𝑧^𝑇_𝑖(𝑠)𝑄_𝑖𝑒^{𝛼(𝑠−𝑡)}𝑧_𝑖(𝑠)𝑉𝑑𝑠,_𝑖3(𝑡)=^𝑡_{𝑡−ℎ𝑀}𝑧^𝑇_𝑖(𝑠)𝑅_𝑖𝑒^{𝛼(𝑠−𝑡)}𝑧_𝑖(𝑠)𝑉𝑑𝑠,_𝑖4(𝑡)=ℎ_𝑚⁰_−ℎ𝑚^𝑡_𝑡+𝜃𝑧̇^𝑇_𝑖(𝑠)𝑈_𝑖𝑒^{𝛼(𝑠−𝑡)}𝑧̇_𝑖(𝑠)𝑉𝑑𝑠𝑑𝜃,_𝑖5(𝑡)=ℎ_𝑀⁰_−ℎ𝑀^𝑡_𝑡+𝜃𝑦^𝑇_𝑖(𝑠)𝑆_𝑖𝑒^{𝛼(𝑠−𝑡)}𝑦_𝑖(𝑠)𝑉𝑑𝑠𝑑𝜃,_𝑖6(𝑡)=𝛿^−ℎ𝑚_−ℎ𝑀^𝑡_𝑡+𝜃𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑒^{𝛼(𝑠−𝑡)}𝑧̇_𝑖(𝑠)𝜉𝑑𝑠𝑑𝜃,_𝑖(𝑡)=𝑧^𝑇_𝑖(𝑡)𝑦^𝑇_𝑖(𝑡)^𝑡_{(𝑡)𝑡−ℎ}𝑦^𝑇_𝑖(𝑠)𝑑𝑠^𝑇,𝐸=𝐼00000^𝑇,𝑃_𝑖=𝑃_𝑖1𝑃00_𝑖2𝑃_𝑖3𝑃_𝑖4. (23) For given 𝛼>0, the following lemma provides an estimation of 𝑉_𝑖(𝑡).

Lemma 8. Assume that the time-varying delay ℎ(𝑡) satisfies 0≤ℎ_𝑚≤ℎ(𝑡)≤ℎ_𝑀 and let 𝛼>0. If there exist symmetric positive definite matrices 𝑃_𝑖1>0, 𝑄_𝑖>0, 𝑅_𝑖>0, 𝑆_𝑖>0, 𝑈_𝑖>0, and 𝑊_𝑖>0 and appropriately dimensioned matrices 𝑃_𝑖2, 𝑃_𝑖3, and 𝑃_𝑖4 for 𝑖=2,3,…,𝑁, such that the following LMIs hold: Σ_𝑖1=Σ_𝑖−00𝐼−𝐼00^𝑇×𝑒^−𝛼𝛿𝑊_𝑖Σ00𝐼−𝐼00<0,_𝑖2=Σ_𝑖−000𝐼−𝐼0^𝑇×𝑒^−𝛼𝛿𝑊_𝑖000𝐼−𝐼0<0, (24)(25)𝑖=2,…,𝑁, where Σ_𝑖=Σ_𝑖11Σ_𝑖12Σ_𝑖13Σ_𝑖140Σ_𝑖16∗Σ_𝑖220Σ_𝑖240Σ_𝑖26∗∗Σ_𝑖33Σ_𝑖3400∗∗∗Σ_𝑖44Σ_𝑖45Σ_𝑖46∗∗∗∗Σ_𝑖550∗∗∗∗∗Σ_𝑖66,Σ_𝑖11=𝑃^𝑇_𝑖2𝐴_𝑖+𝐵_𝑖1+𝐴_𝑖+𝐵_𝑖1^𝑇𝑃_𝑖2+𝑄_𝑖+𝑅_𝑖−𝑒^{−𝛼ℎ𝑚}𝑈_𝑖+𝛼𝑃_𝑖1,Σ_𝑖12=𝑃^𝑇_𝑖1−𝑃^𝑇_𝑖2+𝐴_𝑖+𝐵_𝑖1^𝑇𝑃_𝑖3,Σ_𝑖13=𝑒^{−𝛼ℎ𝑀}𝑈_𝑖,Σ_𝑖14=𝑃^𝑇_𝑖2𝐵_𝑖2,Σ_𝑖16=−𝑃^𝑇_𝑖2𝐵_𝑖1−𝐴_𝑖+𝐵_𝑖1^𝑇𝑃_𝑖4,Σ_𝑖22=ℎ²_𝑚𝑈_𝑖+ℎ²_𝑀𝑆_𝑖+𝛿²𝑊_𝑖−𝑃^𝑇_𝑖3−𝑃_𝑖3,Σ_𝑖24=𝑃^𝑇_𝑖3𝐵_𝑖2,Σ_𝑖26=−𝑃^𝑇_𝑖3𝐵_𝑖1−𝑃_𝑖4,Σ_𝑖33=−𝑒^{−𝛼ℎ𝑚}𝑄_𝑖−𝑒^{−𝛼ℎ𝑚}𝑈_𝑖−𝑊_𝑖,Σ_𝑖34=𝑒^−𝛼𝛿𝑊_𝑖,Σ_𝑖44=−2𝑒^−𝛼𝛿𝑊_𝑖,Σ_𝑖45=𝑒^−𝛼𝛿𝑊_𝑖,Σ_𝑖46=𝐵_𝑖2𝑃^𝑇_𝑖4,Σ_𝑖55=−𝑒^{−𝛼ℎ𝑀}𝑅_𝑖−𝑒^−𝛼𝛿𝑊_𝑖,Σ_𝑖66=−𝑒^{−𝛼ℎ𝑀}𝑆_𝑖−𝑃^𝑇_𝑖4𝐵_𝑖1−𝐵^𝑇_𝑖1𝑃_𝑖4. (26) Then, along the trajectory of system (17), we have 𝑉_𝑖(𝑡)≤𝑒^{−𝛼(𝑡−𝑡0)}𝑉_𝑖𝑡₀,𝑡≥𝑡₀≥0. (27)

