Mean-Square Exponential Synchronization of Stochastic Complex Dynamical Networks with Switching Topology by Impulsive Control

Abstract

This paper investigates the mean-square exponential synchronization issues of delayed stochastic complex dynamical networks with switching topology and impulsive control. By using the Lyapunov functional method, impulsive control theory, and linear matrix inequality (LMI) approaches, some sufficient conditions are derived to guarantee the mean-square exponential synchronization of delay complex dynamical network with switch topology, which are independent of the network size and switch topology. Numerical simulations are given to illustrate the effectiveness of the obtained results in the end.

1. Introduction

Complex networks are everywhere in nature and our daily life, such as the Internet, ecosystems, social networks, World Wide Web, and neural networks. A complex network can be described by a set of nodes and edges interconnecting these nodes together. During the last two decades, complex networks have been focused on by scientists from various fields, such as mathematical, engineering, and social and economic science. There are many literatures concerning the collective behaviors of complex networks [1–11]. Among them, synchronization is the most interesting phenomenon [3–11], because the synchronization is a kind of typical collective behaviors exhibited in many natural systems.

Due to many complex systems may experience abrupt changes in their connection caused by some phenomena such as link failures, component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbance, and so forth. Thus, the synchronization of complex network with switching topology has been studied in [12–17]. In [12], Yao et al. studied the synchronization of a general complex dynamical network with switching topology and the time-varying coupling is unknown but bounded. Wang et al. [13] investigated the synchronization issues of complex dynamical networks with switching topology. Yu et al. [14] explored the synchronization of switched linearly coupled neural networks with delay. Some sufficient conditions were given to guarantee the global synchronization. In [15], Liu et al. studied the local and global exponential synchronization of complex dynamical network with switching topology and time-varying coupling delays. In [16], some sufficient conditions were given to guarantee the synchronization of leader following issues with switching connective network and coupling delay. In [17], authors investigated the consensus problem in mean square for uncertain multiagent systems with stochastic measurement noises and symmetric or asymmetric time-varying delays.

Uncertainties commonly exist in the real world, such as stochastic forces on the physical systems and noisy measurements caused by environmental uncertainties. Thus, a stochastic behavior should be produced instead of a deterministic one [18]. In fact, signals transmitted between nodes of complex networks are unavoidably subject to stochastic perturbations from environment, which may cause information contained in these signals to be lost [19]. Therefore, stochastic perturbations should be considered [18–23]. In [19–21], stochastic perturbations are all one-dimensional, which means that the signal transmitted by nodes is influenced by the same noise. In [18, 22], the authors considered stochastic synchronization of coupled neural networks, in which noise perturbations are vector forms. Vector-form perturbation means that different nodes are influenced by different noise, which is more practical in the real world. In [23], authors investigated the mean-square exponential synchronization of stochastic complex networks with Markovian switching and time-varying delays by using the pinning control method.

In many systems, the impulsive effects are common phenomena due to instantaneous perturbations at certain moments [24–28]. In the past several years, impulsive control strategies have been widely used to stabilize and synchronize coupled complex dynamical system, such as signal processing system, computer networks, automatic control systems, and telecommunications. In [24], Cai et al. investigated the robust impulsive synchronization of complex delayed dynamical networks. Yang et al. [25] studied the exponential synchronization of complex dynamical network with a coupling delay and impulsive control. Zhu et al. gave some global impulsive exponential synchronization criteria of time-delayed coupled chaotic systems [27]. In [28], some sufficient conditions were given to guarantee the consensus of nonlinear multiagent systems with switching topology.

Based on the above discussion, studying the synchronization problem of the complex dynamical network with switch topology and impulsive effects is very useful and meaningful. It should be pointed out that exponential synchronization of the coupling delay complex dynamical networks with switch topology and impulsive effects has received very little research attention.

In this paper, we investigate the problem of exponential synchronization of coupling delay complex dynamical network with switching topology and impulsive effects, basing on the Lyapunov theory and impulsive control theory and by assuming that there exists finite connective topology of the complex dynamical network, and the connective topology may be switched (or jumped) from one to another at different moments. For controlling, the synchronization state can be any weighting average of the network states. It means that each node of the dynamical network can contribute to synchronization of the network in its weight. Finally, some sufficient conditions are given to guarantee the synchronization of the complex dynamical network, which are independent of the network size and switch topology.

