Impulsive Pinning Markovian Switching Stochastic Complex Networks with Time-Varying Delay

Abstract

The synchronization problem of stochastic complex networks with Markovian switching and time-varying delays is investigated by using impulsive pinning control scheme. The complex network possesses noise perturbations, Markovian switching, and internal and outer time-varying delays. Sufficient conditions for synchronization are obtained by employing the Lyapunov-Krasovskii functional method, Itö's formula, and the linear matrix inequality (LMI). Numerical examples are also given to demonstrate the validity of the theoretical results.

1. Introduction and Model Description

Collective behaviors in complex networks and systems have attracted increasing attention in recent years due to their wide applications in physics, mathematics, engineering, biology, and so forth (see [1, 2] and references therein). While complex networks are ubiquitously found in nature and in the modern world, such as neural networks, socially interacting animal species, power networks, wireless sensor networks, Internet, and the World Wide Web.

In the past few decades, the synchronization problems in complex networks have attracted increasing attention. Many synchronization patterns have been studied, including complete synchronization [3], cluster synchronization [4–6], phase synchronization [7], and partial synchronization [8]. There are several control methods to guide the dynamics of a complex network to a desired state, such as adaptive control [9], feedback control [10], intermittent control [11], fuzzy control [12], impulsive control [13, 14], and pinning control [5, 6, 15, 16]. Synchronization of complex networks holds particular promise for applications to many fields [17–21].

Synchronization in complex dynamical networks is realized via information exchanges among the interconnect nodes [22]. The signal traveling along real physical system is usually perturbed randomly by the environmental elements, such as noises, the structures of the interconnections, time delays, and the positions of nodes [9]. One popular model is the Markovian switching model driven by continuous-time Markov chains in the sciences and industries (see [23–26] and references therein). In [23, 24], Mao et al. studied stability and controllability of stochastic differential delay equations with Markovian switching, while [25, 26] discussed the exponential stability of stochastic delayed neural networks. Liu et al. [26], on the other hand, investigated the synchronization of discrete-time stochastic complex networks with Markovian jumping and mode-dependent mixed time delays. In [16], Wang et al. investigated the mean-square exponential synchronization of stochastic complex networks with Markovian switching and time-varying delays by using the pinning control method, which is described as 𝑑𝑥_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖((𝑡+𝑡−𝜏))_{𝑁𝑗=1,𝑖≠𝑗}𝑎_𝑖𝑗(𝑟(𝑡))Σ𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+^𝑁_{𝑗=1,𝑖≠𝑗}𝑏_𝑖𝑗(𝑟(𝑡))Σ𝑥_𝑗𝑡−𝜏_𝑐(𝑡)−𝑥_𝑖𝑡−𝜏_𝑐(𝑡)+𝑢_𝑖(𝑡)𝑑𝑡+𝜎_𝑖𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟(𝑡)𝑑𝑤_𝑖(𝑡),𝑖=1,2,…,𝑁, (1) where 𝑢_𝑖(𝑡)(𝑖=1,2,…,𝑁) are the linear state feedback controllers that are defined by 𝑢_𝑖(𝑡)=−𝜀_𝑖𝑥_𝑖(𝑡)−𝑠(𝑡),𝑖=1,2,…,𝑙,0,𝑖=𝑙+1,𝑙+2,…,𝑁, (2) and 𝜀_𝑖>0(𝑖=1,2,…,𝑙) are the control gains.

Pinning control has been proved to be effective for the synchronization of complex dynamical networks with continuous state coupling [15, 16, 27]. In many systems, the impulsive effect is a common phenomenon due to instantaneous perturbations at certain moments [27, 28]. Impulsive control strategies have been widely used to stabilize and synchronize coupled complex dynamical systems, such as signal processing system, computer networks, automatic control systems, and telecommunications [13]. In [27], pinning impulsive strategy is proposed for the synchronization of stochastic dynamical networks with nonlinear coupling. Zhou et al. studied synchronization in complex delayed dynamical networks with impulsive effects in [28]. And Zhu et al., in [29], investigated the exponential stability of a class of stochastic neural networks with both Markovian jump parameters and mixed fixed time delays. Can the stochastic dynamical network with Markovian switching and time-varying delays be synchronized by impulsive pinning control? This paper is devoted to solving this problem.

In this paper, we study the synchronization of stochastic complex networks with Markovian switching by using the impulse control method. We consider a kind of stochastic complex networks with internal time-varying delayed couplings, Markovian switching, and Wiener processes. By applying the Lyapunov-Krasovskii functional method, Itö’s formula and the linear matrix inequality (LMI), some sufficient conditions for synchronization of these networks are derived. Numerical examples are finally given to demonstrate the effectiveness of the proposed impulsive pinning strategy.