Proof. By taking the derivative of Lyapunov-Krasovskii functional candidate (22) along the trajectory of the system (21), we obtain 𝑉̇_𝑖(𝑡)=𝑉̇_𝑖1(𝑡)+𝑉̇_𝑖2(𝑡)+𝑉̇_𝑖3(𝑡)+𝑉̇_𝑖4(𝑡)+𝑉̇_𝑖5(𝑡)+𝑉̇_𝑖6(𝑡), (28) where 𝑉̇_𝑖1(𝑡)=2𝑧^𝑇_𝑖(𝑡)𝑃_𝑖1𝑧̇_𝑖(𝑡)=2𝜉^𝑇_𝑖(𝑡)𝑃^𝑇_𝑖𝑧̇_𝑖(𝑡)0,𝑉̇_𝑖2(𝑡)=−𝛼𝑉_𝑖2(𝑡)+𝑧^𝑇_𝑖(𝑡)𝑄_𝑖𝑧_𝑖(𝑡)−𝑧^𝑇_𝑖𝑡−ℎ_𝑚𝑄_𝑖𝑒^{−𝛼ℎ𝑚}𝑧_𝑖𝑡−ℎ_𝑚,𝑉̇_𝑖3(𝑡)=−𝛼𝑉_𝑖3(𝑡)+𝑧^𝑇_𝑖(𝑡)𝑅_𝑖𝑧_𝑖(𝑡)−𝑧^𝑇_𝑖𝑡−ℎ_𝑀𝑅_𝑖𝑒^{−𝛼ℎ𝑀}𝑧_𝑖𝑡−ℎ_𝑀,𝑉̇_𝑖4(𝑡)≤−𝛼𝑉_𝑖4(𝑡)+𝑦^𝑇_𝑖(𝑡)ℎ²_𝑚𝑈_𝑖𝑦_𝑖(𝑡)−ℎ_𝑚𝑒^{−𝛼ℎ𝑚𝑡}_{𝑡−ℎ𝑚}𝑧̇^𝑇_𝑖(𝑠)𝑈_𝑖𝑧̇_𝑖(𝑠)𝑉̇𝑑𝑠,_𝑖5(𝑡)≤−𝛼𝑉_𝑖5(𝑡)+𝑦^𝑇_𝑖(𝑡)ℎ²_𝑀𝑆_𝑖𝑦_𝑖(𝑡)−ℎ_𝑀𝑒^{−𝛼ℎ𝑀𝑡}_{𝑡−ℎ𝑀}𝑦^𝑇_𝑖(𝑠)𝑆_𝑖𝑦_𝑖(𝑠)𝑉̇𝑑𝑠,_𝑖6(𝑡)≤−𝛼𝑉_𝑖6(𝑡)+𝑦^𝑇_𝑖(𝑡)𝛿²𝑊_𝑖𝑦_𝑖(𝑡)−𝛿𝑒^{−𝛼𝛿𝑡−ℎ𝑚}_{𝑡−ℎ𝑀}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠. (29) Using Lemma 6, we obtain −ℎ_𝑚^𝑡_{𝑡−ℎ𝑚}𝑧̇^𝑇_𝑖(𝑠)𝑈_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠≤−^𝑡_{𝑡−ℎ𝑚}𝑧̇^𝑇_𝑖(𝑠)𝑑𝑠^𝑇𝑈_𝑖^𝑡_{𝑡−ℎ𝑚}𝑧̇^𝑇_𝑖(𝑠)𝑧𝑑𝑠=−_𝑖(𝑡)−𝑧_𝑖𝑡−ℎ_𝑚^𝑇𝑈_𝑖𝑧_𝑖(𝑡)−𝑧_𝑖𝑡−ℎ_𝑚=−𝑧^𝑇_𝑖(𝑡)𝑈_𝑖𝑧_𝑖(𝑡)+2𝑧^𝑇_𝑖(𝑡)𝑈_𝑖𝑧_𝑖𝑡−ℎ_𝑚−𝑧^𝑇_𝑖𝑡−ℎ_𝑚𝑈_𝑖𝑧_𝑖𝑡−ℎ_𝑚,−ℎ_𝑀^𝑡_{𝑡−ℎ𝑀}𝑦^𝑇_𝑖(𝑠)𝑆_𝑖𝑦_𝑖(𝑠)(𝑡)𝑑𝑠≤−ℎ^𝑡_{(𝑡)𝑡−ℎ}𝑦^𝑇_𝑖(𝑠)𝑆_𝑖𝑦_𝑖(𝑠)𝑑𝑠≤−^𝑡_{(𝑡)𝑡−ℎ}𝑦^𝑇_𝑖(𝑠)𝑑𝑠^𝑇𝑆_𝑖^𝑡_{(𝑡)𝑡−ℎ}𝑦^𝑇_𝑖(𝑠)𝑑𝑠. (30)(31) On the other hand, we have−𝛿^{𝑡−ℎ𝑚}_{𝑡−ℎ𝑀}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠=−𝛿^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠−𝛿^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)ℎ𝑑𝑠=−_𝑀−ℎ(𝑡)^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)−ℎ(𝑡)𝑑𝑠−ℎ_𝑚^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)−ℎ𝑑𝑠_𝑀−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)−ℎ(𝑡)𝑑𝑠−ℎ_𝑚^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠. (32) Let 𝛾=(ℎ(𝑡)−ℎ_𝑚)/𝛿. Then, we obtain −ℎ(𝑡)−ℎ_𝑚^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠=−𝛾^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝛿𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠≤−𝛾^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}ℎ_𝑀−ℎ(𝑡)𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)−ℎ𝑑𝑠,_𝑀−ℎ(𝑡)^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠=−1−𝛾^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}𝛿𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠≤−1−𝛾^{𝑡−ℎ𝑚}_{(𝑡)𝑡−ℎ}ℎ(𝑡)−ℎ_𝑚𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑑𝑠. (33) From Lemma 6, we have −𝛿^{𝑡−ℎ𝑚}_{𝑡−ℎ𝑀}𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑧̇_𝑖(𝑠)𝑧𝑑𝑠≤−_𝑖𝑡−ℎ_𝑚−𝑧_𝑖(𝑡−ℎ(𝑡))^𝑇×𝑊_𝑖𝑧_𝑖𝑡−ℎ_𝑚−𝑧_𝑖((𝑡−𝑧𝑡−ℎ))_𝑖(𝑡−ℎ(𝑡))−𝑧_𝑖𝑡−ℎ_𝑀^𝑇×𝑊_𝑖𝑧_𝑖(𝑡−ℎ(𝑡))−𝑧_𝑖𝑡−ℎ_𝑀−𝛾𝑧_𝑖(𝑡−ℎ(𝑡))−𝑧_𝑖𝑡−ℎ_𝑀^𝑇×𝑊_𝑖𝑧_𝑖(𝑡−ℎ(𝑡))−𝑧_𝑖𝑡−ℎ_𝑀−1−𝛾𝑧_𝑖𝑡−ℎ_𝑚−𝑧_𝑖(𝑡−ℎ(𝑡))^𝑇×𝑊_𝑖𝑧_𝑖𝑡−ℎ_𝑚−𝑧_𝑖(𝑡−ℎ(𝑡)). (34) Therefore, from (21) and (30)–(34), it follows that 𝑉̇_𝑖(𝑡)+𝛼𝑉_𝑖(𝑡)≤𝜉^𝑇₁(𝑡)Σ_𝑖𝜉₁(𝑡)−𝛾𝑧_𝑖(𝑡−ℎ(𝑡))−𝑧_𝑖𝑡−ℎ_𝑀^𝑇×𝑒^−𝛼𝛿𝑊_𝑖𝑧_𝑖(𝑡−ℎ(𝑡))−𝑧_𝑖𝑡−ℎ_𝑀−1−𝛾𝑧_𝑖𝑡−ℎ_𝑚−𝑧_𝑖(𝑡−ℎ(𝑡))^𝑇×𝑒^−𝛼𝛿𝑊_𝑖𝑧_𝑖𝑡−ℎ_𝑚−𝑧_𝑖(𝑡−ℎ(𝑡))=𝜉^𝑇₁(𝑡)𝛾Σ_𝑖1+1−𝛾Σ_𝑖2𝜉₁(𝑡), (35) where Σ_𝑖1, Σ_𝑖2, and Σ_𝑖 are defined in (24), (25), and (26), respectively, and 𝜉₁(𝑡)=𝑧^𝑇_𝑖(𝑡)𝑦^𝑇_𝑖(𝑡)𝑧^𝑇_𝑖𝑡−ℎ_𝑚𝑧^𝑇_𝑖((𝑡𝑧𝑡−ℎ))_𝑇𝑖𝑡−ℎ_𝑀∫^𝑡_{(𝑡)𝑡−ℎ}𝑦^𝑇_𝑖(𝑠)𝑑𝑠^𝑇. (36) Since 0≤𝛾≤1, 𝛾Σ_𝑖1+(1−𝛾)Σ_𝑖2 is a convex combination of Σ_𝑖1 and Σ_𝑖2. Therefore, 𝛾Σ_𝑖1+(1−𝛾)Σ_𝑖1<0 is equivalent to Σ_𝑖1<0 and Σ_𝑖2<0. Thus, it follows from (24), (25), and (35) that 𝑉̇_𝑖(𝑡)+𝛼𝑉_𝑖(𝑡)≤0,𝑡≥𝑡₀≥0. (37) By integrating this inequality from 𝑡₀ to 𝑡, we obtain (27).