2. Model and Preliminaries

The switching complex dynamical networks investigated in this paper consist of 𝑁 nodes, whose state is described as follows: 𝑑𝑥_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑎_𝑖𝑗(𝑟(𝑡))Σ𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗(𝑟(𝑡))Σ𝑥_𝑗𝑡−𝜏_𝑐(𝑡)𝑑𝑡+𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟(𝑡)𝑑𝑤_𝑖(𝑡),𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝜖_𝑖𝑘𝑥_𝑖𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,𝑖=1,2,…,𝑁, (1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛 is the state vector of the 𝑖th node of the network, 𝑓(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))=[𝑓₁(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))),𝑓₂(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))),…,𝑓_𝑛(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))]^𝑇 is a continuous vector-valued function, Σ=diag(𝜚₁,𝜚₂,…,𝜚_𝑛) is an inner coupling of the networks that satisfies 𝜚_𝑗>0, 𝑗=1,2,…,𝑛, and 𝑟(𝑡)∶[0,∞)→ℵ={1,2,…,𝑀} is a switching signal, which is a piecewise constant function. Here, 𝐴(𝑟(𝑡))=[𝑎_𝑖𝑗(𝑟(𝑡))]∈ℝ^𝑛×𝑛 and 𝐵(𝑟(𝑡))=[𝑏_𝑖𝑗(𝑟(𝑡))]∈ℝ^𝑛×𝑛 are the outer-coupling matrices of the network at time 𝑡 at nodes 𝑟(𝑡), 𝑡−𝜏_𝑐(𝑡), and 𝑟(𝑡), respectively, such that 𝑎_𝑖𝑗(𝑟(𝑡))≥0 for 𝑖≠𝑗, 𝑎_𝑖𝑖(𝑟(𝑡))=−∑^𝑁_{𝑗=𝑖,𝑗≠𝑖}𝑎_𝑖𝑗(𝑟(𝑡)), 𝑏_𝑖𝑗(𝑟(𝑡))≥0 for 𝑖≠𝑗, and 𝑏_𝑖𝑖(𝑟(𝑡))=−∑^𝑁_{𝑗=𝑖,𝑗≠𝑖}𝑏_𝑖𝑗(𝑟(𝑡)). 𝜏(𝑡) is the inner time-varying delay satisfying 𝜏≥𝜏(𝑡)≥0 and 𝜏_𝑐(𝑡) is the coupling time-varying delay satisfying 𝜏_𝑐≥𝜏_𝑐(𝑡)≥0. Finally, 𝜎_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡−𝜏_𝑐(𝑡)),𝑟(𝑡))=𝜎_𝑖(𝑡,𝑥₁(𝑡),…,𝑥_𝑛(𝑡),𝑥₁(𝑡−𝜏(𝑡)),…,𝑥_𝑛(𝑡−𝜏(𝑡)),𝑥₁𝜏(𝑡−_𝑐(𝑡)),…,𝑥_𝑛(𝑡−𝜏_𝑐(𝑡)),𝑟(𝑡))∈ℝ^𝑛×𝑛 and 𝑤_𝑖(𝑡)=(𝑤_𝑖1(𝑡),𝑤_𝑖2(𝑡),…,𝑤_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛 is a bounded vector-form Weiner process, satisfying 𝔼𝑤_𝑖𝑗(𝑡)=0,𝔼𝑤²_𝑖𝑗(𝑡)=1,𝔼𝑤_𝑖𝑗(𝑡)𝑤_𝑖𝑗(𝑠)=0(𝑠≠𝑡). (2)𝜖_𝑖𝑘 is the 𝑖th node impulsive gain at 𝑡=𝑡_𝑘. The discrete set {𝑡_𝑘} satisfies 𝑡_𝑘=𝑡_𝑘−1+𝑇, 𝑡₀≥0, 𝑡_𝑘→+∞ as 𝑘→+∞, note 𝑥(𝑡⁻_𝑘)=lim_𝑡→𝑡⁻_𝑘𝑥(𝑡), and 𝑥(𝑡⁺_𝑘)=lim_𝑡→𝑡⁺_𝑘𝑥(𝑡)=𝑥(𝑡_𝑘). In this paper, 𝐴(𝑟(𝑡)) is assumed to be irreducible in the sense that there are no isolated nodes.

The initial conditions associated with (1) are 𝑥_𝑖(𝑠)=𝜉_𝑖(𝑠),−𝜏^̌‌≤𝑠≤0,𝑖=1,2,…,𝑁, (3) where 𝜏^̌‌=max{𝜏(𝑡),𝜏_𝑐(𝑡)}, 𝜉_𝑖∈𝐶^𝑏_ℱ0([−𝜏^̌‌,0],ℝ^𝑛) with the norm ‖𝜉_𝑖‖²=sup_{−̌‌𝜏≤𝑠≤0}𝜉_𝑖(𝑠)^𝑇𝜉_𝑖(𝑠) and our objective is to control system (1) so that it stays in the trajectory 𝑠(𝑡)∈ℝ^𝑛 of the system 𝑑𝑠(𝑡)=𝑓(𝑡,𝑠(𝑡),𝑠(𝑡−𝜏(𝑡)))𝑑𝑡. (4)

