Global Synchronization of Complex Networks with Discrete Time Delays on Time Scales

Abstract

This paper studies the global synchronization problem for a class of complex networks with discrete time delays. By using the theory of calculus on time scales, the properties of Kronecker product, and Lyapunov method, some sufficient conditions are obtained to ensure the global synchronization of the complex networks with delays on time scales. These sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). The main contribution of the result is that the global synchronization problems with both discrete time and continuous time are unified under the same framework.

1. Introduction

As well known, complex dynamical networks have been a subject of high importance and increasing interest within the science and technology communities. The synchronization is one of the most typical phenomena in complex networks, which is ubiquitous in the real world, such as secure communication, chaos generators design, and harmonic oscillation generation, ([1–8], and references cited therein).

During the past many years, the synchronization of complex networks has received increasing research attention. There are lots of the papers studying the continuous time and the discrete time dynamical systems. However, most of the investigations are restricted to the continuous or discrete systems, respectively, [9–22]. For avoiding this trouble, it is meaningful to study this problem on time scales which can unify the continuous and discrete dynamical systems under the unified framework.

The theory of time scale calculus was initiated by Hilger in 1988, developed, and consummated by Bohner and Peterson [23–25], which has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, population dynamics, biotechnology, economics, and so on [26, 27]. This novel and fascinating type of mathematics is more general and versatile than the traditional theories of differential and difference equations as it can, under one framework, mathematically describe continuous and discrete hybrid processes and hence is the optimal way forward for accurate and malleable mathematical modeling. The field of dynamic equations on time scales contains, links, and extends the classical theory of differential and difference equations.

However, to the best of our knowledge, there are few works investigating the synchronization problem of complex networks with delays on time scales.

Notations. Throughout this paper, ℝ^𝑛 and ℝ^𝑛×𝑚 denote the n-dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices, respectively. 𝕋 is a time scale, which is an arbitrary nonempty closed subset of the real number ℝ with the topology and ordering inherited from ℝ, and assume that 0∈𝕋, and 𝕋 is unbounded above, that is, sup𝕋=∞. Set [𝑎,𝑏]_𝕋∶={𝑡∈𝕋,𝑎≤𝑡≤𝑏}. 𝕋⁺={𝑡∈𝕋,𝑡≥0}. 𝑃>0 means that matrix 𝑃 is real, symmetric, and positive definite. 𝐼 and 𝑂 denote the identity matrix and the zero matrix with compatible dimensions, respectively; and diag{⋯} stands for a block-diagonal matrix. The superscript “T” stands for a matrix transposition. The Kronecker product of matrices 𝑄∈ℝ^𝑚×𝑛 and 𝑅∈ℝ^𝑝×𝑞 is a matrix in ℝ^{𝑚𝑝×𝑛𝑞} and denoted as 𝑄⊗𝑅. Let 𝜏>0 and 𝐶([−𝜏,0]_𝕋;ℝ^𝑛) denote the family of continuous functions 𝜑 from [−𝜏,0]_𝕋 to ℝ^𝑛 with the norm ‖𝜑‖=sup_{−𝜏≤𝜃≤0}‖𝜑(𝜃)‖, where ‖⋅‖ is the Euclidean norm in ℝ^𝑛.

The rest of this paper is organized as follows. In Section 2, some preliminaries on time scale are briefly outlined. In Section 3, by utilizing the approach of the Lyapunov functional method on time scale and the LMI [28], our main result for ensuring the global synchronization is derived. In Section 4, an example is given to illustrate the effectiveness of our main result. Finally, in Section 5, this paper is concluded.

2. Preliminaries

In this paper, the global synchronization problem is investigated for a class of complex networks with discrete time delays which is described by the following dynamic equation on time scale 𝕋:𝑥^Δ_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡)+𝐵_𝜏𝑓𝑥_𝑖𝑡−𝜏₁+^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗𝑡−𝜏₂+Λ_𝑖,𝑖=1,2,…,𝑁, (2.1) where 𝑡∈𝕋, 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛 is the state vector of the 𝑖th network at time 𝑡. 𝐴 denotes a known connection matrix, 𝐵 and 𝐵_𝜏 denote the connection weight matrices, Γ∈ℝ^𝑛×𝑛 is the matrix describing the innercoupling between the subsystems at time 𝑡, 𝐺=(𝐺_𝑖𝑗)_𝑁×𝑁 is the outercoupling configuration matrix representing the coupling strength and the topological structure of the complex networks, and Λ_𝑖 is the external input. The constants 𝜏₁ and 𝜏₂ stand for the constant time delays, and 𝑓(𝑥_𝑖(𝑡))=(𝑓₁(𝑥_𝑖1(𝑡)),𝑓₂(𝑥_𝑖2(𝑡)),…,𝑓_𝑛(𝑥_𝑖𝑛(𝑡)))^𝑇 is an unknown but sector-bounded nonlinear function.

The initial conditions associated with system (2.1) are given by𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑠∈[−ℎ,0]_𝕋,𝑖=1,2,…,𝑁, (2.2) where 𝜑_𝑖(𝑠) is rd-continuous and the corresponding state trajectory is denoted as 𝑥_𝑖(𝑡,𝜑_𝑖).

Throughout this paper, the following assumptions are needed.

Assumption 2.1. The outer-coupling configuration matrix of the complex networks (2.1) satisfies 𝐺_𝑖𝑗=𝐺_𝑗𝑖≥0𝑖≠𝑗,𝐺_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗𝑖,𝑗=1,2,…,𝑁. (2.3)

Assumption 2.2. For all 𝑢,𝑣∈ℝ^𝑛, the nonlinear function 𝑓(⋅) is assumed to satisfy the following sector-bounded condition: 𝑓(𝑢)−𝑓(𝑣)−𝐿_𝑓(𝑢−𝑣)^𝑇𝑓(𝑢)−𝑓(𝑣)−𝐿^𝑓(𝑢−𝑣)≤0, (2.4) where 𝐿_𝑓 and 𝐿^𝑓 are real constant matrices with 𝐿^𝑓−𝐿_𝑓 being symmetric and positive definite.