Remark 9. If 𝛼=0 in (24) and (25), then it follows from Lemma 8 that the network system (17) is asymptotically stable.

Next, we define 𝑉(𝑡) as follows: 𝑉_𝑖(𝑡)=𝑉_𝑖1(𝑡)+𝑉_𝑖2(𝑡)+𝑉_𝑖3(𝑡)+𝑉_𝑖4(𝑡)+𝑉_𝑖5(𝑡)+𝑉_𝑖6(𝑡), (38) where 𝑉_𝑖1(𝑡)=𝜉^𝑇_𝑖(𝑡)𝐸𝑃_𝑖𝜉_𝑖(𝑡)=𝑧^𝑇_𝑖(𝑡)𝑃_𝑖1𝑧_𝑖(𝑡),𝑉_𝑖2(𝑡)=^𝑡_{𝑡−ℎ𝑚}𝑧^𝑇_𝑖(𝑠)𝑄_𝑖𝑒^{𝛽(𝑡−𝑠)}𝑧_𝑖(𝑠)𝑉𝑑𝑠,_𝑖3(𝑡)=^𝑡_{𝑡−ℎ𝑀}𝑧^𝑇_𝑖(𝑠)𝑅_𝑖𝑒^{𝛽(𝑡−𝑠)}𝑧_𝑖(𝑠)𝑉𝑑𝑠,_𝑖4(𝑡)=ℎ_𝑚⁰_−ℎ𝑚^𝑡_𝑡+𝜃𝑧̇^𝑇_𝑖(𝑠)𝑈_𝑖𝑒^{𝛽(𝑡−𝑠)}𝑧̇_𝑖(𝑠)𝑉𝑑𝑠𝑑𝜃,_𝑖5(𝑡)=ℎ_𝑀⁰_−ℎ𝑀^𝑡_𝑡+𝜃𝑦^𝑇_𝑖(𝑠)𝑆_𝑖𝑒^{𝛽(𝑡−𝑠)}𝑦_𝑖(𝑠)𝑉𝑑𝑠𝑑𝜃,_𝑖6(𝑡)=𝛿^−ℎ𝑚_−ℎ𝑀^𝑡_𝑡+𝜃𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑒^{𝛽(𝑡−𝑠)}𝑧̇_𝑖(𝑠)𝜉𝑑𝑠𝑑𝜃,_𝑖(𝑡)=𝑧^𝑇_𝑖(𝑡)𝑦^𝑇_𝑖(𝑡)∫^𝑡_{(𝑡)𝑡−ℎ}𝑦^𝑇_𝑖(𝑠)𝑑𝑠^𝑇,𝐸=𝐼00000^𝑇,𝑃_𝑖=𝑃_𝑖1𝑃00_𝑖2𝑃_𝑖3𝑃_𝑖4. (39) For given 𝛼>0, the following lemma provides an estimation of 𝑉_𝑖(𝑡).