Remark 1. In general, the synchronization state 𝑠(𝑡) may be an equilibrium point, a periodic orbit, or a chaotic attractor.
The following assumptions will be used throughout this paper for establishing the synchronization conditions.   (H1) 𝜏(𝑡) and 𝜏_𝑐(𝑡) are bounded and continuously differentiable functions such that 0<𝜏(𝑡)≤𝜏, 𝜏̇(𝑡)<𝜏<1, 0<𝜏_𝑐(𝑡)≤𝜏_𝑐, and 𝜏̇_𝑐(𝑡)<𝜏_𝑐<1. Let 𝜏^̌‌=max{𝜏,𝜏_𝑐} and 𝜏^̌‌=max{𝜏,𝜏_𝑐}. (H2) Let 𝜎(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒(𝑡−𝜏_𝑐(𝑡)),𝑟)=𝜎(𝑡,𝑒₁(𝑡),…,𝑒_𝑁(𝑡),𝑒₁(𝑡−𝜏(𝑡)),…,𝑒_𝑁(𝑡−𝜏(𝑡)),𝑒₁(𝑡−𝜏_𝑐(𝑡)),…,𝑒_𝑁(𝑡−𝜏_𝑐(𝑡)),𝑟). Then there exist positive definite constant matrices Υ^𝑟_𝑖1, Υ^𝑟_𝑖2, and Υ^𝑟_𝑖3 for 𝑖=1,2,…,𝑁 and 𝑟=1,2,…,𝑀 such that Tr𝜎_𝑖𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡),𝑟^𝑇×𝜎_𝑖𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡)≤,𝑟_𝑁𝑗=1𝑒_𝑗(𝑡)^𝑇Υ^𝑟_𝑖1𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑒_𝑗(𝑡−𝜏(𝑡))^𝑇Υ^𝑟_𝑖2𝑒_𝑗((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑒_𝑗𝑡−𝜏_𝑐(𝑡)^𝑇Υ^𝑟_𝑖3𝑒_𝑗𝑡−𝜏_𝑐(𝑡). (5) (H3) 𝜎_𝑖(𝑡,𝑠(𝑡),…,𝑠(𝑡),𝑠(𝑡−𝜏(𝑡)),…,𝑠(𝑡−𝜏(𝑡)),𝑠(𝑡−𝜏_𝑐(𝑡)),…,𝑠(𝑡−𝜏_𝑐(𝑡)),𝑟(𝑡))=0.

Define error state 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡), 𝑑𝑒_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖((𝑡((𝑡)((𝑡+𝑡−𝜏))−𝑓𝑡,𝑠,𝑠𝑡−𝜏)))_𝑁𝑗=1𝑎_𝑖𝑗(𝑟(𝑡))Σ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗(𝑟(𝑡))Σ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)+𝑢_𝑖(𝑡)𝑑𝑡+𝜎_𝑖𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡),𝑟(𝑡)𝑑𝑤_𝑖(𝑡),𝑡≠𝑡_𝑘,Δ𝑒_𝑖𝑡_𝑘=𝑒_𝑖𝑡⁺_𝑘−𝑒_𝑖𝑡⁻_𝑘=𝜖_𝑖𝑘𝑒_𝑖𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,𝑖=1,2,…,𝑁. (6)

Definition 2. The complex network (1) is said to be exponentially synchronized in mean square if the trivial solution of system (6) is such that ^𝑁_𝑖=1𝔼𝑒_𝑖(𝑡,𝑡₀,𝜉_𝑖)²≤𝐾𝑒^−𝜅𝑡, (7) for some 𝐾>0 and some 𝜅>0 for any initial data 𝜉_𝑖∈𝒞^𝑏_ℱ0([−𝜏,0];ℝ^𝑛).

Definition 3. A continuous function 𝑓(𝑡,𝑥,𝑦)∶[0,+∞]×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 is said to belong to the function class QUAD, denoted by 𝑓∈QUAD(𝑃,Δ,𝜂,𝜃) for some given matrix Σ=diag{𝜚₁,𝜚₂,…,𝜚_𝑛} if there exist a positive definite diagonal matrix 𝑃=diag{𝑝₁,𝑝₂,…,𝑝_𝑛}, a diagonal matrix Δ=diag{𝛿₁,𝛿₂,…,𝛿_𝑛}, and constants 𝜂>0,𝜃>0 such that 𝑓(⋅) satisfies the condition 𝑥−𝑦^𝑇𝑃𝑓(𝑡,𝑥,𝑧)−𝑓𝑡,𝑦,𝑤−ΔΣ𝑥−𝑦≤−𝜂𝑥−𝑦^𝑇𝑥−𝑦+𝜃(𝑧−𝑤)^𝑇(𝑧−𝑤) (8) for all 𝑥,𝑦,𝑧,𝑤∈ℝ^𝑛.