Assumption 2.3. 0≤𝜏_𝑖<ℎ(𝑖=1,2).

Remark 2.4. System (2.1) is a general model of a class of complex networks. Its one special case with continuous time system is the following: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝐴𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡)+𝐵_𝜏𝑓𝑥_𝑖𝑡−𝜏₁+^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗𝑡−𝜏₂+Λ_𝑖,𝑖=1,2,…,𝑁 (2.5) for 𝑡∈[𝑡₀,+∞), and its another special case with discrete time system is the following: Δ𝑥_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡)+𝐵_𝜏𝑓𝑥_𝑖𝑡−𝜏₁+^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗𝑡−𝜏₂+Λ_𝑖,𝑖=1,2,…,𝑁 (2.6) for 𝑡∈𝑁, where Δ𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡+1)−𝑥_𝑖(𝑡) is the forward difference operator.
The continuous-time system (2.5) and the discrete-time system (2.6) are unified as system (2.1). The main objective of this paper is to study the synchronization problem of system (2.1) under the same framework.

In order to obtain the main results, some preliminary results are presented in this section.

Definition 2.5 (see [23]). A time scale 𝕋 is an arbitrary nonempty closed subset of the real set ℝ with the topology and ordering inherited from ℝ. The forward and backward jump operators 𝜎,𝜌∶𝕋→𝕋 and the graininess 𝜇∶𝕋→ℝ⁺ are defined, respectively, by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡};𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡};𝜇(𝑡)∶=𝜎(𝑡)−𝑡. We put inf∅:= sup𝕋 and sup∅:= inf𝕋, where ∅ denotes the empty set. A point 𝑡 is said to be left-dense if 𝑡>inf𝕋 and 𝜌(𝑡)=𝑡, right-dense if 𝑡<sup𝕋 and 𝜎(𝑡)=𝑡, left-scattered if 𝜌(𝑡)<𝑡, and right-scattered if 𝜎(𝑡)>𝑡. If 𝕋 has a left-scattered maximum 𝑚, then we define 𝕋^𝑘 to be 𝕋−{𝑚}. Otherwise 𝕋^𝑘=𝕋.

Definition 2.6 (see [23]). A function 𝑓∶𝕋→ℝ is called right-dense continuous provided it is continuous at right-dense point of 𝕋 and the left side limit exists (finite) at left-dense point of 𝕋. The set of all right-dense continuous functions on 𝕋 is defined by 𝐶_rd=C_rd(𝕋)=𝐶_rd(𝕋,ℝ).

Definition 2.7 (see [23]). For a function 𝑓∶𝕋→ℝ,𝑡∈𝕋^𝑘, the delta derivative of 𝑓(𝑡), 𝑓^Δ(𝑡), is the number (if it exists) with the property that, for a given 𝜀>0, there exists a neighborhood 𝑈 of 𝑡 such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]<𝜀|𝜎(𝑡)−𝑠| (2.7) for all 𝑠∈𝑈.
If 𝑡 is right-scattered and 𝑓 is continuous at 𝑡, then 𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜎(𝑡).−𝑡 (2.8) If 𝑡 is right-dense, then 𝑓^Δ(𝑡)=lim_𝑠→𝑡𝑓(𝜎(𝑡))−𝑓(𝑠)𝜎(𝑡)−𝑠=lim_𝑠→𝑡𝑓(𝑡)−𝑓(𝑠)𝑡−𝑠. (2.9) For all 𝑡∈𝕋^𝑘, one can get 𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡). (2.10) If 𝑓^Δ≥0, then 𝑓 is not decreasing on time scale.

Lemma 2.8 (see [23]). If 𝑓 and 𝑔 are two differentiable functions, then the product rule for the derivative of product 𝑓⋅𝑔 is that 𝑓⋅𝑔^Δ=𝑓^Δ⋅𝑔+𝑓^𝜎⋅𝑔^Δ=𝑓^Δ⋅𝑔^𝜎+𝑓⋅𝑔^Δ. (2.11)

Definition 2.9 (see [23]). A function 𝐹∶𝕋^𝑘→ℝ is called a delta-antiderivative of 𝑓∶𝕋→ℝ provided 𝐹^Δ=𝑓 holds for all 𝑡∈𝕋^𝑘. In this case, we define the integral of 𝑓 by ^𝑡_𝑎𝑓(𝑠)Δ𝑠=𝐹(𝑡)−𝐹(𝑎),for𝑡∈𝕋, (2.12) and we have the following formula: _𝑡^𝜎(𝑡)𝑓(𝑠)Δ𝑠=𝜇(𝑡)𝑓(𝑡),for𝑡∈𝕋^𝑘. (2.13)

Definition 2.10 (see [23]). A function 𝑓∶𝕋→ℝ is called regressive provided 1+𝜇(𝑡)𝑓(𝑡)≠0,∀𝑡∈𝕋. (2.14)

The addition “⊕” is defined by 𝑝⊕𝑞∶=𝑝+𝑞+𝜇𝑝𝑞. The set of all regressive functions on a time scale 𝕋 forms an Abelian group under the addition “⊕”. The additive inverse in this group is denoted by ⊖𝑝∶=−𝑝/(1+𝜇𝑝). Then the subtraction ⊖ on the set of regressive functions is defined by 𝑝⊖𝑞∶=𝑝⊕(⊖𝑝). It can be shown easily that 𝑝⊖𝑞=−(𝑝−𝑞)/(1+𝜇𝑞). The set of all regressive and right-dense continuous functions will be denoted by ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ).

We denote that ℛ⁺=ℛ⁺(𝕋,ℝ)={𝑓∈ℛ∶1+𝜇(𝑡)𝑓(𝑡)>0forall𝑡∈𝕋}. Obviously ℛ⁺ is the set of all positively regressive elements of ℛ. One can easily verify that if 𝑓∈ℛ⁺, then ⊖𝑓∈ℛ⁺.