Lemma 10. Assume that the time-varying delay ℎ(𝑡) satisfies 0≤ℎ_𝑚≤ℎ(𝑡)≤ℎ_𝑀 and let 𝛽>0. If there exist symmetric positive definite matrices 𝑃_𝑖1>0, 𝑄_𝑖>0, 𝑅_𝑖>0, 𝑆_𝑖>0, 𝑈_𝑖>0, and 𝑊_𝑖>0 and appropriately dimensioned matrices 𝑃_𝑖2, 𝑃_𝑖3, and 𝑃_𝑖4 for 𝑖=2,3,…,𝑁, such that the following symmetric LMIs hold: Σ_𝑖1=Σ_𝑖−00𝐼−𝐼00^𝑇×𝑊_𝑖Σ00𝐼−𝐼00<0,_𝑖2=Σ_𝑖−000𝐼−𝐼0^𝑇×𝑊_𝑖000𝐼−𝐼0<0, (40)
𝑖=2,…,𝑁, where Σ_𝑖=Σ_𝑖11Σ_𝑖12Σ_𝑖13Σ_𝑖140Σ_𝑖16∗Σ_𝑖220Σ_𝑖240Σ_𝑖26∗∗Σ_𝑖33Σ_𝑖3400∗∗∗Σ_𝑖44Σ_𝑖45Σ_𝑖46∗∗∗∗Σ_𝑖550∗∗∗∗∗Σ_𝑖66,Σ_𝑖11=𝑃^𝑇_𝑖2𝐴_𝑖+𝐵_𝑖1+𝐴_𝑖+𝐵_𝑖1^𝑇𝑃_𝑖2+𝑄_𝑖+𝑅_𝑖−𝑈_𝑖−𝛽𝑃_𝑖1,Σ_𝑖12=𝑃^𝑇_𝑖1−𝑃^𝑇_𝑖2+𝐴_𝑖+𝐵_𝑖1^𝑇𝑃_𝑖3,Σ_𝑖13=𝑈_𝑖,Σ_𝑖14=𝑃^𝑇_𝑖2𝐵_𝑖2,Σ_𝑖16=−𝑃^𝑇_𝑖2𝐵_𝑖1−𝐴_𝑖+𝐵_𝑖1^𝑇𝑃_𝑖4,Σ_𝑖22=ℎ²_𝑚𝑈_𝑖+ℎ²_𝑀𝑆_𝑖+𝛿²𝑊_𝑖−𝑃^𝑇_𝑖3−𝑃_𝑖3,Σ_𝑖24=𝑃^𝑇_𝑖3𝐵_𝑖2,Σ_𝑖26=−𝑃^𝑇_𝑖3𝐵_𝑖1−𝑃_𝑖4,Σ_𝑖33=−𝑄_𝑖−𝑈_𝑖−𝑊_𝑖,Σ_𝑖34=𝑊_𝑖,Σ_𝑖44=−2𝑊_𝑖,Σ_𝑖45=𝑊_𝑖,Σ_𝑖46=𝐵_𝑖2𝑃^𝑇_𝑖4,Σ_𝑖55=−𝑅_𝑖−𝑊_𝑖,Σ_𝑖66=−𝑆_𝑖−𝑃^𝑇_𝑖4𝐵_𝑖1−𝐵^𝑇_𝑖1𝑃_𝑖4. (41) Then, along the trajectory of system (17), we have 𝑉_𝑖(𝑡)≤𝑒^{𝛽(𝑡−𝑡0)}𝑉_𝑖𝑡₀,𝑡≥𝑡₀≥0. (42)

Proof. With the same argument as in Lemma 8, by taking the derivative of Lyapunov-Krasovskii functional candidate (38) along the trajectory of the system (21), we obtain 𝑉̇_𝑖(𝑡)−𝛽𝑉_𝑖(𝑡)≤0,𝑡≥𝑡₀≥0. (43) Integrating this inequality from 𝑡₀ to 𝑡, we get (42).

We are now ready to derive some new sufficient conditions for exponential synchronization of the switched network 𝑧̇_𝑖(𝑡)=𝐴_𝑖𝑧_𝑖(𝑡)+𝐵_{𝑖𝜎(𝑡)}𝑧_𝑖(𝑡−ℎ(𝑡)),𝑖=2,3,…,𝑁. (44) We rewrite system (44) in the following equivalent descriptor system form: 𝑧̇_𝑖(𝑡)=𝑦_𝑖(𝑡),0=−𝑦_𝑖(𝑡)+𝐴_𝑖+𝐵_{𝑖𝜎(𝑡)1}𝑧_𝑖(𝑡)+𝐵_{𝑖𝜎(𝑡)2}𝑧_𝑖(𝑡−ℎ(𝑡))−𝐵_{𝑖𝜎(𝑡)1}^𝑡_{(𝑡)𝑡−ℎ}𝑦_𝑖(𝑠)𝑑𝑠. (45)

We consider the case when synchronizable and non-synchronizable subnetworks coexist. Without loss of generality, we suppose that each subnetwork 𝑘∈𝑆_𝑠={1,2,…,𝑟} is synchronizable, where 1≤𝑟<𝑚, and each subnetwork 𝑙∈𝑆_𝑛={𝑟+1,𝑟+2,…,𝑚} is non-synchronizable.

For each subnetwork of the system (45) which satisfies Lemma 8, the Lyapunov-Krasovskii functional candidate can be chosen as 𝑉_𝑖𝑘(𝑡)=𝑧^𝑇_𝑖(𝑡)𝐸𝑃_𝑖𝑘1𝑧_𝑖(𝑡)+^𝑡_{𝑡−ℎ𝑚}𝑧^𝑇_𝑖(𝑠)𝑄_𝑖𝑘𝑒^{𝛼(𝑠−𝑡)}𝑧_𝑖(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ𝑀}𝑧^𝑇_𝑖(𝑠)𝑅_𝑖𝑘𝑒^{𝛼(𝑠−𝑡)}𝑧_𝑖(𝑠)𝑑𝑠+ℎ_𝑚⁰_−ℎ𝑚^𝑡_𝑡+𝜃𝑧̇^𝑇_𝑖(𝑠)𝑈_𝑖𝑘𝑒^{𝛼(𝑠−𝑡)}𝑧̇_𝑖(𝑠)𝑑𝑠𝑑𝜃+ℎ_𝑀⁰_−ℎ𝑀^𝑡_𝑡+𝜃𝑦^𝑇_𝑖(𝑠)𝑆_𝑖𝑘𝑒^{𝛼(𝑠−𝑡)}𝑦_𝑖(𝑠)𝑑𝑠𝑑𝜃+𝛿^−ℎ𝑚_−ℎ𝑀^𝑡_𝑡+𝜃𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑘𝑒^{𝛼(𝑠−𝑡)}𝑧̇_𝑖(𝑠)𝑑𝑠𝑑𝜃, (46) where 𝑖=2,3,…,𝑁, 𝑘∈𝑆_𝑠, 𝑃_𝑖𝑘1, 𝑄_𝑖𝑘, 𝑅_𝑖𝑘, 𝑈_𝑖𝑘, 𝑆_𝑖𝑘, and 𝑊_𝑖𝑘 are positive definite matrices and 𝜉_𝑖(𝑡)=𝑧^𝑇_𝑖(𝑡)𝑦^𝑇_𝑖(𝑡)∫^𝑡_{(𝑡)𝑡−ℎ}𝑦^𝑇_𝑖(𝑠)𝑑𝑠^𝑇,𝐸=𝐼00000^𝑇,𝑃_𝑖𝑘=𝑃_𝑖𝑘1𝑃00_𝑖𝑘2𝑃_𝑖𝑘3𝑃_𝑖𝑘4. (47) Similarly, for each subnetwork of the system (45) which satisfies Lemma 10, the Lyapunov-Krasovskii functional candidate can be chosen as 𝑉_𝑖𝑙(𝑡)=𝑧^𝑇_𝑖(𝑡)𝐸𝑃_𝑖𝑙1𝑧_𝑖(𝑡)+^𝑡_{𝑡−ℎ𝑚}𝑧^𝑇_𝑖(𝑠)𝑄_𝑖𝑙𝑒^{𝛽(𝑡−𝑠)}𝑧_𝑖(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ𝑀}𝑧^𝑇_𝑖(𝑠)𝑅_𝑖𝑙𝑒^{𝛽(𝑡−𝑠)}𝑧_𝑖(𝑠)𝑑𝑠+ℎ_𝑚⁰_−ℎ𝑚^𝑡_𝑡+𝜃𝑧̇^𝑇_𝑖(𝑠)𝑈_𝑖𝑙𝑒^{𝛽(𝑡−𝑠)}𝑧̇_𝑖(𝑠)𝑑𝑠𝑑𝜃+ℎ_𝑀⁰_−ℎ𝑀^𝑡_𝑡+𝜃𝑦^𝑇_𝑖(𝑠)𝑆_𝑖𝑙𝑒^{𝛽(𝑡−𝑠)}𝑦_𝑖(𝑠)𝑑𝑠𝑑𝜃+𝛿^−ℎ𝑚_−ℎ𝑀^𝑡_𝑡+𝜃𝑧̇^𝑇_𝑖(𝑠)𝑊_𝑖𝑙𝑒^{𝛽(𝑡−𝑠)}𝑧̇_𝑖(𝑠)𝑑𝑠𝑑𝜃, (48) where 𝑖=2,3,…,𝑁, 𝑙∈𝑆_𝑛, 𝑃_𝑖𝑙1, 𝑄_𝑖𝑙, 𝑅_𝑖𝑙, 𝑈_𝑖𝑙, 𝑆_𝑖𝑙, and 𝑊_𝑖𝑙 are positive definite matrices.