Remark 4. The function class QUAD includes almost all the well-known chaotic systems with or without delays such as the Lorenz system, the Rössler system, the Chen system, the delayed Chua’s circuit, the logistic delayed differential system, the delayed Hopfield neural network, and the delayed CNNs. We shall simply write 𝑝^̌‌=max𝑝₁,𝑝₂,…,𝑝_𝑛,𝑝=min𝑝₁,𝑝₂,…,𝑝_𝑛,_̌‌𝛿=max𝛿₁,𝛿₂,…,𝛿_𝑛. (9)

In order to derive the main results, it is necessary to propose the following lemmas.

Lemma 5 (see [16]). The following linear matrix inequality 𝑆𝑄𝑆_𝑇𝑅>0, (10) where 𝑄=𝑄^𝑇,𝑅=𝑅^𝑇, is equivalent to one of the following conditions: (i)𝑄>0,𝑅−𝑆^𝑇𝑄⁻¹𝑆>0. (ii)𝑅>0,𝑄−𝑆^𝑇𝑅⁻¹𝑆>0.

3. Main Result

In this section, we investigate the exponential stability condition of the error system (6). Some new criteria are presented for the exponential synchronization of network (1) based on the Lyapunov functional method, linear matrix inequality approach, and impulsive control theory.

Theorem 6. Let assumptions (H1) and (H2) be true and let 𝑓∈QUAD(𝑃,Δ,𝜂,𝜃). If there exist positive constants 𝛼_𝑟 and 𝛽_𝑟 such that 𝐴(𝑟)^𝑠+𝛿^̌‌𝐼_𝑁−𝛼_𝑟𝐼_𝑁𝐵(𝑟)2𝐵(𝑟)^𝑇2−𝛽_𝑟𝐼_𝑁≤0,𝑓𝑜𝑟𝑟=1,2,…,𝑀,𝛾+𝑎^̌‌+𝑏1−𝜏𝑒^𝛾𝜏+^̌‌𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐×1+𝑇𝜏^̌‌𝑇+2ln|1+𝜖|𝑇−𝛾≤−𝜂, (11)(12) where 𝑎_𝑟=𝜆_min−2𝜂𝐼_𝑛+𝑝^̌‌∑^𝑁_𝑖=1Υ^𝑟_𝑖1+2𝛼_𝑟𝑃Σ_̌‌𝑝,𝑎^̌‌=max_𝑟∈𝑆𝑎_𝑟,𝑏_𝑟=𝜆_max∑^𝑁_𝑖=1𝑃Υ^𝑟_𝑖2+2𝜃𝐼_𝑁𝑝,𝑏^̌‌=max_𝑟∈𝑆𝑏_𝑟,𝑐_𝑟=𝜆_max∑^𝑁_𝑖=1𝑃Υ^𝑟_𝑖3+2𝛽_𝑟𝑃Σ𝑝,𝑐^̌‌=max_𝑟∈𝑆𝑐_𝑟,|1+𝜖|=max_{𝑖=1,2,…,𝑁,𝑘∈𝑍+}1+𝜖_𝑖𝑘, (13) then the solutions 𝑥₁(𝑡),𝑥₁(𝑡),…,𝑥_𝑁(𝑡) of system (6) are globally and exponentially stable.