Notations. Throughout this paper, ℝ^𝑛 will denote the 𝑛-dimensional Euclidean space and ℝ^𝑛×𝑛 the set of all 𝑛×𝑛 real matrices. The superscript 𝑇 will denote the transpose of a matrix or a vector. And Tr(⋅) stands for the trace of the corresponding matrix. 𝟏_𝑛=(1,1,…,1)^𝑇∈ℝ^𝑛, and 𝐼_𝑛 is the 𝑛-dimensional identity matrix. For square matrices 𝑀, the notation 𝑀>0 (<0) will mean that 𝑀 is a positive-definite (negative-definite) matrix. 𝜆_max(𝐴) and 𝜆_min(𝐴) will denote the greatest and least eigenvalues of a symmetric matrix, respectively. 𝑝̌=max{𝑝₁,𝑝₂,…,𝑝_𝑛}, and 𝑝=min{𝑝₁,𝑝₂,…,𝑝_𝑛}.

2. Preliminaries

Let (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝒫) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 that is right continuous with ℱ₀ containing all the 𝒫-null sets. 𝐶([−𝜏,0];ℝ^𝑛) will denote the family of continuous functions 𝜙 from [−𝜏,0] to ℝ^𝑛 with the uniform norm ‖𝜙‖²=sup_{−𝜏≤𝑠≤0}𝜙(𝑠)^𝑇𝜙(𝑠). And 𝐶²_ℱ0([−𝜏,0];ℝ^𝑛) denotes the family of all ℱ₀ measurable, 𝐶([−𝜏,0];ℝ^𝑛)-valued stochastic variables 𝜉={𝜉(𝜃)∶−𝜏≤𝜃≤0} such that ∫⁰_−𝜏𝔼‖𝜉(𝑠)‖²𝑑𝑠≤∞, where 𝔼 stands for the correspondent expectation operator with respect to the given probability measure 𝒫.

Consider a complex network consisting of 𝑁 identical nodes with Markovian switching 𝑑𝑥_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑥_𝑖((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑎^[𝑟(𝑡)]_𝑖𝑗Σ𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑏^[𝑟(𝑡)]_𝑖𝑗Σ𝑥_𝑗𝑡−𝜏_𝑐(𝑡)𝑑𝑡+𝜎_𝑖^[𝑟(𝑡)]𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)𝑑𝑤(𝑡),𝑖=1,2,…,𝑁, (3) 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(𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))=[𝑓₁(𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))),𝑓₂(𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))),…,𝑓_𝑛(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 matrix of the networks that satisfies 𝜚_𝑗>0, 𝑗=1,2,…,𝑛, and 𝑟(𝑡) are the continuous-time Markov processes that describe the evolution of the nodes at time 𝑡. Here, 𝐴^[𝑟(𝑡)]=[𝑎^[𝑟(𝑡)]_𝑖𝑗]∈ℝ^𝑛×𝑛 and 𝐵^[𝑟(𝑡)]=[𝑏^[𝑟(𝑡)]_𝑖𝑗]∈ℝ^𝑛×𝑛 are the outer coupling matrices of the network at time 𝑡 at nodes 𝑟(𝑡), 𝑡−𝜏_𝑐(𝑡), respectively, such that 𝑎^[𝑟(𝑡)]_𝑖𝑗≥0 for 𝑖≠𝑗, 𝑎^[𝑟(𝑡)]_𝑖𝑖=−∑^𝑁_{𝑗=𝑖,𝑗≠𝑖}𝑎^[𝑟(𝑡)]_𝑖𝑗, 𝑏^[𝑟(𝑡)]_𝑖𝑗≥0 for 𝑖≠𝑗 and 𝑏^[𝑟(𝑡)]_𝑖𝑖=−∑^𝑁_{𝑗=𝑖,𝑗≠𝑖}𝑏^[𝑟(𝑡)]_𝑖𝑗. Figure 1 shows the topology structures of the switching networks for 5 nodes.  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^[𝑟(𝑡)](𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡−𝜏_𝑐(𝑡)))=𝜎_𝑖^[𝑟(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^𝑛×𝑛 and 𝑤(𝑡)=(𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑛(𝑡))^𝑇∈ℝ^𝑛 is a bounded vector-form Weiner process, satisfying 𝔼𝑤_𝑗(𝑡)=0,𝔼𝑤²_𝑗(𝑡)=1,𝔼𝑤_𝑗(𝑡)𝑤_𝑗(𝑠)=0(𝑠≠𝑡). (4) Let 𝑟(𝑡),𝑡>0 be a right-continuous Markov chain on a probability space that takes values in a finite state space 𝑆=1,2,…,𝑀 with a generator Γ=[𝛾_𝑖𝑗]∈ℝ^𝑀×𝑀 given by 𝑃𝑟()(𝑡)=𝛾𝑡+Δ=𝑗∣𝑟=𝑖^𝑖𝑗Δ+𝑜(Δ)if𝑖≠𝑗,1+𝛾_𝑖𝑖Δ+𝑜(Δ)if𝑖=𝑗, (5) for some Δ>0. Here 𝛾_𝑖𝑗≥0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 and 𝛾_𝑖𝑖=−∑_𝑖≠𝑗𝛾_𝑖𝑗, In this paper, 𝐴^[𝑟] is assumed to be symmetric and irreducible, and 𝐵^[𝑟] is assumed to be symmetric, for 𝑟=1,2,…,𝑀.