Consider the following piecewise Lyapunov-Krasovskii functional candidate: 𝑉_𝑖(𝑡)=𝑉_{𝑖𝜎(𝑡)},𝑖=2,3,…,𝑁,𝜎(𝑡)∈𝑀. (49)

From Lemmas 8 and 10, the following properties of the Lyapunov-Krasovskii functional candidate (49) are obtained.(i)There exist 𝑎,𝑏>0 such that 𝑎𝑧_𝑖(𝑡)²≤𝑉_{𝑖𝑘,𝑖𝑙}(𝑡)≤𝑏𝑧_𝑖𝑡₀²_cl,𝑖=2,3,…,𝑁,𝑘,𝑙∈𝑀. (50)(ii)There exists a constant 𝜇≥1 such that 𝑉_𝑖𝑘(𝑡)≤𝜇𝑉_𝑖𝑙(𝑡),𝑖=2,3,…,𝑁,𝑘,𝑙∈𝑀. (51)(iii)The Lyapunov functional candidate (49) satisfies 𝑉_𝑖(𝑡)≤𝑒^{−𝛼(𝑡−𝑡0)}𝑉_𝑖𝑘𝑡₀,if𝜎(𝑡)=𝑘∈𝑆_𝑠,𝑒^{𝛽(𝑡−𝑡0)}𝑉_𝑖𝑙𝑡₀,if𝜎(𝑡)=𝑙∈𝑆_𝑛. (52)

Now, for any piecewise constant switching signal 𝜎(𝑡) and any 0≤𝑡₀<𝑡, we let 𝑇⁻(𝑡₀,𝑡) (𝑇⁺(𝑡₀,𝑡) resp.) denote the total activation time of the synchronizable subnetworks (the ones of non-synchronizable subnetworks resp.) during (𝑡₀,𝑡). Then, we choose a scalar 𝛼^∗∈(0,𝛼) arbitrarily to propose the following switching law. (S1): Determine the switching signal 𝜎(𝑡) so that inf_𝑡≥𝑡0𝑇⁻𝑡₀,𝑡𝑇⁺𝑡₀≥,𝑡𝛽+𝛼^∗𝛼−𝛼^∗ (53)holds on time interval (𝑡₀,𝑡). Meanwhile, we choose 𝛼^∗<𝛼 as the average dwell time scheme: for any 𝑡>𝑡₀, 𝑁_𝜎𝑡₀,𝑡≤𝑁₀+𝑡−𝑡₀𝑇_𝑎,𝑇_𝑎≥𝑇^∗_𝑎=ln𝜇𝛼^∗. (54)

Theorem 11. For a given constant 𝛼>0, 𝛽>0 and time-varying delay satisfying (3), suppose that the subnetwork (1≤𝑘≤𝑟) of switched network (1) satisfies the conditions of Lemma 8, and the others satisfy the conditions of Lemma 10. Assume that there exists 𝜇≥1 such that 𝑃_𝑖𝑝1≤𝜇𝑃_𝑖𝑞1,𝑄_𝑖𝑝≤𝜇𝑄_𝑖𝑞,𝑅_𝑖𝑝≤𝜇𝑅_𝑖𝑞,𝑈_𝑖𝑝≤𝜇𝑈_𝑖𝑞,𝑆_𝑖𝑝≤𝜇𝑆_𝑖𝑞,𝑊_𝑖𝑝≤𝜇𝑊_𝑖𝑞,∀𝑝,𝑞∈𝑀. (55) Then the synchronous manifold (5) of network (1) is locally exponentially stable for switching signal satisfying (53) and (54), and the state decay estimate is given by 𝑧_𝑖(𝑡)≤𝑐₀𝑏_𝑖𝑎_𝑖𝑒^{−𝜆(𝑡−𝑡0)}𝑧_𝑖(𝑡₀)_cl,𝑖=2,3,…,𝑁, (56) where 𝑐₀=𝑒^𝑁0ln𝜇1,𝜆=2𝛼^∗−ln𝜇𝑇_𝑎,𝑎_𝑖=min_{∀𝑘∈𝑀}𝜆_min𝑃_𝑖𝑘1,𝑏_𝑖=max_{∀𝑘∈𝑀}𝜆_max𝑃_𝑖𝑘1+ℎ_𝑚max_{∀𝑘∈𝑀}𝜆_max𝑄_𝑖𝑘+ℎ_𝑀max_{∀𝑘∈𝑀}𝜆_max𝑅_𝑖𝑘+ℎ²_𝑚2max_{∀𝑘∈𝑀}𝜆_max𝑈_𝑖𝑘+ℎ²_𝑀2max_{∀𝑘∈𝑀}𝜆_max𝑆_𝑖𝑘+𝛿²2max_{∀𝑘∈𝑀}𝜆_max𝑊_𝑖𝑘. (57)