Proof. Define the Lyapunov-Krasovskii function 𝑉((𝑡=1𝑡,𝑒))2^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡) (14) and let 𝑒^∼𝑘(𝑡)=(𝑒_1𝑘(𝑡),𝑒_2𝑘(𝑡),…,𝑒_𝑁𝑘(𝑡))^𝑇, 𝑘=1,2,…,𝑛. For 𝑟(𝑡)=𝑟, we have ((𝑡))=ℒ𝑉𝑡,𝑒,𝑟_𝑁𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖((𝑡((𝑡)((𝑡+𝑡−𝜏))−𝑓𝑡,𝑠,𝑠𝑡−𝜏)))^𝑁_𝑗=1𝑎_𝑖𝑗(𝑟)Σ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗(𝑟)Σ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)+12^𝑁_𝑖=1Tr𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟^𝑇×𝑃𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)=,𝑟_𝑁𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖((𝑡((𝑡)((𝑡𝑡−𝜏))−𝑓𝑡,𝑠,𝑠𝑡−𝜏)))−ΔΣ𝑒_𝑖(𝑡+_𝑁𝑖=1)𝑒_𝑖(𝑡)^𝑇𝑃ΔΣ𝑒_𝑖(𝑡)+^𝑁𝑁_𝑖=1𝑗=1𝑎_𝑖𝑗(𝑟)𝑒_𝑖(𝑡)^𝑇𝑃Σ𝑒_𝑗(𝑡)+^𝑁𝑁_𝑖=1𝑗=1𝑏_𝑖𝑗(𝑟)𝑒_𝑖(𝑡)^𝑇𝑃Σ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)+12^𝑁_𝑖=1Tr𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟^𝑇×𝑃𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)≤,𝑟−𝜂^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+𝜃^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇𝑒_𝑖((𝑡+𝑡−𝜏))_𝑛𝑘=1𝑝_𝑘𝜚_𝑘𝛿𝑒_𝑘^∼𝑘(𝑡)𝑒^𝑇∼𝑘(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝜚𝑒_𝑘^∼𝑘(𝑡)^𝑇𝐴(𝑟)𝑒^∼𝑘(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝜚𝑒_𝑘^∼𝑘(𝑡)^𝑇𝐵(𝑟)𝑒^∼𝑘𝑡−𝜏_𝑐(𝑡)+12^̌‌𝑝^𝑁_𝑗=1^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇Υ^𝑟_𝑗1𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇Υ^𝑟_𝑗2𝑒_𝑖((𝑡+𝑡−𝜏))_𝑁𝑖=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇Υ^𝑟_𝑗3𝑒_𝑖𝑡−𝜏_𝑐(𝑡)=−𝜂^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+𝜃^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇𝑒_𝑖((𝑡+1𝑡−𝜏))2^̌‌𝑝^𝑁𝑁_𝑖=1𝑗=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇Υ^𝑟_𝑗3𝑒_𝑖𝑡−𝜏_𝑐(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝜚𝑒_𝑘^∼𝑘(𝑡)^𝑇𝐴(𝑟)𝑒^∼𝑘(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝜚𝑒_𝑘^∼𝑘(𝑡)^𝑇𝐵(𝑟)𝑒^∼𝑘𝑡−𝜏_𝑐(𝑡)+12^̌‌𝑝^𝑁𝑁_𝑖=1𝑗=1𝑒_𝑖(𝑡)^𝑇Υ^𝑟_𝑗1𝑒_𝑖(𝑡)+^𝑁𝑁_𝑖=1𝑗=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇Υ^𝑟_𝑗2𝑒_𝑖((𝑡+𝑡−𝜏))_𝑛𝑘=1𝑝_𝑘𝜚_𝑘𝛿𝑒_𝑘^∼𝑘(𝑡)𝑒^𝑇∼𝑘(𝑡)=^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇−𝜂𝐼_𝑁+12^̌‌𝑝^𝑁_𝑗=1Υ^𝑟_𝑗1+𝛼_𝑟𝑃Σ𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇1𝜃+2^̌‌𝑝^𝑁_𝑗=1Υ^𝑟_𝑗2𝑒_𝑖((𝑡+𝑡−𝜏))_𝑁𝑖=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇12^̌‌𝑝^𝑁_𝑗=1Υ^𝑟_𝑗3+𝛽_𝑟𝑃Σ×𝑒_𝑖𝑡−𝜏_𝑐(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝜚𝑒_𝑘^∼𝑘(𝑡)^𝑇𝐴(𝑟)+𝛿^̌‌−𝛼_𝑟𝐼𝑒_𝑁^∼𝑘(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝜚𝑒_𝑘^∼𝑘(𝑡)^𝑇𝐵(𝑟)𝑒^∼𝑘𝑡−𝜏_𝑐(𝑡)−^𝑛_𝑘=1𝑝_𝑘𝜚𝑒_𝑘^∼𝑘𝑡−𝜏_𝑐(𝑡)^𝑇𝛽𝑒_𝑟^∼𝑘𝑡−𝜏_𝑐(𝑡≤_𝑁𝑖=1)𝑒_𝑖(𝑡)^𝑇−𝜂𝐼_𝑁+12^̌‌𝑝^𝑁_𝑗=1Υ^𝑟_𝑗1+𝛼_𝑟𝑃Σ𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇𝜃𝐼_𝑁+12^̌‌𝑝^𝑁_𝑗=1Υ^𝑟_𝑗2𝑒_𝑖((𝑡+𝑡−𝜏))_𝑁𝑖=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇12^̌‌𝑝^𝑁_𝑗=1Υ^𝑟_𝑗3+𝛽_𝑟𝑃Σ×𝑒_𝑖𝑡−𝜏_𝑐(𝑡). (15) Then