(a)

(b)

Figure 1 

The topology structures of the switching networks for 5 nodes; (A1) and (A2) the topology structures of the coupling matrix 𝐴1 and 𝐴2, respectively; (B1) and (B2) the topology structures of the coupling matrix 𝐵1 and 𝐵2, respectively.

The initial conditions associated with (3) are 𝑥_𝑖(𝑠)=𝜉_𝑖(𝑠),−𝜏̌≤𝑠≤0,𝑖=1,2,…,𝑁, (6) where 𝜏̌=max{𝜏(𝑡),𝜏_𝑐(𝑡)}, 𝜉_𝑖∈𝐶^𝑏_ℱ0([−𝜏̌,0],ℝ^𝑛) with the norm ‖𝜉_𝑖‖²=sup_{−̌𝜏≤𝑠≤0}𝜉_𝑖(𝑠)^𝑇𝜉_𝑖(𝑠).

The impulse controllers are defined by Δ𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝜖_𝑖𝑘𝑥_𝑖𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,𝑖=1,2,…,𝑁, (7) where 𝜖_i𝑘 are constants, and {𝑡₁,𝑡₂,𝑡₃,…} is the impulsive sequence of strictly increasing impulsive instants satisfying lim_𝑘→∞𝑡_𝑘=+∞, and 𝑡_𝑘−𝑡_𝑘−1=𝑇 for 𝑘>1.

In the case that system (3) reaches synchronization, that is, 𝑥₁(𝑡)=𝑥₂(𝑡)=⋯=𝑥_𝑁(𝑡)=𝑠(𝑡), we have the following synchronized state equation: (𝑡)(𝑠(𝑡)((𝑡(𝑠(𝑡)((𝑡(𝑡).𝑑𝑠=𝑓,𝑠𝑡−𝜏)))𝑑𝑡+𝜎,𝑠𝑡−𝜏)))𝑑𝑤 (8)

In the paper, we would control the system (3) to the desired trajectory 𝑠(𝑡). Define 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡)(𝑖=1,2,…,𝑁) as the synchronization error. Then, according to the controller (7), the error system is 𝑑𝑒_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑥_𝑖((𝑡(𝑠(𝑡)((𝑡+𝑡−𝜏))−𝑓,𝑠𝑡−𝜏)))_𝑁𝑗=1𝑎^[𝑟(𝑡)]_𝑖𝑗Σ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏^[𝑟(𝑡)]_𝑖𝑗Σ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)𝑑𝑡+𝜎_𝑖^[𝑟(𝑡)]𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡)𝑑𝑤(𝑡),𝑡≠𝑡_𝑘,𝑘∈𝑍⁺,𝑖=1,2,…,𝑁,Δ𝑒_𝑖𝑡_𝑘=𝜖_𝑖𝑘𝑒_𝑖𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍⁺,𝑖=1,2,…,𝑁, (9) where 𝜎_𝑖^[𝑟(𝑡)](𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒(𝑡−𝜏_𝑐(𝑡)))=𝜎_𝑖^[𝑟(𝑡)](𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡−𝜏_𝑐(𝑡)))−𝜎(𝑠(𝑡),𝑠(𝑡−𝜏(𝑡))).

Definition 1 (see [16, 27]). The complex network (3) is said to be exponentially synchronized in mean square if the trivial solution of system (9) is such that ^𝑁_𝑖=1𝔼𝑒_𝑖(𝑡,𝑡₀,𝜉_𝑖)²≤𝐾𝑒^−𝜅𝑡, (10) for some 𝐾>0 and 𝜅>0 under any initial data 𝜉_𝑖∈𝒞^𝑏_ℱ0([−𝜏̌,0];ℝ^𝑛).

Definition 2 (see [9, 11, 16]). A continuous function 𝑓(𝑥,𝑦)∶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 exists 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(𝑥,𝑧)−𝑓𝑦,𝑤−ΔΣ𝑥−𝑦≤−𝜂𝑥−𝑦^𝑇𝑥−𝑦+𝜃(𝑧−𝑤)^𝑇(𝑧−𝑤) (11) for all 𝑥,𝑦,𝑧,𝑤∈ℝ^𝑛.

The following assumptions are usual and will be used throughout this paper for establishing the synchronization conditions [9, 11, 16].(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 𝜏̌‌=max{𝜏,𝜏_𝑐}.(H2) 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_𝑖^[𝑟]𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡)^𝑇×𝜎_𝑖^[𝑟]𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡)≤^𝑁_𝑗=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). (12)

Remark 3. Considering Definition 2 and assumption (H2), there exists a unique solution of (9) under the initial data 𝜉_𝑖∈𝒞^𝑏_ℱ0([−𝜏,0];ℝ^𝑛) (see [23, 24]).