Proof. Suppose that 𝑡∈[𝑡_𝑘,𝑡_𝑘+1). For piecewise Lyapunov-Krasovskii functional candidate (49), along trajectory of network system (45), we have 𝑉_𝑖(𝑡)≤𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑉_𝑖𝑘𝑡_𝑘,if𝜎(𝑡)=𝑘∈𝑆_𝑠,𝑒^{𝛽(𝑡−𝑡𝑘)}𝑉_𝑖𝑙𝑡_𝑘,if𝜎(𝑡)=𝑙∈𝑆_𝑛. (58) Since 𝑉_{𝑖𝜎(𝑡𝑘)}(𝑡_𝑘)≤𝜇𝑉_{𝑖𝜎(𝑡}⁻_𝑘)(𝑡⁻_𝑘) is true from (51) at the switching point 𝑡_𝑘, where 𝑡⁻_𝑘=lim_{𝑡→𝑡𝑘}𝑡 and from 𝑘=𝑁_𝜎(𝑡₀,𝑡)≤𝑁₀+((𝑡−𝑡₀)/𝑇_𝑎), we obtain 𝑉_𝑖(𝑡)≤𝑒^{𝛽𝑇+(𝑡𝑘,𝑡)−𝛼𝑇−(𝑡𝑘,𝑡)}𝑉_{𝑖𝜎(𝑡𝑘)}𝑡_𝑘≤𝑒^{𝛽𝑇+(𝑡𝑘,𝑡)−𝛼𝑇−(𝑡𝑘,𝑡)}𝜇𝑉_{𝑖𝜎(𝑡}⁻_𝑘)𝑡⁻_𝑘≤𝑒^{𝛽𝑇+(𝑡𝑘,𝑡)−𝛼𝑇−(𝑡𝑘,𝑡)}𝜇𝑒^{𝛽𝑇+(𝑡𝑘−1,𝑡𝑘)−𝛼𝑇−(𝑡𝑘−1,𝑡𝑘)}×𝑉_{𝑖𝜎(𝑡𝑘−1)}𝑡_𝑘−1≤𝜇𝑒^{𝛽𝑇+(𝑡𝑘−1,𝑡)−𝛼𝑇−(𝑡𝑘−1,𝑡)}𝑉_{𝑖𝜎(𝑡𝑘−1)}𝑡_𝑘−1⋮≤𝜇^𝑘𝑒^{𝛽𝑇+(𝑡0,𝑡)−𝛼𝑇−(𝑡0,𝑡)}𝑉_{𝑖𝜎(𝑡0)}𝑡₀≤𝑒^{𝛽𝑇+(𝑡0,𝑡)−𝛼𝑇−(𝑡0,𝑡)+(𝑁0+((𝑡−𝑡0)/𝑇𝑎))ln𝜇}𝑉_{𝑖𝜎(𝑡0)}𝑡₀. (59) Under the switching law (𝑆1) for any 𝑡₀, 𝑡, we have 𝛽𝑇⁺𝑡₀,𝑡−𝛼𝑇⁻𝑡₀,𝑡≤−𝛼^∗𝑇⁺𝑡₀,𝑡+𝑇⁻𝑡₀,𝑡=−𝛼^∗𝑡−𝑡₀. (60) Thus, 𝑉_𝑖(𝑡)≤𝑒^𝑁0ln𝜇𝑒^{−(𝛼∗−((ln𝜇)/𝑇𝑎))(𝑡−𝑡0)}𝑉_{𝑖𝜎(𝑡0)}𝑡₀≤𝑐₀𝑒^{−2𝜆(𝑡−𝑡0)}𝑉_{𝑖𝜎(𝑡0)}𝑡₀, (61) where 𝑐₀=𝑒^𝑁0ln𝜇, 𝜆=(1/2)(𝛼^∗−((ln𝜇)/𝑇_𝑎)). According to (49), we have 𝑎_𝑖𝑧_𝑖(𝑡)²≤𝑉_𝑖(𝑡),𝑉_{𝑖𝜎(𝑡0)}𝑡₀≤𝑏_𝑖𝑧_𝑖𝑡₀²_cl. (62) Combining (61) and (62) leads to 𝑧_𝑖(𝑡)²≤1𝑎_𝑖𝑉_𝑖(𝑡)≤𝑏_𝑖𝑎_𝑖𝑐₀𝑒^{−2𝜆(𝑡−𝑡0)}𝑧_𝑖𝑡₀²_cl. (63) Therefore, 𝑧_𝑖(𝑡)≤𝑐₀𝑏_𝑖𝑎_𝑖𝑒^{−𝜆(𝑡−𝑡0)}𝑧_𝑖𝑡_0cl. (64) which means that synchronous solution to (1) is locally exponentially stable. The proof is completed.

4. Numerical Examples

In this section, we give examples to show the effectiveness of theoretical results obtained in Lemma 8 and Theorem 11.

Example 12. Consider a nonswitched network model with 5 nodes, where each node is a three-dimensional stable linear system described by 𝑥̇_𝑖1(𝑡)=−𝑥_𝑖1(𝑡)𝑥̇_𝑖2(𝑡)=−2𝑥_𝑖2(𝑡)𝑥̇_𝑖3(𝑡)=−3𝑥_𝑖3(𝑡), (65) which is asymptotically stable at the equilibrium point 𝑠(𝑡)=0, and its Jacobin matrices are 𝐽(𝑡)=diag{−1,−2,−3}, 𝐽_ℎ(𝑡−ℎ(𝑡))=0. Assume that the inner-coupling matrix is 𝐴=diag{1,1,1}, and the outer-coupling matrix is given by the following irreducible symmetric matrix satisfying condition (2): 𝐺=−210011−311001−210011−311001−2. (66) The eigenvalues of 𝐺 are 𝜆_𝑖={0,−1.382,−2.382,−3.618,−4.618}, 𝑖=1,2,…,5. Therefore, if the delayed subsystem in (17) is asymptotically stable, then the synchronized state 𝑠(𝑡) is asymptotically stable.

In this example, we have 𝐴_𝑖=diag{−1,−2,−3}, 𝐵_𝑖=diag{𝑐𝜆_𝑖,𝑐𝜆_𝑖,𝑐𝜆_𝑖}, and 𝑖=2,…,5. Let us decompose the matrix 𝐵_𝑖=𝐵_𝑖1+𝐵_𝑖2, where 𝐵_𝑖1=0.39𝑐𝜆_𝑖0000.39𝑐𝜆_𝑖0000.39𝑐𝜆_𝑖,𝐵_𝑖2=0.61𝑐𝜆_𝑖0000.61𝑐𝜆_𝑖0000.61𝑐𝜆_𝑖,𝑖=2,…,5. (67)

By using Lemma 8, we will give the maximum upper bounds ℎ_𝑀 of the time-varying delay for different lower bounds ℎ_𝑚 and coupling strength 𝑐. Applying Lemma 8 with 𝛼=0, we obtain the maximum upper bound of delays ℎ_𝑀 as shown in Table 1. We see that, Lemma 8 provides a less conservative result than those obtained via the methods of [8, 12]. When ℎ_𝑚≠0 especially, the result in [8] is not discussed while Lemma 8 in this paper also considers the case ℎ_𝑚≠0. Note that we use MATLAB LMI Control Toolbox in order to solve the LMI in (24) and (25).