we have ℒ𝑉(𝑡,𝑒(𝑡))≤𝑎_𝑟𝑉(𝑡)+𝑏_𝑟𝑉(𝑡−𝜏(𝑡))+𝑐_𝑟𝑉𝑡−𝜏_𝑐(𝑡). (16) So we have ℒ𝑉(𝑡)≤𝑎^̌‌𝑉(𝑡)+𝑏^̌‌𝑉(𝑡−𝜏(𝑡))+𝑐^̌‌𝑉𝑡−𝜏_𝑐(𝑡). (17) Define 𝑊(𝑡)=𝑒^𝛾𝑡𝑉(𝑡) (18) and use (17) to compute the operator ℒ𝑊(𝑡)=𝑒^𝛾𝑡𝛾𝑉(𝑡)+ℒ𝑉(𝑡)≤𝑒^𝛾𝑡𝛾𝑉(𝑡)+𝑎^̌‌𝑉(𝑡)+𝑏^̌‌𝑉((𝑡+𝑡−𝜏))_̌‌𝑐𝑉𝑡−𝜏_𝑐(𝑡), (19) which after applying the generalized Itô’s formula, gives 𝑒^𝛾𝑡𝔼𝑉(𝑡)=𝑒^𝛾𝑡0𝔼𝑉𝑡₀+𝔼^𝑡_𝑡0ℒ𝑊(𝑠)𝑑𝑠 (20) for any 𝑡_𝑘>𝑡>𝑡₀>𝑡_𝑘−1≥0. Hence we have 𝑒^𝛾𝑡𝔼𝑉(𝑡)≤𝑒^𝛾𝑡0𝔼𝑉𝑡₀+^𝑡_𝑡0𝑒^𝛾𝑠𝛾𝑉(𝑠)+𝑎^̌‌𝑉(𝑠)+𝑏^̌‌𝑉(𝑠−𝜏(𝑠))+𝑐^̌‌𝑉𝑠−𝜏_𝑐(𝑠)𝑑𝑠≤𝑒^𝛾𝑡0𝔼𝑉𝑡₀+𝛾+𝑎^̌‌𝑡_𝑡0𝑒^𝛾𝑠(𝑠)+𝔼𝑉𝑑𝑠_̌‌𝑏𝑒^𝛾𝜏𝑡_𝑡0𝑒^{𝛾(𝑠−𝜏(𝑠))}((𝑠+𝔼𝑉𝑠−𝜏))𝑑𝑠_̌‌𝑐𝑒^{𝛾𝜏𝑐𝑡}_𝑡0𝑒^{𝛾(𝑠−𝜏𝑐(𝑠))}𝔼𝑉𝑠−𝜏_𝑐(𝑠)𝑑𝑠. (21) By changing variable 𝑠−𝜏(𝑠)=𝑢, we have ^𝑡_𝑡0𝑒^{𝛾(𝑠−𝜏(𝑠))}((𝑠=𝔼𝑉𝑠−𝜏))𝑑𝑠_{𝑡𝑡−𝜏(𝑡)0−𝜏(𝑡0)}𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢1−𝜏̇(𝑡)≤^𝑡_𝑡0−𝜏𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢1−𝜏≤11−𝜏^𝑡_𝑡0−𝜏𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢. (22) Similarly, we have ^𝑡_𝑡0𝑒^{𝛾(𝑠−𝜏𝑐(𝑠))}((𝑠1𝔼𝑉𝑠−𝜏))𝑑𝑠≤1−𝜏_𝑐^𝑡_{𝑡0−𝜏𝑐}𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢. (23) Substituting (22) and (23) into (21), we get 𝑒^𝛾𝑡𝔼𝑉(𝑡)≤𝑒^𝛾𝑡0𝔼𝑉𝑡₀+𝛾+𝑎^̌‌+^̌‌𝑏𝑒^𝛾𝜏1−𝜏+^̌‌𝑐𝑒^𝛾𝜏𝑐1−𝜏_𝑐^𝑡_𝑡0−^̌‌_𝜏𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢. (24) By using Gronwall’s inequality, we get 𝑒^𝛾𝑡𝔼𝑉(𝑡)≤𝑒^𝛾𝑡0𝔼𝑉𝑡₀𝑒^{(𝛾+̌‌𝑎+(̌‌𝑏/(1−𝜏))𝑒𝛾𝜏+(̌‌𝑐/(1−𝜏𝑐))𝑒𝑐𝛾𝜏)(𝑡−𝑡0+̌‌𝜏)}. (25)
On the other hand, from the construction of 𝑉(𝑡), we have 𝑉𝑡_𝑘≤1+𝜖_𝑘²𝑉𝑡⁻_𝑘, (26) where |1+𝜖_𝑘|=max_{𝑖=1,2,…,𝑁}|1+𝜖_𝑖𝑘|.
According to (8)–(11), for any 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), we get 𝑒^𝛾𝑡𝔼𝑉(𝑡)≤𝑒^{𝛾𝑡𝑘−1}𝔼𝑉𝑡_𝑘−1𝑒^{(𝛾+̌‌𝑎+(𝑏/(1−𝜏))𝑒𝛾𝜏+(̌‌𝑐/(1−𝜏𝑐))𝑒𝑐𝛾𝜏)(𝑡−𝑡𝑘−1+̌‌𝜏)}≤𝑒^{𝛾𝑡𝑘−1}𝔼𝑉𝑡⁻_𝑘−1×𝑒^{(𝛾+̌‌𝑎+(̌‌𝑏/(1−𝜏))𝑒𝛾𝜏+(̌‌𝑐/(1−𝜏𝑐))𝑒𝑐𝛾𝜏)(𝑡−𝑡𝑘−1+̌‌𝜏)+2ln|1+𝜖𝑘−1|}≤⋯≤𝑒^𝛾𝑡0𝔼𝑉𝑡₀×𝑒^{(𝛾+̌‌𝑎+(̌‌𝑏/(1−𝜏))𝑒𝛾𝜏+(̌‌𝑐/(1−𝜏𝑐))𝑒𝑐𝛾𝜏)(𝑡−𝑡0+𝑘̌‌𝜏)+∑𝑘−1𝑣=12ln|1+𝜖𝑣|}. (27)
Let |1+𝜖|=max_𝑣∈𝑍⁺|1+𝜖_𝑣|. Because of 𝑘=⌊(𝑡−𝑡₀)/𝑇⌋, we have(𝑡)𝑡𝔼𝑉≤𝔼𝑉₀×𝑒^{(𝛾+̌‌𝑎+(̌‌𝑏/(1−𝜏))𝑒𝛾𝜏+(̌‌𝑐/(1−𝜏𝑐))𝑒𝑐𝛾𝜏)(𝑡−𝑡0+𝑘̌‌𝜏)+2(𝑘−1)ln|1+𝜖𝑣|−(𝑡−𝑡0)𝛾}≤𝔼𝑉𝑡₀×𝑒^{(𝛾+̌‌𝑎+(̌‌𝑏/(1−𝜏))𝑒𝛾𝜏+(̌‌𝑐/(1−𝜏𝑐))𝑒𝑐𝛾𝜏)(1+𝑇)̌‌𝜏/𝑇(𝑡−𝑡0)+(2(𝑡−𝑡0)/𝑇−4)ln|1+𝜖𝑣|−(𝑡−𝑡0)𝛾}. (28)
Using Condition (12) of Theorem 6, we get 𝔼𝑉(𝑡)≤𝔼𝑉(𝑡₀)𝑒^{−𝜂(𝑡−𝑡0)}. Hence, 𝔼||𝑒_𝑖(𝑡)||≤(𝑉(𝑡₀)/𝑝^̌‌)^1/2𝑒^{−(𝜂/2)(𝑡−𝑡0)}. The proof of Theorem 6 is completed.