Lemma 2.11 (see [23]). Assume 𝑣∶𝕋→ℝ is strictly increasing and 𝕋=𝑣(𝕋) is a time scale. If 𝑓∶𝕋→ℝ is an rd-continuous function and 𝑣 is differentiable with rd-continuous derivative, then for 𝑎,𝑏∈𝕋, ^𝑏_𝑎𝑓(𝑡)𝑣^Δ(𝑡)Δ𝑡=_𝑣(𝑎)^𝑣(𝑏)𝑓∘𝑣⁻¹(𝑠)Δ𝑠. (2.15)

Definition 2.12. Let 𝐴 be an 𝑚×𝑛-matrix-valued function on 𝕋. We say that 𝐴 is rd-continuous on 𝕋 if each entry of 𝐴 is rd-continuous on 𝕋, and the class of all such rd-continuous 𝑚×𝑛-matrix-valued function on 𝕋 is denoted, similar to the scalar case, by 𝐶_rd=𝐶_rd(𝕋)=𝐶_rd(𝕋,ℝ^𝑚×𝑛).
We say that 𝐴 is differentiable on 𝕋 provided each entry of A is differentiable on 𝕋. In this case, we put 𝐴^Δ=𝑎^Δ_{𝑖𝑗1≤𝑖≤𝑚,1≤𝑗≤𝑛}, (2.16) where 𝐴=(𝑎_𝑖𝑗)_{1≤𝑖≤𝑚,1≤𝑗≤𝑛}. And denote that 𝐴^𝜎=(𝑎^𝜎_𝑖𝑗).

Lemma 2.13 (see [23]). Suppose A and B are differentiable 𝑛×𝑛-matrix-valued functions. Then (1)(𝐴+𝐵)^Δ=𝐴^Δ+𝐵^Δ; (2)(𝛼𝐴)^Δ=𝛼𝐴^Δ𝑖𝑓𝛼isaconstant; (3)(𝐴𝐵)^Δ=𝐴^Δ𝐵^𝜎+𝐴𝐵^Δ.

Let 𝑥(𝑡)=𝑥^𝑇₁(𝑡),𝑥^𝑇₂(𝑡),…,𝑥^𝑇_𝑁(𝑡)^𝑇,Λ=Λ^𝑇₁,Λ^𝑇₂,…,Λ^𝑇_𝑁^𝑇,𝐹(𝑥(𝑡))=𝑓^𝑇𝑥₁(𝑡),𝑓^𝑇𝑥₂(𝑡),…,𝑓^𝑇𝑥_𝑁(𝑡)^𝑇, (2.17) together with the Kronecker product “⊗” for matrices, system (2.1) can be recasted into𝑥^Δ(𝑡)=𝐼_𝑁⊗𝐴𝑥(𝑡)+𝐼_𝑁⊗𝐵𝐹(𝑥(𝑡))+𝐼_𝑁⊗𝐵_𝜏𝐹𝑥𝑡−𝜏₁+(𝐺⊗Γ)𝑥𝑡−𝜏₂+Λ. (2.18)

Definition 2.14. The complex system (2.18) is said to be globally synchronized, if lim_𝑡→∞‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖=0, (2.19) hold for all 𝜑_𝑖(𝑠),𝜑_𝑗(𝑠)∈𝐶_𝑟𝑑([−ℎ,0]_𝕋,ℝ^𝑛),𝑖,𝑗=1,2,…𝑁.

Lemma 2.15 (see [29]). The Kronecker product has the following properties: (1)(𝛼𝐴)⊗𝐵=𝐴⊗(𝛼𝐵), (2)(𝐴+𝐵)⊗𝐶=𝐴⊗𝐶+𝐵⊗𝐶, (3)(𝐴⊗𝐵)(𝐶⊗𝐷)=(𝐴𝐶)⊗(𝐵𝐷), (4)(𝐴⊗𝐵)^𝑇=𝐴^𝑇⊗𝐵^𝑇.

In the mean of time scale, Schwarz inequality holds similarly.

Lemma 2.16. Suppose 𝑓,𝑔∶𝕋→𝑅 are two rd-continuous functions, and 0,𝑡∈𝕋,𝑡>0. Then ^𝑡₀𝑓(𝑠)𝑔(𝑠)Δ𝑠²≤^𝑡₀𝑓²(𝑠)Δ𝑠^𝑡₀𝑔²(𝑠)Δ𝑠. (2.20)

Remark 2.17. One can prove Lemma 2.16 easily by using analysis knowledge. The proof is not given here for the purpose of space saving.

Corollary 2.18. Suppose 𝑓∶𝕋→ℝ is an rd-continuous function and 0,𝑡∈𝕋,>0. Then ^𝑡₀𝑓(𝑠)Δ𝑠²≤𝑡^𝑡₀𝑓²(𝑠)Δ𝑠. (2.21)

Lemma 2.19. Assume 0,𝑡∈𝕋,𝑡>0. 𝑀∈ℝ^𝑛×𝑛 is a symmetric and positive semidefinite matrix, and 𝑓∶[0,𝑡]_𝕋→ℝ^𝑛 is a vector function. If the integrations concerned are well defined, then the following inequality holds: ^𝑡₀𝑓(𝑠)Δ𝑠^𝑇𝑀^𝑡₀𝑓(𝑠)Δ𝑠≤𝑡^𝑡₀𝑓^𝑇(𝑠)𝑀𝑓(𝑠)Δ𝑠. (2.22)