Table 1 

Comparison of the maximum value ℎ_𝑀 for the asymptotic stability of system (65) by different methods.

The numerical simulations are carried out using the explicit Runge-Kutta-like method (dde45), interpolation and extrapolation by spline of the third order. Figure 1 shows the synchronization state curves 𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),𝑥_𝑖3(𝑡),𝑖=1,2,…,5 with 𝑐₁=0.5, ℎ(𝑡)=0.5+0.715|cos𝑡| and the initial function 𝜙₁(𝑡)=[],𝜙−0.2cos𝑡,−0.5cos𝑡,1.1cos𝑡₂(𝑡)=[],𝜙0.5cos𝑡,0.7cos𝑡,−0.7cos𝑡₃(𝑡)=[],𝜙0.3cos𝑡,1.2cos𝑡,0.4cos𝑡₄(𝑡)=[],𝜙0.35cos𝑡,−0.5cos𝑡,−1.25cos𝑡₅(𝑡)=[−0.4cos𝑡,−1.1cos𝑡,0.8cos𝑡]. (68)

(a)

(b)

(c)

Figure 1 

Synchronization state curves for the dynamic nodes delayed network (65) with 𝑐=0.5 and ℎ(𝑡)=0.5+0.715|cos𝑡|.

Example 13. Consider a linear switched network model with 5 nodes where each node is described by the following delayed dynamical system: 𝑥̇_𝑖1(𝑡)=−1𝑥_𝑖1(𝑡)−𝑥_𝑖1((𝑡𝑥̇𝑡−ℎ))_𝑖2(𝑡)=−2𝑥_𝑖2+𝑥_𝑖1(𝑡−ℎ(𝑡))−𝑥_𝑖2((𝑡𝑥̇𝑡−ℎ))_𝑖3(𝑡)=−1.5𝑥_𝑖3−𝑥_𝑖3(𝑡−ℎ(𝑡)), (69) which is asymptotically stable at the equilibrium point 𝑠(𝑡)=0, 𝑠(𝑡−ℎ(𝑡))=0. Jacobian matrices 𝐽(𝑡) and 𝐽_ℎ(𝑡−ℎ(𝑡)) are 𝐽(𝑡)=−1000−2000−1.5,𝐽_ℎ(𝑡−ℎ(𝑡))=−1001−1000−1. (70) Assume that the inner-coupling matrices are 𝐴₁=diag{0.5,0.5,0.5}, 𝐴₂=diag{−1,−1,−1} and the coupling strengths are 𝑐₁=0.3 and 𝑐₂=0.6, and the outer-coupling matrices are 𝐺₁=,𝐺−210011−311001−210011−311001−2₂=.−411111−411111−411111−411111−4 (71) The eigenvalues of 𝐺₁ and 𝐺₂ are 𝜆_𝑖1={0,−1.382,−2.382,−3.618,−4.618}, 𝜆_𝑖2={0,−5,−5,−5,−5}, and 𝑖=1,2,…,5, respectively. Obviously, there exists a unitary matrix 𝑈 which diagonalizes 𝐺₁ and 𝐺₂ simultaneously. The subnetwork associated with {𝑐₁,𝐴₁,𝐺₁} is synchronizable and the subnetwork associated with {𝑐₂,𝐴₂,𝐺₂} is non-synchronizable.
From the above conditions, we obtain matrices 𝐴_𝑖=,𝐵−1000−2000−1.5_𝑖𝑘=−1+𝑐_𝑘𝑎^𝑘₁₁𝜆_𝑖𝑘001−1+𝑐_𝑘𝑎^𝑘₂₂𝜆_𝑖𝑘000−1+𝑐_𝑘𝑎^𝑘₃₃𝜆_𝑖𝑘, (72)𝑖=2,…,5, and 𝑘=1,2. Let us decompose the matrix 𝐵_𝑖𝑘=𝐵_𝑖𝑘1+𝐵_𝑖𝑘2, where 𝐵_𝑖11=0.93𝐵_𝑖1,𝐵_𝑖12=0.07𝐵_𝑖1,𝐵_𝑖21=0.86𝐵_𝑖2,𝐵_𝑖22=0.14𝐵_𝑖2,𝑖=2,…,5. (73)
Given ℎ₁=0.3, ℎ₂=0.68, 𝛼=1.25, and 𝛽=0.6, it is found that LMIs (24), (25), (42), and (42) have feasible solutions. We may choose 𝜇=1.35. Let 𝛼^∗=0.55<𝛼. Then, from the switching law (𝑆1), it is required that inf_𝑡≥𝑡0(𝑇⁻(𝑡₀,𝑡)/𝑇⁺(𝑡₀,𝑡))≥(𝛽+𝛼^∗)/(𝛼−𝛼^∗)=1.6429 and the average dwell time is computed as 𝑇_𝑎≥𝑇^∗_𝑎=(ln𝜇)/𝛼^∗=0.5456.
Let ℎ(𝑡)=0.3+0.38|sin𝑡| and the initial function 𝜙₁(𝑡)=[],𝜙−0.4cos𝑡,−0.5cos𝑡,1.1cos𝑡₂(𝑡)=[],𝜙0.5cos𝑡,0.7cos𝑡,−0.7cos𝑡₃(𝑡)=[],𝜙0.89cos𝑡,1.2cos𝑡,0.4cos𝑡₄(𝑡)=[],𝜙1.35cos𝑡,−1.5cos𝑡,−1.25cos𝑡₅(𝑡)=[−0.8cos𝑡,−1.1cos𝑡,0.8cos𝑡]. (74) Figure 2 shows the structure of complex networks. Figures 3 and 4 show the synchronization state curves of the synchronizable subnetwork and non-synchronizable subnetwork, respectively. Figure 5 shows the synchronization state curves 𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),𝑥_𝑖3(𝑡),𝑖=1,2,…,5 of switched network with 𝜎(𝑡) satisfying (𝑆1). Figure 6 shows the switching 𝜎(𝑡) of the switched delay network with average dwell time. We see that the synchronization state converges to zero under the above conditions.