Remark 7. Theorem 6 provides a sufficient condition for exponential synchronization of coupling delay switched stochastic dynamical networks with impulsive effects. If the time 𝑇 is sufficiently small and the impulsive gains 𝜖_𝑖𝑘, then exponential synchronization of the network (1) could be achieved.

If the switching signal 𝜎(𝑡)≡1, then the network (1) has only one coupling matrix 𝐺. Suppose 𝐺 is irreducible and 𝜉^𝑇=(𝜉₁,𝜉₂,…,𝜉_𝑁) is the left eigenvector of coupling matrix 𝐺 corresponding to eigenvalue 0. Let impulsive gains are 𝜖_𝑖𝑘=𝑏_𝑘. By the proof of Theorem 6, we can derive the exponential synchronization criteria of the network (1) with only one topology, which is given as follows.

Corollary 8. Let assumptions (H1) and (H2) be true and let 𝑓∈QUAD(𝑃,Δ,𝜂,𝜃). If there exist positive constants 𝛼 and 𝛽 such that 𝐴^𝑠+𝛿^̌‌𝐼_𝑁−𝛼𝐼_𝑁𝐵2𝐵^𝑇2−𝛽𝐼_𝑁𝑏≤0,𝛾+𝑎+1−𝜏𝑒^𝛾𝜏+𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐×1+𝑇𝜏^̌‌𝑇+2ln|1+𝜖|𝑇−𝛾≤−𝜂, (29) where 𝜆𝑎=^min−2𝜂𝐼_𝑛+𝑝^̌‌∑^𝑁_𝑖=1Υ_𝑖1+2𝛼𝑃Σ_̌‌𝑝,𝜆𝑏=^max∑^𝑁_𝑖=1𝑃Υ_𝑖2+2𝜃𝐼_𝑁𝑝,𝜆𝑐=^max∑^𝑁_𝑖=1𝑃Υ_𝑖3+2𝛽𝑃Σ𝑝,|1+𝜖|=max_{𝑖=1,2,…,𝑁,𝑘∈𝑍+}1+𝜖_𝑖𝑘. (30) Then the solutions 𝑥₁(𝑡),𝑥₁(𝑡),…,𝑥_𝑁(𝑡) of system (6) are globally and exponentially stable.