Lemma 4 (see [23, 24]). Consider a stochastic delayed differential equation with Markovian switching of the form (𝑡)(𝑥(𝑡)()(𝑡(𝑥(𝑡)()(𝑡(𝑡)𝑑𝑥=𝑓,𝑥𝑡−𝜏,𝑟))𝑑𝑡+𝜎,𝑥𝑡−𝜏,𝑟))𝑑𝜔 (13) on 𝑡≥0 with initial value 𝑥₀=𝜉∈𝐶^𝑏_𝐹0([−𝜏,0];ℝ^𝑛), where 𝑓∶ℝ^𝑛×ℝ^𝑛×𝑆→ℝ^𝑛,𝜎∶ℝ^𝑛×ℝ^𝑛×𝑆→ℝ^𝑛×𝑚. (14)
Let 𝐶^2,1(ℝ₊×ℝ^𝑛;ℝ₊) be the family of all the nonnegative functions 𝑉(𝑡,𝑥,𝑟) on ℝ₊×ℝ^𝑛×𝑆 that are twice continuously differentiable in 𝑥 and once differentiable in 𝑡. Let 𝑉∈𝐶^2,1(ℝ₊×ℝ^𝑛×𝑆;ℝ₊). Define an operator ℒ𝑉 from ℝ^𝑛×ℝ₊×𝑆 to ℝ^𝑛 by ℒ𝑉(𝑡,𝑥,𝑟)=𝑉_𝑡(𝑡,𝑥,𝑟)+𝑉_𝑥()𝑓()+1𝑡,𝑥,𝑟𝑥,𝑟2Tr𝜎(𝑥,𝑟)^𝑇𝑉_𝑥𝑥𝜎(+𝑥,𝑟)_𝑀𝑗=1𝛾_𝑖𝑗𝑉𝑡,𝑥,𝑗, (15) where 𝑉_𝑡(𝑡,𝑥,𝑟)=𝜕𝑉(𝑡,𝑥,𝑟)/𝜕𝑡, 𝑉_𝑥(𝑡,𝑥,𝑟)=(𝜕𝑉(𝑡,𝑥,𝑟)/𝜕𝑥₁,…,𝜕𝑉(𝑡,𝑥,𝑟)/𝜕𝑥_𝑛), and 𝑉_𝑥𝑥(𝑡,𝑥,𝑟)=(𝜕²𝑉(𝑡,𝑥,𝑟)/𝜕𝑥_𝑖𝑥_𝑗)_𝑛×𝑛. If  𝑉∈𝐶^2,1(ℝ₊×ℝ^𝑛×𝑆;ℝ₊), and then 𝔼𝑉(𝑡,𝑥(𝑡),𝑟)=𝔼𝑉𝑡₀,𝑥𝑡₀,𝑟+𝔼^𝑡_𝑡0ℒ𝑉(𝑠,𝑥(𝑠),𝑟)𝑑𝑠 (16) for all ∞>𝑡>𝑡₀≥0, as long as the expectations of the integrals exist.

3. Main Result

In this section, we will deduce our main results.

Theorem 5. 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]+𝛿̌𝐼_𝑁−𝛼_𝑟𝐼_𝑁𝐵^[𝑟]2𝐵^[𝑟]2−𝛽_𝑟𝐼_𝑁̌̌()̌≤0,𝑓𝑜𝑟𝑟=1,2,…,𝑀,𝜏≤𝜃𝑇,𝜏≤1−𝜃𝑇,0≤𝑞̌̌𝑐̌𝜏≤1−𝑏+,𝜑(̌)𝑞̌1+𝜃𝜏+𝑇+2ln𝑞1+𝜖_𝑣1−𝛾𝑇<0,𝑏₁+𝑐₁,1𝑏₂+𝑐₂1,…,𝑏_𝑀+𝑐_𝑀^𝑇>Γ⁻¹𝟏_𝑀, (17) where ̌𝑏̌𝑞̌𝜑=1+𝜃+𝛾𝑞+1−𝜏𝑒^𝛾𝜏+𝑐𝑞̌̌1−𝜏_𝑐𝑒^𝛾𝜏𝑐,Γ=diag𝑎₁,𝑎₂,…,𝑎_𝑀̌+Γ,𝑎=max_𝑟∈𝑆𝑎_𝑟,𝑎_𝑟=𝜆_max−2𝜂𝐼_𝑛+𝑝̌∑^𝑁_𝑖=1Υ^[𝑟]_𝑖1+2𝛼_𝑟𝑃Σ𝑝̌,𝑏̌=max_𝑟∈𝑆𝑏_𝑟,𝑏_𝑟=𝜆_max∑^𝑁_𝑖=1𝑃Υ^[𝑟]_𝑖2+2𝜃𝐼_𝑁𝑝,𝑐̌=max_𝑟∈𝑆𝑐_𝑟,𝑐_𝑟=𝜆_max∑^𝑁_𝑖=1𝑃Υ^[𝑟]_𝑖3+2𝛽_𝑟𝑃Σ𝑝, (18) then the solutions 𝑒₁(𝑡),𝑒₂(𝑡),…, and 𝑒_𝑁(𝑡) of system (9) are exponentially stable in mean square. It means that the complex dynamical network (3) can be exponentially controlled to the objective trajectory 𝑠(𝑡) under the controllers (7).