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Figure 2 

The structure of complex networks with 𝑁=5.

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Figure 3 

Synchronization state curves for the synchronizable subnetwork (69).

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Figure 4 

Synchronization state curves for the nonsynchronizable subnetwork (69).

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Figure 5 

Synchronization state curves for the switched delay network (69).

Figure 6 

The switching signal 𝜎(𝑡) of the switched delay network (69) with average dwell time.

Example 14. We consider nonlinear switched network model with 5 nodes in which each node is a Lorenz chaotic system with time-varying delay described by [35] as 𝑥̇_𝑖1(𝑡)=𝑎𝑥_𝑖2(𝑡)−𝑥_𝑖1(𝑡),𝑥̇_𝑖2(𝑡)=𝑐𝑥_𝑖1−𝑥_𝑖2(𝑡)−𝑥_𝑖1(𝑡)𝑥_𝑖3((𝑡,𝑥̇𝑡−ℎ))_𝑖3(𝑡)=𝑥_𝑖1(𝑡)𝑥_𝑖2(𝑡−ℎ(𝑡))−𝑏𝑥_𝑖3(𝑡−ℎ(𝑡)), (75) where 𝑎=10, 𝑏=1.3, and 𝑐=−28. It is asymptotically stable at the equilibrium point 𝑠(𝑡)=0, 𝑠(𝑡−ℎ(𝑡))=0 and its Jacobian matrices are 𝐽(𝑡)=−10100−28−10000,𝐽_ℎ(𝑡−ℎ(𝑡))=00000−100−1.3. (76)
Assume that the inner-coupling matrices are 𝐴₁=diag{0.4,0.4,0.4} and 𝐴₂=diag{−0.2,−0.2,−0.2}, the coupling strengths 𝑐₁=0.5, and 𝑐₂=0.3, and the outer-coupling matrices 𝐺₁, and 𝐺₂ are the same as in Example 13. The subnetwork associated with {𝑐₁,𝐴₁,𝐺₁} is synchronizable and the subnetwork associated with {𝑐₂,𝐴₂,𝐺₂} is non-synchronizable.
From the above conditions, we obtain 𝐴_𝑖=,𝐵−10100−28−10000_𝑖𝑘=𝑐_𝑘𝑎^𝑘₁₁𝜆_𝑖𝑘000𝑐_𝑘𝑎^𝑘₂₂𝜆_𝑖𝑘−100−1.3+𝑐_𝑘𝑎^𝑘₃₃𝜆_𝑖𝑘, (77)𝑖=2,…,5, and 𝑘=1,2. Let us decompose the matrix 𝐵_𝑖𝑘=𝐵_𝑖𝑘1+𝐵_𝑖𝑘2, where 𝐵_𝑖11=0.8𝐵_𝑖1,𝐵_𝑖12=0.2𝐵_𝑖1,𝐵_𝑖21=0.86𝐵_𝑖2,𝐵_𝑖22=0.14𝐵_𝑖2,𝑖=2,…,5. (78)
Given ℎ₁=0.2, ℎ₂=0.55, 𝛼=0.9, and 𝛽=0.3, it is found that LMIs (24), (25), (42), and (42) have feasible solutions. We may choose 𝜇=1.253. Let 𝛼^∗=0.45<𝛼. Then, from the switching law (𝑆1), it is required that inf_𝑡≥𝑡0(𝑇⁻(𝑡₀,𝑡)/𝑇⁺(𝑡₀,𝑡))≥(𝛽+𝛼^∗)/(𝛼−𝛼^∗)=1.6667 and the average dwell time is computed as 𝑇_𝑎≥𝑇^∗_𝑎=(ln𝜇)/𝛼^∗=0.5012.
Let ℎ(𝑡)=0.2+0.35|cos𝑡| and the initial function 𝜙₁(𝑡)=[],𝜙15cos𝑡,20cos𝑡,5cos𝑡₂(𝑡)=[],𝜙5cos𝑡,15cos𝑡,−2cos𝑡₃(𝑡)=[],𝜙10cos𝑡,5cos𝑡,3cos𝑡₄(𝑡)=[],𝜙20cos𝑡,10cos𝑡,−1cos𝑡₅(𝑡)=[15cos𝑡,20cos𝑡,8cos𝑡]. (79) Figures 7 and 8 show the synchronization errors between the states of node 𝑖 and node 𝑖+1, 𝑒_𝑖𝑗(𝑡)=𝑥_𝑖𝑗−𝑥_(𝑖+1)𝑗,𝑖=1,…,4, and 𝑗=1,…,3 of the synchronizable subnetwork and non-synchronizable subnetwork, respectively. Figure 9 shows the synchronization errors between the states of node 𝑖 and node 𝑖+1 of switched network with 𝜎(𝑡) satisfying (𝑆1). Figure 10 shows the switching 𝜎(𝑡) of the switched delay network with average dwell time. We see that the synchronization state converges to zero under the above conditions.

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Figure 7 

Synchronization errors between the states of node 𝑖 and node 𝑖+1 for the synchronizable subnetwork (75).

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Figure 8 

Synchronization errors between the states of node 𝑖 and node 𝑖+1 for the non-synchronizable subnetwork (75).

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Figure 9 

Synchronization errors between the states of node 𝑖 and node 𝑖+1 for the switched delay network (75).

Figure 10 

The switching signal 𝜎(𝑡) of the switched delay network (75) with average dwell time.

Remark 15. The advantage of Examples 13 and 14 is the lower bound of the delay ℎ_𝑚≠0. Moreover, in these examples we still investigate interval time-varying delays in the dynamical nodes and the switched coupling term simultaneously, hence the synchronization conditions derived in [33] cannot be applied to these examples.

5. Conclusion

In this paper, the synchronization problem has been investigated for complex dynamical networks with interval time-varying and switched coupling delays. Both interval time-varying delays in dynamical nodes and interval time-varying delays in switched couplings have been considered. We transformed the synchronization problem of the switched network into the stability analysis of linear switched systems. By using the average dwell time approach and piecewise Lyapunov-Krasovskii functionals which are constructed based on the descriptor model transformation and decomposition technique of coefficient matrix, new delay-dependent synchronization criteria was derived in terms of linear matrix inequalities. Numerical examples were given to illustrate that the derived criteria are less conservative than some existing results.

Acknowledgments

The first author is supported by the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education. The second author is supported by the Center of Excellence in Mathematics, Thailand and Commission for Higher Education, Thailand. The authors also wish to thank the National Research University Project under Thailand’s Office of the Higher Education Commission for financial support. They would like to thank Professor Xinzhi Liu for the valuable comments and suggestions.