Proof. Since matrix 𝑀 is symmetric and positive semidefinite, there exists a reversible orthogonal matrix 𝑃=(𝑝_𝑖𝑗)∈ℝ^𝑛×𝑛, such that 𝑀=𝑃⁻¹Λ𝑃=𝑃^𝑇Λ𝑃, (2.23) where Λ=diag{𝜆₁,𝜆₂,…,𝜆_𝑛},𝜆_𝑖≥0(𝑖=1,2,…,𝑛) are the eigenvalues of 𝑀. Then, one has ^𝑡₀𝑓(𝑠)Δ𝑠^𝑇𝑀^𝑡₀𝑓(𝑠)Δ𝑠=^𝑡₀𝑓(𝑠)Δ𝑠^𝑇𝑃^𝑇Λ𝑃^𝑡₀𝑓(𝑠)=𝑃Δ𝑠_𝑡0𝑓(𝑠)Δ𝑠^𝑇Λ𝑃^𝑡₀𝑓(𝑠)Δ𝑠=^𝑡₀𝑃𝑓(𝑠)Δ𝑠^𝑇Λ^𝑡₀(𝑠)=𝑃𝑓Δ𝑠_{𝑡0𝑛𝑖=1}𝑝_1𝑖𝑓_𝑖(𝑠)_𝑛𝑖=1𝑝_2𝑖𝑓_𝑖(𝑠)_𝑛𝑖=1⋮𝑝_𝑛𝑖𝑓_𝑖(𝑠)Δ𝑠^𝑇𝜆₁𝜆₂⋱𝜆_𝑛^{𝑡0𝑛𝑖=1}𝑝_1𝑖𝑓_𝑖(𝑠)_𝑛𝑖=1𝑝_2𝑖𝑓_𝑖(𝑠)_𝑛𝑖=1⋮𝑝_𝑛𝑖𝑓_𝑖(𝑠)Δ𝑠=𝜆₁^𝑡₀^𝑛_𝑖=1𝑝_1𝑖𝑓_𝑖(𝑠)Δ𝑠²+𝜆₂^𝑡₀^𝑛_𝑖=1𝑝_2𝑖𝑓_𝑖(𝑠)Δ𝑠²+⋯+𝜆_𝑛^𝑡₀^𝑛_𝑖=1𝑝_𝑛𝑖𝑓_𝑖(𝑠)Δ𝑠². (2.24) By using Corollary 2.18, one can obtain ^𝑡₀^𝑛_𝑖=1𝑝_𝑗𝑖𝑓_𝑖(𝑠)Δ𝑠²≤𝑡^𝑡₀^𝑛_𝑖=1𝑝_𝑗𝑖𝑓_𝑖(𝑠)²Δ𝑠,𝑗=1,2,…,𝑛; (2.25) then ^𝑡₀𝑓(𝑠)Δ𝑠^𝑇𝑀^𝑡₀𝑓(𝑠)Δ𝑠≤𝜆₁𝑡^𝑡₀^𝑛_𝑖=1𝑝_1𝑖𝑓_𝑖(𝑠)²Δ𝑠+𝜆₂𝑡^𝑡₀^𝑛_𝑖=1𝑝_2𝑖𝑓_𝑖(𝑠)²Δ𝑠+⋯+𝜆_𝑛𝑡^𝑡₀^𝑛_𝑖=1𝑝_𝑛𝑖𝑓_𝑖(𝑠)²Δ𝑠=𝑡^𝑡₀^𝑛_𝑖=1𝑝_1𝑖𝑓_𝑖(𝑠)𝜆₁^𝑛_𝑖=1𝑝_1𝑖𝑓_𝑖(𝑠)+⋯+^𝑛_𝑖=1𝑝_𝑛𝑖𝑓_𝑖(𝑠)𝜆_𝑛^𝑛_𝑖=1𝑝_𝑛𝑖𝑓_𝑖(𝑠)Δ𝑠=𝑡^𝑡₀𝑃𝑓(𝑠)^𝑇Λ𝑃𝑓(𝑠)Δ𝑠=𝑡^𝑡₀𝑓^𝑇(𝑠)𝑃^𝑇Λ𝑃𝑓(𝑠)Δ𝑠=𝑡^𝑡₀𝑓^𝑇(𝑠)𝑀𝑓(𝑠)Δ𝑠. (2.26) The proof is completed.

Lemma 2.20 (See [30]). Let 𝑈=(𝛼_𝑖𝑗)_𝑁×𝑁, 𝑃∈ℝ^𝑛×𝑛, 𝑥=(𝑥^𝑇₁,𝑥^𝑇₂,…,𝑥^𝑇_𝑁)^𝑇 where 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛)^𝑇∈ℝ^𝑛 and 𝑦=(𝑦^𝑇₁,𝑦^𝑇₂,…,𝑦^𝑇_𝑁)^𝑇 where 𝑦_𝑖=(𝑦_𝑖1,𝑦_𝑖2,…,𝑦_𝑖𝑛)^𝑇∈ℝ^𝑛(𝑖=1,2,…,𝑁). If 𝑈=𝑈^𝑇 and each row sum of 𝑈 is zero, then 𝑥^𝑇(𝑈⊗𝑃)𝑦=−_{1≤𝑖<𝑗≤𝑁}𝛼_𝑖𝑗𝑥_𝑖−𝑥_𝑗^𝑇𝑃𝑦_𝑖−𝑦_𝑗. (2.27)

3. Main Results and Proofs

In this section, the main results for global synchronization criteria of the delayed complex networks on time scales are presented.