Proof. By (44), there exists a sufficiently small constant 𝜃>0 such that 1𝑏₁+𝑐₁,1𝑏₂+𝑐₂1,…,𝑏_𝑀+𝑐_𝑀^𝑇≥(1+𝜃)Γ⁻¹𝟏. (19) Set (1+𝜃)Γ⁻¹𝟏=𝑞=(𝑞₁,𝑞₂,…,𝑞_𝑀)^𝑇. Then Γ𝑞=(1+𝜃)𝟏_𝑀, that is, 𝑏_𝑟+𝑐_𝑟𝑞_𝑟≤1,𝑎_𝑟𝑞_𝑟+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠=1+𝜃. (20) For 1≤𝑟≤𝑀, define the Lyapunov-Krasovskii function 𝑉(𝑒(𝑡),𝑟)=𝑞_𝑟12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡), (21) and let 𝑒^𝑘̃(𝑡)=(𝑒_1𝑘(𝑡),𝑒_2𝑘(𝑡),…,𝑒_𝑁𝑘(𝑡))^𝑇, 𝑘=1,2,…,𝑛. For any 𝑡∈(𝑡_𝑘−1,𝑡_𝑘], 𝑘=1,2,…, by Lemma 4, we haveℒ𝑉(𝑒(𝑡),𝑟)=𝑞_𝑟^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓𝑥_𝑖(𝑡),𝑥_𝑖((𝑡(𝑠(𝑡)((𝑡+𝑡−𝜏))−𝑓,𝑠𝑡−𝜏)))^𝑁_𝑗=1𝑎^[𝑟]_𝑖𝑗Σ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏^[𝑟]_𝑖𝑗Σ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)+12𝑞_𝑟^𝑁_𝑖=1Tr𝜎_𝑖𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡−𝜏_𝑐(𝑡)),𝑟^𝑇×𝑃𝜎_𝑖𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)+,𝑟_𝑀𝑠=1𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)≤𝑞_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_𝑘𝛿_𝑘𝑒^𝑘̃(𝑡)^𝑇𝑒^𝑘̃(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐴^[𝑟]𝑒^𝑘̃(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐵^[𝑟]𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)+12𝑝̌^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)+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)=𝑞_𝑟^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇−𝜂𝐼_𝑁+12𝑝̌^𝑁_𝑗=1Υ^[𝑟]_𝑗1+𝛼_𝑟𝑃Σ𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇1𝜃+2𝑝̌^𝑁_𝑗=1Υ^[𝑟]_𝑗2𝑒_𝑖((𝑡+𝑡−𝜏))_𝑁𝑖=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇12𝑝̌^𝑁_𝑗=1Υ^[𝑟]_𝑗3+𝛽_𝑟𝑃Σ𝑒_𝑖𝑡−𝜏_𝑐(𝑡+_𝑀𝑠=1)𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)+𝑞_𝑟^𝑛_𝑘=1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐴^[𝑟]+𝛿̌−𝛼_𝑟𝐼_𝑁𝑒^𝑘̃(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐵^[𝑟]𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)−^𝑛_𝑘=1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)^𝑇𝛽_𝑟𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)≤𝑞_𝑟^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇−𝜂𝐼_𝑁+12𝑝̌^𝑁_𝑗=1Υ^[𝑟]_𝑗1+𝛼_𝑟𝑃Σ𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇𝜃𝐼_𝑁+12𝑝̌^𝑁_𝑗=1Υ^[𝑟]_𝑗2𝑒_𝑖((𝑡+𝑡−𝜏))_𝑁𝑖=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇12𝑝̌^𝑁_𝑗=1Υ^[𝑟]_𝑗3+𝛽_𝑟𝑃Σ𝑒_𝑖𝑡−𝜏_𝑐(𝑡)+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡). (22)Let 𝐸(𝑡)=12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡); (23) then we have ℒ𝑉(𝑡)≤𝑎_𝑟𝑞_𝑟𝐸(𝑡)+𝑏_𝑟𝑞_𝑟𝐸(𝑡−𝜏(𝑡))+𝑐_𝑟𝑞_𝑟𝐸𝑡−𝜏_𝑐(𝑡)+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠𝐸(𝑡), (24) and by (20), we have ℒ𝑉(𝑡)≤(1+𝜃)𝐸(𝑡)+𝑏̌𝑞𝐸̌(𝑡−𝜏(𝑡))+𝑐𝑞̌𝐸̌𝑡−𝜏_𝑐(𝑡). (25) Define 𝑊(𝑡)=𝑒^𝛾𝑡𝑉(𝑡). (26) Use (25) to compute the operator ℒ𝑊(𝑡)=𝑒^𝛾𝑡𝛾𝑉(𝑡)+ℒ𝑉(𝑡)≤𝑒^𝛾𝑡𝛾̌(𝑡)+()𝐸(𝑡)+𝑏̌̌((𝑡+𝑐̌̌𝑞𝐸1+𝜃𝑞𝐸𝑡−𝜏))𝑞𝐸𝑡−𝜏_𝑐(𝑡). (27) The generalized Itô’s formula gives 𝑒^𝛾𝑡𝔼𝑉(𝑡)=𝑒^𝛾𝑡0𝔼𝑉𝑡₀+𝔼^𝑡_𝑡0ℒ𝑊(𝑠)𝑑𝑠, (28) for any 𝑡_𝑘>𝑡>𝑡₀>𝑡_𝑘−1≥0. Hence we have 𝑒^𝛾𝑡𝔼𝑉(𝑡)≤𝑒^𝛾𝑡0𝔼𝑉𝑡₀+𝔼^𝑡_𝑡0𝑒^𝛾𝑠𝛾̌(𝑠)+()𝐸(𝑠)+𝑏̌̌((𝑠+𝑐̌̌𝑞𝐸1+𝜃𝑞𝐸𝑠−𝜏))𝑞𝐸𝑠−𝜏_𝑐(𝑠)≤̌𝑑𝑠𝑞𝑒^𝛾𝑡0𝔼𝐸𝑡₀+𝛾𝑞̌+1+𝜃×^𝑡_𝑡0𝑒^𝛾𝑠(𝑠)+𝑏̌̌𝔼𝐸𝑑𝑠𝑞𝑒^𝛾𝜏𝑡_𝑡0𝑒^{𝛾(𝑠−𝜏(𝑠))}((𝑠+𝑐̌̌𝔼𝐸𝑠−𝜏))𝑑𝑠𝑞𝑒^{𝛾𝜏𝑐𝑡}_𝑡0𝑒^{𝛾(𝑠−𝜏𝑐(𝑠))}𝔼𝐸𝑠−𝜏_𝑐(𝑠)𝑑𝑠. (29) By changing variable 𝑠−𝜏(𝑠)=𝑢, we have ^𝑡_𝑡0𝑒^{𝛾(𝑠−𝜏(𝑠))}((𝑠=𝔼𝐸𝑠−𝜏))𝑑𝑠_{𝑡𝑡−𝜏(𝑡)0𝑡−𝜏0}𝑒^𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢𝜏̇(𝑡)≤11−1−𝜏^𝑡_𝑡0−𝜏𝑒^𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢. (30) Similarly, we have ^𝑡_𝑡0𝑒^{𝛾(𝑠−𝜏𝑐(𝑠))}((𝑠1𝔼𝐸𝑠−𝜏))𝑑𝑠≤1−𝜏_𝑐^𝑡_{𝑡0−𝜏𝑐}𝑒^𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢. (31) Substituting (30) and (31) into (29), we get 𝑒^𝛾𝑡𝔼𝑉(𝑡)≤𝑞_𝑟𝑒^𝛾𝑡0𝔼𝑉𝑡₀+𝜑^𝑡_{𝑡0−̌𝜏}𝑒^𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢. (32) By using Gronwall inequality, we have (𝑡)≤𝑞̌𝔼𝐸𝑞𝔼𝐸𝑡₀𝑒^{𝜑(𝑡−𝑡0+̌𝜏)+𝛾(𝑡0−𝑡)}. (33) On the other hand, from the construction of 𝐸(𝑡), we have 𝐸𝑡_𝑘≤1+𝜖_𝑘²𝐸𝑡⁻_𝑘, (34) where |1+𝜖_𝑘|=max_{𝑖=1,2,…,𝑁}|1+𝜖_𝑖𝑘|.
According to (33)-(34), let 𝑘=⌊(𝑡−𝑡₀)/𝑇⌋, for any 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), and we get (𝑡)≤𝑞̌𝔼𝑉𝑞𝔼𝐸𝑡_𝑘𝑒^{𝜑(𝑡−𝑡𝑘+̌𝜏)+𝛾(𝑡𝑘−𝑡)}≤𝑞̌𝑞𝔼𝑉𝑡⁻_𝑘−1𝑒^{𝜑(𝑡−𝑡𝑘−1+̌𝜏)+𝛾(𝑡𝑘−𝑡)+2ln|1+𝜖𝑘−1|}𝑞̌≤⋯≤𝑞^𝑘−1𝔼𝐸(0)𝑒^{̌𝜑(𝑡+𝑘𝜏)−𝛾𝑡+∑𝑘−1𝑣=12ln|1+𝜖𝑣|}. (35)
Let |1+𝜖|=max_𝑣∈𝑍⁺|1+𝜖_𝑣|, and we have 𝔼𝐸(𝑡)≤𝔼𝐸(0)𝑒^{̌̌𝜑(𝑡+𝑘𝜏)−𝛾𝑡+2(𝑘−1)ln(𝑞/}𝑞_)|1+𝜖|. (36)
Using condition (43) of Theorem 5, there exist a number 𝜂 such that 𝔼𝐸(𝑡)≤𝔼𝐸(𝑡₀)𝑒^−𝜂𝑡. Hence, 𝔼‖𝑒_𝑖(𝑡)‖≤(𝐸(𝑡₀)/𝑝)̌^1/2𝑒^{−(𝜂/2)(𝑡−𝑡0)}. The proof of Theorem 5 is completed.

