Adaptive Pinning Synchronization of Complex Networks with Stochastic Perturbations

Abstract

The adaptive pinning synchronization is investigated for complex networks with nondelayed and delayed couplings and vector-form stochastic perturbations. Two kinds of adaptive pinning controllers are designed. Based on an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are developed to guarantee the synchronization of the proposed complex networks even if partial states of the nodes are coupled. Furthermore, three examples with their numerical simulations are employed to show the effectiveness of the theoretical results.

1. Introduction

Recently, synchronization of all dynamical nodes in a network is one of the hot topics in the investigation of complex networks. It is well known that there are many useful synchronization phenomena in real life, such as the synchronous transfer of digital or analog signals in communication networks. Adaptive feedback control has witnessed its effectiveness in synchronizing a complex network [1–4]. By using the adaptive feedback control scheme, Chen and Zhou [1] studied synchronization of complex nondelayed networks, Cao et al. [2] investigated the complete synchronization in an array of linearly stochastically coupled identical networks with delays. In [3], Zhou et al. considered complex dynamical networks with uncertain couplings. In [4], Lu et al. studied the synchronization in arrays of delay-coupled neural networks. However, it is assumed that all the nodes need to be controlled in [1–4]. As we know, the real-world complex networks normally have a large number of nodes; it is usually impractical and impossible to control a complex network by adding the controllers to all nodes. To overcome this difficulty, pinning control, in which controllers are only applied to a small fraction of nodes, has been introduced in recent years [5–9]. By using adaptive pinning control method, Zhou et al. [10] studied local and global synchronizations of complex networks without delays; authors of [11, 12] considered the global synchronizations of the complex networks with nondelayed and delayed couplings. Note that, in most of existing results of complex networks' synchronization, all of the states are coupled for connected nodes. However, adaptive pinning synchronization results in which only partial states of the nodes are coupled are few. Hence, in this paper, we consider two different adaptive pinning controllers, which synchronize complex networks with partial or complete couplings of the nodes' states.

In the process of studying synchronization of complex networks, two main factors should be considered: time delays and stochastic perturbations. Time delays commonly exist in the real world and even vary according to time. In the subsystems, time delay can give rise to chaos, such as delayed neural network and delayed Chua's circuit system in Section 4 of this paper. Moreover, time-delayed couplings between subsystems cannot be ignored, such as in long-distance communication and traffic congestions, and so forth. In [8], authors investigated synchronization of complex networks with delayed subsystems, while in [4, 12, 13], authors investigated synchronization of complex networks with coupling delays. On the other hand, in real world, due to random uncertainties, such as stochastic forces on the physical systems and noisy measurements caused by environmental uncertainties, a stochastic behavior should be produced instead of a deterministic one. In fact, signals transmitted between subsystems of complex networks are unavoidably subject to stochastic perturbations from environment, which may cause information contained in these signals to be lost [2]. Therefore, transmitted signals may not be fully detected and received by other subsystems. This can have a great influence on the behavior of complex networks. There are some works in the field of synchronization of complex networks [2, 14–16]. Noise perturbations in [2, 14, 15] are all one-dimensional, which means that the signal transmitted by subsystems is influenced by the same noise. In [16], Yang and Cao considered stochastic synchronization of coupled neural networks with intermittent control, in which noise perturbations are vector forms. Vector-form perturbation means that different subsystem is influenced by different noise, and hence is practical in the real world.

Based on the above analysis, in this paper, we study the synchronization of complex networks with time-varying delays and vector-form stochastic elements. Two different adaptive pinning controllers according to the different properties of inner couplings are considered. To obtain our main results, we first formulate a new complex network with nondelayed and delayed couplings and vector-form Wiener processes. Then, by virtue of an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, we develop several theoretical results guaranteeing the synchronization of the new complex networks. Our adaptive pinning controllers are simple. The coupling matrices can be symmetric or asymmetric. Numerical simulations testify the effectiveness of our theoretical results.