Theorem 3.1. Suppose Assumptions 2.1 and 2.2 hold. The global synchronization of system (2.18) is achieved if there exist 𝑛×𝑛 positive matrices 𝑃>0, 𝑃₁>0, 𝑃₂>0, 𝑄₁>0, 𝑄₂>0, 𝑅>0, and matrices 𝑀₁, 𝑀₂, 𝑆, and positive scalars 𝜀₁, 𝜀₂, such that the following LMI holds for all 1≤𝑖<𝑗≤𝑁: 𝔻_𝑖𝑗=𝐷₁+𝐷₂+𝐷^𝑇₂+𝐷₃+𝐷^𝑇₃<0, (3.1) where 𝐷₁=diag2𝑃𝐴+𝑄₁+𝑄₂+2𝑀₁+2𝑀₂−𝜀₁𝐿,0_𝑛×𝑛,−𝑄₁−𝜀₂𝐿,−𝑄₂,𝑅−2𝜀₁𝐼_𝑛,−𝑅−2𝜀₂𝐼_𝑛,ℎ𝑃₁+ℎ𝑃₂−𝑆−𝑆^𝑇1,−ℎ𝑃₁1,−ℎ𝑃₂,𝐷₂=𝑂_𝑛×𝑛𝑃𝐴−𝑀₁−𝑀₂𝑂_𝑛×𝑛𝑂_𝑛×𝑛𝐴^𝑇𝑆^𝑇−𝑀₁−𝑀₂𝑂_2𝑛×𝑛−𝑁𝐺_𝑖𝑗Γ^𝑇𝑆^𝑇𝑂_8𝑛×6𝑛𝐵^𝑇𝑆^𝑇𝑂_8𝑛×2n𝐵^𝑇_𝜏𝑆^𝑇𝑂_3𝑛×𝑛,𝐷₃=−𝑁𝐺_𝑖𝑗(𝑃Γ)𝑃𝐵+𝜀₁𝐿̆𝑃𝐵_𝜏−𝑁𝐺_𝑖𝑗(𝑃Γ)𝑃𝐵𝑃𝐵_𝜏𝑂_9𝑛×3𝑛𝑂_𝑛×𝑛𝑂_𝑛×𝑛𝜀₂𝐿̆𝑂_9𝑛×3𝑛𝑂_6𝑛×𝑛𝑂_6𝑛×𝑛𝑂_6𝑛×𝑛 (3.2) with 𝐿=𝐿^𝑇_𝑓𝐿^𝑓+𝐿^𝑓𝑇𝐿_𝑓,𝐿̆=𝐿^𝑇_𝑓+𝐿^𝑓𝑇. (3.3)