Remark 6. The stochastic networks studied before are without topological switch, and the time delays are always assumed to be fixed. However, for the sake of applications in the real work, these two points above should be taken into consideration. Of course, it will enhance the difficulties of the investigations on this network. For example, if the network has Markovian switching topology, the structure of the network is fast varying and the Lyapunov function is hard to be determined. By using the Lyapunov-Krasovskii functional, Itö’s formula, and LMI, the exponential stability criterion of the pinning impulsive controlled Markovian switching stochastic dynamical network with time-varying delays was obtained. This also showed that the impulsive pinning control is a kind of cheap control strategy for guiding complex dynamical networks to the objective trajectory.

To make Theorem 5 more applicative, we give the following corollaries.

When complex dynamic networks (3) are considered without coupled delay time (𝐵^[𝑟(𝑡)]=𝐵), we can get the following corollary.

Corollary 7. 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]+𝛿̌𝐼_𝑁−𝛼_𝑟𝐼_𝑁()≤0,𝑓𝑜𝑟𝑟=1,2,…,𝑀,𝜏≤𝜃𝑇,𝜏≤1−𝜃𝑇,0≤𝑞𝑏̌̌𝜏≤1−,𝜑(̌)𝑞̌1+𝜃𝜏+𝑇+2ln𝑞||11+𝜖−𝛾𝑇<0,𝑏₁,1𝑏₂1,…,𝑏_𝑀^𝑇>Γ⁻¹𝟏_𝑀, (37) where ̌𝑏̌𝑞̌𝜑=1+𝜃+𝛾𝑞+1−𝜏𝑒^𝛾𝜏,Γ=diag𝑎₁,𝑎₂,…,𝑎_𝑀̌+Γ,𝑎=max_𝑟∈𝑆𝑎_𝑟,𝑎_𝑟=𝜆_max−2𝜂𝐼_𝑛+𝑝̌∑^𝑁_𝑖=1Υ^[𝑟]_𝑖1+2𝛼_𝑟𝑃Σ𝑝̌,𝑏̌=max_𝑟∈𝑆𝑏_𝑟,𝑏_𝑟=𝜆_max∑^𝑁_𝑖=1𝑃Υ^[𝑟]_𝑖2+2𝜃𝐼_𝑁𝑝, (38) then the solutions 𝑒₁(𝑡),𝑒₁(𝑡),…, and 𝑒_𝑁(𝑡) of system (9) are exponentially stable in mean square.

In another case, when we consider the system (3) without Markov switching, that is, 𝐴^[𝑟(𝑡)]=𝐴, 𝐵^[𝑟(𝑡)]=𝐵, and 𝜎_𝑖^[𝑟(𝑡)](𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒(𝑡−𝜏_𝑐(𝑡)))=𝜎_𝑖(𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒(𝑡−𝜏_𝑐(𝑡))), we can get another corollary.

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+𝛿̌𝐼_𝑁−𝛼𝐼_𝑁𝐵2𝐵2−𝛽𝐼_𝑁̌̌()̌≤0,𝜏≤𝜃𝑇,𝜏≤1−𝜃𝑇,0≤𝜏≤1−𝑏+𝑐𝑎,𝜑(𝜏̌+𝑇)+2ln1+𝜖_𝑣−𝛾𝑇<0,𝑏+𝑐<𝑎, (39) where 𝑏𝜑=𝑎+𝛾+1−𝜏𝑒^𝛾𝜏+𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐,𝜆𝑎=^max−2𝜂𝐼_𝑛+𝑝̌∑^𝑁_𝑖=1Υ_𝑖1+2𝛼𝑃Σ𝑝̌,𝜆𝑏=^max∑^𝑁_𝑖=1𝑃Υ_𝑖2+2𝜃𝐼_𝑁𝑝,𝜆𝑐=^max∑^𝑁_𝑖=1𝑃Υ_𝑖3+2𝛽_𝑟𝑃Σ𝑝, (40) then the solutions 𝑒₁(𝑡),𝑒₂(𝑡),…, and 𝑒_𝑁(𝑡) of system (9) are exponentially stable in mean square.

Furthermore, when we address the system (3) with fixed time delays, that is, 𝜏(𝑡)=𝜏, 𝜏_𝑐(𝑡)=𝜏_𝑐, the following corollary also can be obtained.

Corollary 9. 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]+𝛿̌𝐼_𝑁−𝛼_𝑟𝐼_𝑁𝐵^[𝑟]2𝐵^[𝑟]2−𝛽_𝑟𝐼_𝑁̌̌()𝜑(̌)𝑞̌≤0,𝑓𝑜𝑟𝑟=1,2,…,𝑀,𝜏≤𝜃𝑇,𝜏≤1−𝜃𝑇,𝜏+𝑇+2ln𝑞1+𝜖_𝑣1−𝛾𝑇<0,𝑏₁+𝑐₁,1𝑏₂+𝑐₂1,…,𝑏_𝑀+𝑐_𝑀^𝑇>Γ⁻¹𝟏_𝑀, (41)(42)(43)(44) where 𝜑=1+𝜃+𝛾𝑞̌+𝑏̌𝑞𝑒̌^𝛾𝜏+𝑐𝑞̌𝑒̌^𝛾𝜏𝑐,Γ=diag𝑎₁,𝑎₂,…,𝑎_𝑀̌+Γ,𝑎=max_𝑟∈𝑆𝑎_𝑟,𝑎_𝑟=𝜆_max−2𝜂𝐼_𝑛+𝑝̌∑^𝑁_𝑖=1Υ^[𝑟]_𝑖1+2𝛼_𝑟𝑃Σ𝑝̌,𝑏̌=max_𝑟∈𝑆𝑏_𝑟,𝑏_𝑟=𝜆_max∑^𝑁_𝑖=1𝑃Υ^[𝑟]_𝑖2+2𝜃𝐼_𝑁𝑝,𝑐̌=max_𝑟∈𝑆𝑐_𝑟,𝑐_𝑟=𝜆_max∑^𝑁_𝑖=1𝑃Υ^[𝑟]_𝑖3+2𝛽_𝑟𝑃Σ𝑝, (45) then the solutions 𝑒₁(𝑡),𝑒₂(𝑡),…, and 𝑒_𝑁(𝑡) of system (9) are exponentially stable in mean square.