Let impulsive gains 𝜖_𝑖𝑘=𝑏, and choose the synchronization state 𝑠(𝑡)=(1/𝑁)∑^𝑁_𝑖=1𝑥_𝑖(𝑡). By the proof of Theorem 6, we can derive the exponential synchronization criteria of the network (1) with the fixed impulsive gain, which is given as follows.

4. Numerical Simulation

In this section, we give two numerical simulations to illustrate the feasibility and effectiveness of the theoretical results presented in the previous sections.

Consider a three-order Chua’s circuit described as follows: 𝑥̇(𝑡)=𝑓(𝑥(𝑡)), (31) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))^𝑇 and function 𝑓(𝑥(𝑡)) was chosen below: 𝑓(𝑥(𝑡=𝑚𝑥))²−ℎ𝑥₁𝑥₁−𝑥₂+𝑥₃−𝑛𝑥₂, (32) where ℎ(𝑥₁)=(2/7)𝑥₁−(3/14)[|𝑥₁+1|−|𝑥₁−1|], 𝑚=9, and 𝑛=14(2/7).

Consider a network model consists of 5 nodes and 2-connected topology. Each node in the network is a three-order Chua’s circuit described by 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+⁵_𝑗=1𝑔^𝜎(𝑡)_𝑖𝑗Γ𝑥_𝑗(𝑡−𝜏)+𝜎_𝑖(𝑡)((𝑡,𝑥𝑡,𝑥,𝑥𝑡−𝜏))𝑡−𝜏_𝑐(𝑡),𝑟(𝑡)𝑑𝑤_𝑖(𝑡),𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝜖_𝑖𝑘𝑥_𝑖𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑖=1,2,…,5, (33) where 𝜏=1, Γ=𝐼₃, and 𝜎_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡−𝜏_𝑐(𝑡)))=0.1×diag{𝑥_𝑖1(𝑡)−𝑥_𝑖+1,1(𝑡),𝑥_𝑖2(𝑡)−𝑥_𝑖+1,2(𝑡)}.

The coupling matrices are as follows: 𝐺¹=,𝐺−321000−220010−320000−222000−3₂=.−211000−211000−211100−211100−2 (34)

If we choose 𝑃=𝐼₃ and Δ=10𝐼₃, then function 𝑓(𝑥) satisfies the condition of function class QUAD(Δ,𝑃), where 𝛼=0.6218. Let 𝛽=23.7, 𝑄=4.1𝐼₃, 𝜖_𝑖𝑘=−0.8, 𝑇=0.1, 𝜂=−3.488, 𝑎=4.1. 𝑎₁=80.1500, 𝑎₂=62.3271, 𝑏=6.5300, 𝑐₁=0.0307, 𝑐₂=80.0300, 𝛼₁=40.0000, 𝛽₁=0.0002, 𝛼₂=15.6568, 𝛽₂=20.0000, and the synchronization state 𝑠(𝑡)=0.2𝑥₁+0.3𝑥₂+0.1𝑥₃+0.3𝑥₄+0.1𝑥₅; then all the conditions in Theorem 6 are satisfied (by using the Matlab LMI toolbox). The switch time is 𝑡=0.2s. The simulation results are given in Figures 1–3. It can be seen clearly from Figures 1, 2, and 3 that all states of the asymmetric coupled network (21) tend to the synchronization state 𝑠(𝑡).

Figure 1 

Time evolution of synchronization errors 𝑒𝑖1(𝑡),(𝑖=1,2,…,5) of the asymmetric coupled network (21) with random initial values.

Figure 2 

Time evolution of synchronization errors 𝑒𝑖2(𝑡),(𝑖=1,2,…,5) of the asymmetric coupled network (21) with random initial values.

Figure 3 

Time evolution of synchronization errors 𝑒𝑖3(𝑡),(𝑖=1,2,…,5) of the asymmetric coupled network (21) with random initial values.

5. Conclusion

In this paper, the exponential synchronization of the coupling delay stochastic complex networks with switch topology and impulsive effects has been investigated. Based on the Lyapunov stability theory, LMI, and impulsive control theory, some simple, yet generic, criteria for exponential synchronization have been derived. It has shown that criteria can provide an effective control scheme to synchronize for a given coupled delay, the network size, and switch topology. Furthermore, the effectiveness of the presented method has been verified by numerical simulations.

Acknowledgments

This work was supported by the National Science Foundation of China under Grant no. 61070087, the Guangdong Education University Industry Cooperation Projects (2009B090300355), the Shenzhen Basic Research Project (JC200903120040A, JC201006010743A), and the Shenzhen Polytechnic Youth Innovation Project (2210k3010020). The authors are very grateful to the reviewers and editors for their valuable comments and suggestions that improved the presentation of this paper.