Proof. Letting 𝑦(𝑡)=𝐼_𝑁⊗𝐴𝑥(𝑡)+𝐼_𝑁⊗𝐵𝐹(𝑥(𝑡))+𝐼_𝑁⊗𝐵_𝜏𝐹𝑥𝑡−𝜏₁+(𝐺⊗Γ)𝑥𝑡−𝜏₂, (3.4) system (2.18) becomes 𝑥^Δ(𝑡)=𝑦(𝑡)+Λ. (3.5)
Based on the theory of calculus on time scales, we choose the following Lyapunov functional candidate: 𝑉(𝑡)=2𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡)+𝑉₅(𝑡)+𝑉₆(𝑡), (3.6) where 𝑉₁(𝑡)=𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑥(𝑡),𝑉₂(𝑡)=⁰_−𝜏1^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑈⊗𝑃₁𝑦(𝑠)𝑉Δ𝑠Δ𝜃,₃(𝑡)=⁰_−𝜏2^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑈⊗𝑃₂𝑦(𝑠)Δ𝑠Δ𝜃,𝑉₄(𝑡)=^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝑈⊗𝑄₁𝑥(𝑠)𝑉Δ𝑠,₅(𝑡)=^𝑡_𝑡−𝜏2𝑥^𝑇(𝑠)𝑈⊗𝑄₂𝑥(𝑠)Δ𝑠,𝑉₆(𝑡)=^𝑡_𝑡−𝜏1𝐹^𝑇(𝑥(𝑠))(𝑈⊗𝑅)𝐹(𝑥(𝑠))Δ𝑠,𝑈=𝑁−1−1⋯−1−1𝑁−1⋯−1⋯⋯⋯⋯−1−1⋯𝑁−1_𝑁×𝑁 (3.7) Note that 𝑈𝐺=𝐺𝑈=𝑁𝐺. For any matrix 𝐻 with appropriate dimension, one obtains (𝑈⊗𝐻)(𝐺⊗Γ)=(𝑈𝐺)⊗(𝐻Γ)=(𝑁𝐺)⊗(𝐻Γ). (3.8) Calculating the delta derivative 𝑉^Δ(𝑡) along the trajectories of the network (2.1) (or (2.18)), one has 𝑉^Δ₁(𝑡)=𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑥^Δ(𝑡)+𝑥^𝜎𝑇(𝑡)(𝑈⊗𝑃)^𝑇𝑥^Δ(𝑡)=𝑥(𝑡)+𝑥^𝜎(𝑡)^𝑇(𝑈⊗𝑃)𝑥^Δ(𝑡)=𝑥(𝑡)+𝑥^𝜎(𝑡)^𝑇()×𝐼𝑈⊗𝑃_𝑁⊗𝐴𝑥(𝑡)+𝐼_𝑁⊗𝐵𝐹(𝑥(𝑡))+𝐼_𝑁⊗𝐵_𝜏𝐹𝑥𝑡−𝜏₁+(𝐺⊗Γ)𝑥𝑡−𝜏₂=𝑥(𝑡)+Λ+𝑥^𝜎(𝑡)^𝑇(𝑥(𝑡)+((𝐹(𝑥(𝑡+[(𝑈⊗𝑃𝐴))𝑈⊗𝑃𝐵))))𝑈⊗𝑃𝐵_𝜏𝐹𝑥𝑡−𝜏₁+((𝑁𝐺)⊗(𝑃Γ))𝑥𝑡−𝜏₂+(𝑈⊗𝑃)Λ=_{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇(𝑃𝐴)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇()×𝑓𝑥𝑃𝐵_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)+𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑃𝐵_𝜏𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁−𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑁𝐺_𝑖𝑗(𝑃Γ)𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂+_{1≤𝑖<𝑗≤𝑁}𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)^𝑇(𝑃𝐴)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)^𝑇()×𝑓𝑥𝑃𝐵_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)+𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)^𝑇𝑃𝐵_𝜏×𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁−𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)^𝑇𝑁𝐺_𝑖𝑗(𝑃Γ)𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂. (3.9) And synchronously, 𝑉^Δ₁(𝑡)=𝑥^Δ𝑇(𝑡)(𝑈⊗𝑃)^𝑇𝑥(𝑡)+𝑥^Δ𝑇(𝑡)(𝑈⊗𝑃)𝑥^𝜎(𝑡)=𝑥^Δ𝑇(𝑡)(𝑈⊗𝑃)𝑥(𝑡)+𝑥^𝜎(𝑡=𝑥_𝑇)(𝑡)𝐼_𝑁⊗𝐴^𝑇+𝐹(𝑥(𝑡))^𝑇𝐼_𝑁⊗𝐵^𝑇+𝐹𝑥𝑡−𝜏₁^𝑇𝐼_𝑁⊗𝐵_𝜏^𝑇+𝑥^𝑇𝑡−𝜏₂(𝐺⊗Γ)^𝑇+Λ^𝑇(𝑈⊗𝑃)𝑥(𝑡)+𝑥^𝜎(𝑡=𝑥_𝑇)(𝑡)(𝑈⊗(𝑃𝐴))+𝐹(𝑥(𝑡))^𝑇(𝑈⊗(𝑃𝐵))+𝐹𝑥𝑡−𝜏₁^𝑇𝑈⊗𝑃𝐵_𝜏+𝑥^𝑇𝑡−𝜏₂((𝑁𝐺)⊗(𝑃Γ))+Λ^𝑇(𝑈⊗𝑃)𝑥(𝑡)+𝑥^𝜎(𝑡)=_{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇(𝑃𝐴)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇×(𝑃𝐵)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇×𝑃𝐵_𝜏𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂^𝑇𝑁𝐺_𝑖𝑗(𝑃Γ)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+_{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇(𝑃A)𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)+𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇×(𝑃𝐵)𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)+𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇×𝑃𝐵_𝜏𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)−𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂^𝑇𝑁𝐺_𝑖𝑗(𝑃Γ)𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡),𝑉^Δ₂(𝑡)=𝜏₁𝑦^𝑇(𝑡)𝑈⊗𝑃₁𝑦(𝑡)−^𝑡_𝑡−𝜏1𝑦^𝑇(𝑠)𝑈⊗𝑃₁𝑦(𝑠)Δ𝑠≤ℎ𝑦^𝑇(𝑡)𝑈⊗𝑃₁𝑦(𝑡)−^𝑡_𝑡−𝜏1𝑦^𝑇(𝑠)𝑈⊗𝑃₁𝑦(𝑠)Δ𝑠≤ℎ𝑦^𝑇(𝑡)𝑈⊗𝑃₁𝑦(𝑡)−1𝜏₁^𝑡_𝑡−𝜏1𝑦(𝑠)𝑑𝑠^𝑇𝑈⊗𝑃₁^𝑡_𝑡−𝜏1𝑦(𝑠)Δ𝑠≤ℎ_{1≤𝑖<𝑗≤𝑁}𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)^𝑇𝑃₁𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)−1ℎ_{1≤𝑖<𝑗≤𝑁}^𝑡_𝑡−𝜏1(𝑦_𝑖(𝑠)−𝑦_𝑗(𝑠))Δ𝑠^𝑇𝑃₁^𝑡_𝑡−𝜏1𝑦_𝑖(𝑠)−𝑦_𝑗(𝑠)Δ𝑠=ℎ_{1≤𝑖<𝑗≤𝑁}𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)^𝑇𝑃₁𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)−1ℎ_{1≤𝑖<𝑗≤𝑁}𝑌_𝑖⁽¹⁾−𝑌_𝑗^(1)𝑇𝑃₁𝑌_𝑖⁽¹⁾−𝑌_𝑗⁽¹⁾, (3.10)(3.11) where 𝑌_𝑖⁽¹⁾(𝑡)=^𝑡_𝑡−𝜏1𝑦_𝑖(𝑠)Δ𝑠. (3.12) Similarly, one has 𝑉^Δ₃(𝑡)=𝜏₂𝑦^𝑇(𝑡)𝑈⊗𝑃₂𝑦(𝑡)−^𝑡_𝑡−𝜏2𝑦^𝑇(𝑠)𝑈⊗𝑃₂𝑦(𝑠)Δ𝑠≤ℎ_{1≤𝑖<𝑗≤𝑁}𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)^𝑇𝑃₂𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)−1ℎ_{1≤𝑖<𝑗≤𝑁}𝑌_𝑖⁽²⁾−𝑌_𝑗^(2)𝑇𝑃₂𝑌_𝑖⁽²⁾−𝑌_𝑗⁽²⁾, (3.13) where 𝑌_𝑖⁽²⁾(𝑡)=^𝑡_𝑡−𝜏2𝑦_𝑖(𝑠)𝑉Δ𝑠,_Δ4(𝑡)=𝑥^𝑇(𝑡)𝑈⊗𝑄₁𝑥(𝑡)−𝑥^𝑇𝑡−𝜏₁𝑈⊗𝑄₁𝑥𝑡−𝜏₁=_{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑄₁𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−_{1≤𝑖<j≤𝑁}𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁^𝑇𝑄₁𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁. (3.14) Similarly, one has 𝑉^Δ₅(𝑡)=𝑥^𝑇(𝑡)𝑈⊗𝑄₂𝑥(𝑡)−𝑥^𝑇𝑡−𝜏₂𝑈⊗𝑄₂𝑥𝑡−𝜏₂=_{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑄₂𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−_{1≤𝑖<𝑗≤𝑁}𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂^𝑇𝑄₂𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂,𝑉^Δ₆(𝑡)=𝐹^𝑇(𝑥(𝑡))(𝑈⊗𝑅)𝐹(𝑥(𝑡))−𝐹^𝑇𝑥𝑡−𝜏₁(𝑈⊗𝑅)𝐹𝑥𝑡−𝜏₁=_{1≤𝑖<𝑗≤𝑁}𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇𝑅𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)−_{1≤𝑖<𝑗≤𝑁}𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇𝑅𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁. (3.15)(3.16)
By formula (2.18), for any matrices 𝑀₁, the following equality is satisfied 2𝑥^𝑇(𝑡)𝑈⊗𝑀₁𝑥(𝑡)−𝑥𝑡−𝜏₁−^𝑡_𝑡−𝜏1𝑦(𝑠)Δ𝑠=0, (3.17) which can be rewritten as 2_{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑀₁𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁−𝑌_𝑖⁽¹⁾−𝑌_𝑗⁽¹⁾=0. (3.18) Similarly, for any matrices 𝑀₂, one has 2_{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑀₂𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂−𝑌_𝑖⁽²⁾−𝑌_𝑗⁽²⁾=0. (3.19)