Remark 10. In [29], the exponential stability of a class of stochastic neural networks with both Markovian jump parameters and mixed fixed time delays were investigated. Therefore, we could see our results as a further research about the stochastic dynamic network of [29].

4. Numerical Simulation

In this section, we present some numerical simulation results that validate the theorem in the previous section.

Consider the chaotic delayed neural network (𝑡)=(𝑡)(𝑠(𝑡(𝑠((𝑡(𝑠(𝑡)((𝑡(𝑡),𝑑𝑠−𝐶𝑠+𝐴𝑓))+𝐵𝑔𝑡−𝜏)))𝑑𝑡+𝜎,𝑠𝑡−𝜏)))𝑑𝑤 (46) where 𝑓(𝑠)=𝑔(𝑠)=tanh(𝑠), 𝜏(𝑡)=1, 𝜎(𝑠(𝑡),𝑠(𝑡−𝜏(𝑡)))=diag{𝑠₁(𝑡),𝑠₂(𝑡)}, ,.𝐶=1001,𝐴=2−0.1−54.5𝐵=−1.5−0.1−0.2−4 (47) Taking 𝑃=diag{1,2} and Δ=diag{5,11,5}, we have 𝜂=0.15 and 𝜃=3.25 so that condition (11) is satisfied. Thus 𝑑𝑥_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))+⁵_𝑗=1𝑎^[𝑟]_𝑖𝑗Σ𝑥_𝑗(𝑡)+⁵_𝑗=1𝑏^[𝑟]_𝑖𝑗Σ𝑥_𝑗𝑡−𝜏_𝑐(𝑡)𝑑𝑡+𝜎_𝑖^[𝑟]𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)𝑑𝑤_𝑖(𝑡),𝑖=1,2,…,5,𝑟=1,2, (48) and Γ=⁻³³₂₋₂, 𝜏_𝑐(𝑡)=0.1(𝑒^𝑡/(1+𝑒^𝑡)), 𝜎_𝑖^[1]𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)=0.1diag𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡,𝜎_𝑖[2])𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)=0.1diag𝑥_𝑖1(𝑡−𝜏(𝑡)),𝑥_𝑖2(𝑡−𝜏(𝑡)). (49) Computations then yield 𝜏=1, 𝜏=0, 𝜏_𝑐=0.1, 𝜏_𝑐=0.1 and Υ_𝑖𝑗=0.1𝐼₂ for 𝑖=1,2,…,𝑁. Then the solutions of inequalities (41)–(44) are (by using the Matlab LMI toolbox) 𝛼₁=2.500, 𝛽₁=0.001, 𝑎₁=4.305, 𝑏₁=5.055, 𝑐₁=0.109; 𝛼₂=4.006, 𝛽₂=0.009, 𝑎₂=5.105, 𝑏₂=5.030, and 𝑐₂=0.065.

The initial conditions for this simulation are 𝑥_𝑖𝑗(𝑡₀) which are constants, for 𝑖=1,2,…,5,𝑗=1,2 and the trajectories of the impulse control gains are shown in Figure 2. Figure 3 shows the time evolution of the synchronization errors with impulse control.

(a)

(b)

Figure 2 

The trajectories of the state variables of 𝑥𝑖1 and 𝑥𝑖2(𝑖=1,2,…,5) in system (48) by impulse control.

(a)

(b)

Figure 3 

The time evolution of 𝑒𝑖1 and 𝑒𝑖2(𝑖=1,2,…,5) in system (48) by impulse control.

5. Conclusion

In this paper, we investigated the synchronization problem for stochastic complex networks with Markovian switching and nondelayed and time-varying delayed hybrid couplings. We achieved synchronization by applying an impulse control scheme to a small fraction of the nodes and derived sufficient conditions for stability of synchronization. Finally, we considered some numerical examples that illustrate the theoretical analysis.

Acknowledgments

This work was supported by the National Science Foundation of China under Grant nos. 61070087 and 61273220, Guangdong Education University Industry Cooperation Projects (2009B090300355), and Shenzhen Basic Research Project (JC201006010743A and JCYJ20120613105730482). The authors are very grateful to the reviewers and the editor for their valuable comments and suggestions to improve the presentation of the paper.