Notations
In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. 𝐼_𝑁 denotes the 𝑁×𝑁 identity matrix. The Euclidean norm in ℝ^𝑛 is denoted as ‖⋅‖; accordingly, for vector 𝑥∈ℝ^𝑛, ‖𝑥‖=𝑥^𝑇𝑥, where 𝑇 denotes transposition. 𝐺=(𝑔_𝑖𝑗)_𝑁×𝑁 denotes a matrix of 𝑁-dimension, ‖𝐺‖=𝜆_max(𝐺^𝑇𝐺), where 𝜆_max(𝐴) (respectively, 𝜆_min(𝐴)) means the largest (respectively, smallest) eigenvalue of matrix 𝐴. 𝐺^𝑠=(1/2)(𝐺+𝐺^𝑇). 𝐺>0 or 𝐺<0 denotes that the matrix 𝐺 is symmetric and positive or negative definite matrix, respectively. 𝐺_𝑙 denotes the matrix of the first 𝑙 row-column pairs of 𝐺. 𝐺_𝑁−𝑙 denotes the minor matrix of matrix 𝐺 by removing all the first 𝑙 row-column pairs of 𝐺. 𝐸[⋅] is the mathematical expectation.

The rest of this paper is organized as follows. In Section 2, new model of delayed complex networks with nondelayed and delayed couplings and vector-form stochastic Weiner processes is presented. Some necessary assumptions, definitions, and lemmas are also given in this section. Our main results and their rigorous proofs are described in Section 3. In Section 4, two directed networks and one Barabási-Albert (BA) network [17] with their numerical simulations are employed to show the effectiveness of our results. Finally, in Section 5, conclusions are given.

2. Preliminaries

Consider complex a network consisting of 𝑁 identical nodes with nondelayed and delayed linear couplings and vector-form stochastic perturbations, which is described asd𝑥_𝑖(𝑡)=[𝑓(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑑^𝑁_𝑗=1𝑢_𝑖𝑗𝐷𝑥_𝑗(𝑡−𝜏(𝑡))]d𝑡+ℎ_𝑖(𝑥₁(𝑡),…,𝑥_𝑁(𝑡))d𝜔_𝑖(𝑡),𝑖=1,2,…,𝑁, (2.1) where 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡)]^𝑇∈ℝ^𝑛 represents the state vector of the 𝑖th node of the network at time 𝑡, 𝜏(𝑡)≥0 is time-varying delay, and 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-form function. Constants 𝑐>0, 𝑑>0 are nondelayed coupling strength and delayed coupling strength, respectively. Matrices Γ=diag(𝛾₁,𝛾₂,…,𝛾_𝑛) and 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) are inner couplings of the networks at time 𝑡 and 𝑡−𝜏(𝑡), respectively, satisfying 𝛾_𝑗≥0, not all 𝛾_𝑗=0, 𝑑_𝑗≥0, 𝑗=1,2,…,𝑛. Matrices 𝐺=(𝑔_𝑖𝑗)_𝑁×𝑁, and 𝑈=(𝑢_𝑖𝑗)_𝑁×𝑁 are outer couplings of the networks at time 𝑡 and 𝑡−𝜏(𝑡), respectively, satisfying 𝑔_𝑖𝑗≥0 for 𝑖≠𝑗, 𝑔_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑔_𝑖𝑗, and 𝑢_𝑖𝑗≥0 for 𝑖≠𝑗, 𝑢_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑢_𝑖𝑗, 𝑖=1,2,…,𝑁. Matrix ℎ_𝑖(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑁(𝑡))∈ℝ^𝑛×𝑛 describes the unknown coupling of the complex networks satisfying ℎ_𝑖(𝑥(𝑡),𝑥(𝑡),…,𝑥(𝑡))=0 (zero matrix of 𝑛 dimension). 𝜔_𝑖(𝑡)=(𝜔_𝑖1(𝑡),𝜔_𝑖2(𝑡),…,𝜔_𝑖𝑛(𝑡))^𝑇 is a bounded vector-form Weiner process. In this paper, we always assume that 𝜔_𝑖(𝑡) and 𝜔_𝑗(𝑡) are independent processes of one another for 𝑖≠𝑗, and matrix 𝐺=(𝑔_𝑖𝑗)_𝑁×𝑁 is irreducible in the sense that there is no isolated node.