In addition, for any matrix 𝑆, the following equality is always true: 2𝑦^𝑇(𝑡)(𝑈⊗𝑆)𝐼_𝑁⊗𝐴𝑥(𝑡)+𝐼_𝑁𝐹(𝑥(𝑡+𝐼⊗𝐵))_𝑁⊗𝐵_𝜏𝐹𝑥𝑡−𝜏₁+(𝐺⊗Γ)𝑥𝑡−𝜏₂−𝑦(𝑡)=2𝑦^𝑇(𝑡(𝑥(𝑡)+((𝐹(𝑥(𝑡+)[(𝑈⊗𝑆𝐴))𝑈⊗𝑆𝐵))))U⊗𝑆𝐵_𝜏𝐹𝑥𝑡−𝜏₁+(𝑁𝐺⊗(𝑆Γ))𝑥𝑡−𝜏₂−(𝑈⊗𝑆)𝑦(𝑡)=0, (3.20) that is, _{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇2𝐴^𝑇𝑆^𝑇+𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇2𝐵^𝑇𝑆^𝑇+𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇2𝐵^𝑇_𝜏𝑆^𝑇−𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂^𝑇2𝑁𝐺_𝑖𝑗Γ^𝑇𝑆^𝑇−𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)^𝑇𝑆+𝑆^𝑇𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)=0. (3.21)
Moreover, from Assumption 2.2, for 𝜀₁>0, one obtains 𝜀₁𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)𝑓(𝑥_𝑖(𝑡))−𝑓(𝑥_𝑗(𝑡))^𝑇𝐿̆−𝐿̆𝐿−_𝑇𝑥2𝐼^𝑖(𝑡)−𝑥_𝑗(𝑡)𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)≤0, (3.22) where 𝐿=𝐿^𝑇_𝑓𝐿^𝑓+𝐿^𝑓𝑇𝐿_𝑓 and 𝐿̆=𝐿^𝑇_𝑓+𝐿^𝑓𝑇.
Applying ∑_{1≤𝑖<𝑗≤𝑁} on both sides of the above inequality, the following formula can be obtained _{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇2𝜀₁𝐿̆𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)−𝜀₁𝐿𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇2𝜀₁𝐼_𝑛𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)≥0. (3.23)
Similarly, for 𝜀₂>0, one has 𝜀₂𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁𝑓(𝑥_𝑖(𝑡−𝜏₁))−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇𝐿̆−𝐿̆𝐿−_𝑇𝑥2𝐼^𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁≤0, (3.24) and then _{1≤𝑖<𝑗≤𝑁}𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁^𝑇×2𝜀₂𝐿̆𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁−𝜀₂𝐿𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁−𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇2𝜀₂𝐼_𝑛𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁≥0. (3.25) From (3.9)–(3.23), we have 𝑉^Δ(𝑡)≤_{1≤𝑖<𝑗≤𝑁}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇2(𝑃𝐴)𝑄₁+𝑄₂+2𝑀₁+2𝑀₂−𝜀₁𝐿𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇(𝑃𝐵)+2𝜀₁𝐿̆𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)+𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑃𝐵_𝜏𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁−𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑁𝐺_𝑖𝑗(𝑃Γ)+2𝑀₂𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂+𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)^𝑇(𝑃𝐴)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)^𝑇()×𝑓𝑥𝑃𝐵_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡+𝑥_𝜎𝑖)(𝑡)−𝑥^𝜎_𝑗(𝑡)^𝑇𝑃𝐵_𝜏𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁−𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)^𝑇𝑁𝐺_𝑖𝑗(𝑃Γ)𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂+𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇(𝑃𝐵)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡+𝑓𝑥_𝑖)𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇𝑃𝐵_𝜏𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂^𝑇𝑁𝐺_𝑖𝑗(𝑃Γ)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇(𝑃𝐴)𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)+𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇()×𝑥𝑃𝐵_𝜎𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)+𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇𝑃𝐵_𝜏𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)−𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂^𝑇𝑁𝐺_𝑖𝑗(𝑃Γ)𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)+𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)^𝑇ℎ𝑃₁+ℎ𝑃₂−𝑆−𝑆^𝑇𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡−1)ℎ𝑌_𝑖⁽¹⁾−𝑌_𝑗^(1)𝑇𝑃₁𝑌_𝑖⁽¹⁾−𝑌_𝑗⁽¹⁾−1ℎ𝑌_𝑖⁽²⁾−𝑌_𝑗^(2)𝑇𝑃₂𝑌_𝑖⁽²⁾−𝑌_𝑗⁽²⁾−𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁^𝑇𝑄₁+𝜀₂𝐿𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁−𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂^𝑇𝑄₂𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂+𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇𝑅−2𝜀₁𝐼_𝑛𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡−𝑓𝑥_𝑖)𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇𝑅+2𝜀₂𝐼_𝑛×𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁−2(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))^𝑇𝑀₁𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁+𝑌_𝑖⁽¹⁾−𝑌_𝑗⁽¹⁾−2𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑀₂𝑌_𝑖⁽²⁾−𝑌_𝑗⁽²⁾+𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇2𝐴^𝑇𝑆^𝑇+𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇2𝐵^𝑇𝑆^𝑇+𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁^𝑇2𝐵^𝑇_𝜏𝑆^𝑇−𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂^𝑇2𝑁𝐺_𝑖𝑗Γ^𝑇𝑆^𝑇𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)+𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁^𝑇2𝜀₂𝐿̆𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁=_{1≤𝑖<𝑗≤𝑁}𝜎^𝑇_𝑖𝑗(𝑡)𝔻_𝑖𝑗𝜎_𝑖𝑗(𝑡)<0, (3.26) where 𝜎_𝑖𝑗(𝑡)=𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)𝑥^𝜎_𝑖(𝑡)−𝑥^𝜎_𝑗(𝑡)𝑥_𝑖𝑡−𝜏₁−𝑥_𝑗𝑡−𝜏₁𝑥_𝑖𝑡−𝜏₂−𝑥_𝑗𝑡−𝜏₂𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)𝑓𝑥_𝑖𝑡−𝜏₁−𝑓𝑥_𝑗𝑡−𝜏₁𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)𝑌_𝑖⁽¹⁾(𝑡)−𝑌_𝑗⁽¹⁾(𝑡)𝑌_𝑖⁽²⁾(𝑡)−𝑌_𝑗⁽²⁾(𝑡), (3.27) and 𝔻_𝑖𝑗 is as defined in (3.1).
From condition (3.1), it is guaranteed that all the subsystems in (2.1) are globally synchronized for any fixed time delays 𝜏_𝑖∈(0,ℎ](𝑖=1,2). The proof is completed.

Specially, in the case of system (2.5) with continuous time, the following corollary can be obtained.

Corollary 3.2. Suppose Assumptions 2.1 and 2.2 hold. The global synchronization of system (2.5) is achieved if there exist 𝑛×𝑛 positive matrices 𝑃>0, 𝑃₁>0, 𝑃₂>0, 𝑄₁>0, 𝑄₂>0, 𝑅>0, and matrices 𝑀₁, 𝑀₂, 𝑆, and positive scalars 𝜀₁, 𝜀₂, such that the following LMI holds for all 1≤𝑖<𝑗≤𝑁: 𝔻_𝑖𝑗=𝐷₁+𝐷₂+𝐷^𝑇₂+𝐷₃+𝐷^𝑇₃<0, (3.28) where 𝐷₁=diag2𝑃𝐴+𝑄₁+𝑄₂+2𝑀₁+2𝑀₂−𝜀₁𝐿,−𝑄₁−𝜀₂𝐿,−𝑄₂,𝑅−2𝜀₁𝐼_𝑛,−𝑅−2𝜀₂𝐼_𝑛,ℎ𝑃₁+ℎ𝑃₂−𝑆−𝑆^𝑇1,−ℎ𝑃₁1,−ℎ𝑃₂,𝐷₂=𝑂_𝑛×𝑛−𝑀₁−𝑀₂𝑂_𝑛×𝑛𝑂_𝑛×𝑛𝐴^𝑇𝑆^𝑇−𝑀₁−𝑀₂𝑂_𝑛×𝑛−𝑁𝐺_𝑖𝑗Γ^𝑇𝑆^𝑇𝑂_7𝑛×5𝑛𝐵^𝑇𝑆^𝑇𝑂_7𝑛×2𝑛𝐵^𝑇_𝜏𝑆^𝑇𝑂_3𝑛×𝑛,𝐷₃=−𝑁𝐺_𝑖𝑗(𝑃Γ)𝑃𝐵+𝜀₁𝐿̆𝑃𝐵_𝜏𝑂_8𝑛×2𝑛𝑂_𝑛×𝑛𝑂_𝑛×𝑛𝜀₂𝐿̆𝑂_8𝑛×3𝑛𝑂_6𝑛×𝑛𝑂_6𝑛×𝑛𝑂_6𝑛×𝑛 (3.29) with 𝐿=𝐿^𝑇_𝑓𝐿^𝑓+𝐿^𝑓𝑇L_𝑓,𝐿̆=𝐿^𝑇_𝑓+𝐿^𝑓𝑇. (3.30)

4. A Numerical Example

In this part, a numerical example is given to verify the theoretical result.

Consider the following complex networks (4.1) with time delays on time scale 𝕋:𝑥^Δ_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡)+𝐵_𝜏𝑓𝑥_𝑖𝑡−𝜏₁+¹⁰_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗𝑡−𝜏₂+Λ_𝑖,𝑖=1,2,…,10, (4.1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡))^𝑇(𝑖=1,2,…,10) is the state vector of the 𝑖th subsystem. Choose the coupling matrix 𝐺 and the linking matrix Γ as𝐺=,−10000000010−10000000100−10000001000−10000010000−10000100000−10001000000−10010000000−10100000000−11111111111−9Γ=0.350.20.20.4,Λ_𝑖=(1,1)^𝑇. (4.2) The other parameters are as follows:𝐴=−120.20.2−10,𝐵=2−0.1−53,𝐵_𝜏=−1.5−0.1−0.2−2.5,𝜏₁=𝜏₂=2. (4.3)

The nonlinear function is given by 𝑓(𝑥_𝑖(𝑡))=(𝑓₁(𝑥_𝑖1(𝑡)),𝑓₂(𝑥_𝑖2(𝑡)))^𝑇, with 𝑓_𝑗(𝑥_𝑖𝑗(𝑡))=tanℎ(𝑥_𝑖𝑗(𝑡))(𝑗=1,2). It is easy to see that the nonlinear functions satisfy all the assumptions. By using the Matlab LMI Toolbox, LMI (3.1) is feasible. According to Theorem 3.1, one concludes that the complex networks (4.1) with delays on time scale 𝕋 can achieve global synchronization. The synchronization errors of the complex networks (4.1) are plotted in Figure 1.

(a)

(b)

Figure 1 

Synchronization error of system (4.1).

5. Conclusions

In this paper, we have investigated the global synchronization of a kind of delayed complex networks on time scales. Utilizing the theory of calculus on time scales and the properties of Kronecker product, the synchronization conditions have been derived through a suitably Lyapunov functional. The obtained synchronization criterion which is expressed in the form of LMIs can be easily verified by the standard numerical software such as Matlab LMI toolbox. The obtained results are novel since there are few works about the synchronization of delayed complex networks on time scales. In addition, the approach utilized in this paper can be considered as a universal framework for the study of other complex systems on time scales.