The initial condition of system (2.1) is given in the following form: 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),−𝜏≤𝑠≤0,𝑖=1,2,…,𝑁, (2.2) where 𝜏=max{𝜏(𝑡)}, 𝜑_𝑖∈𝐿²_ℱ0([−𝜏,0],ℝ^𝑛), 𝐿²_ℱ0([−𝜏,0],ℝ^𝑛) is the family of all ℱ₀-measurable ℜ([−𝜏,0],ℝ^𝑛)-valued random variables satisfying sup_{−𝜏≤𝑠≤0}𝐸[‖𝜑_𝑖(𝑠)‖]<∞, and ℜ([−𝜏,0],ℝ^𝑛) denotes the family of all continuous ℝ^𝑛-valued functions 𝜑_𝑖(𝑠) on [−𝜏,0] with the norm ‖𝜑_𝑖‖=sup_{−𝜏≤𝑠≤0}𝜑^𝑇_𝑖(𝑠)𝜑_𝑖(𝑠). We always assume that system (2.1) has a unique solution with respect to initial condition.

Our objective of synchronization is to control network (2.1) to the trajectory 𝑠(𝑡)∈ℝ^𝑛 of the uncoupled system:d𝑠(𝑡)=𝑓(𝑡,𝑠(𝑡),𝑠(𝑡−𝜏(𝑡)))d𝑡, (2.3) where 𝑠(𝑡)=(𝑠₁(𝑡),𝑠₂(𝑡),…,𝑠_𝑛(𝑡))^𝑇 can be any desired state: equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit. To achieve this goal, some adaptive pinning controllers will be added to part of its nodes. Without loss of generality, rearrange the order of the nodes in the network, and let the first 𝑙 nodes be controlled. Thus, the pinning controlled network can be described byd𝑥_𝑖(𝑡)=[𝑓(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑑^𝑁_𝑗=1𝑢_𝑖𝑗𝐷𝑥_𝑗(𝑡−𝜏(𝑡))]d𝑡+ℎ_𝑖(𝑡,𝑥₁(𝑡),…,𝑥_𝑁(𝑡))d𝜔_𝑖(𝑡)+𝑅_𝑖d𝑡,𝑖=1,2,…,𝑙,d𝑥_𝑖(𝑡)=[𝑓(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑑^𝑁_𝑗=1𝑢_𝑖𝑗𝐷𝑥_𝑗(𝑡−𝜏(𝑡))]d𝑡+ℎ_𝑖(𝑡,𝑥₁(𝑡),…,𝑥_𝑁(𝑡))d𝜔_𝑖(𝑡),𝑖=𝑙+1,2,…,𝑁, (2.4) where 𝑅_𝑖, 𝑖=1,2,…,𝑙, are control inputs.

For convenience of writing, in the sequel, we denote ℎ_𝑖(𝑡,𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑁(𝑡)) as ℎ_𝑖(𝑡,𝑥(𝑡)). Accordingly, denote ℎ_𝑖(𝑡,𝑠(𝑡),𝑠(𝑡),…,𝑠(𝑡)) as ℎ_𝑖(𝑡,𝑠(𝑡)). Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡), 𝑒(𝑡)=(𝑒^𝑇₁(𝑡),𝑒^𝑇₂(𝑡),…,𝑒^𝑇_𝑁(𝑡))^𝑇, 𝑖=1,2,…,𝑁.

Subtracting (2.3) from (2.4) yields the following error dynamical system:d𝑒_𝑖(𝑡)=[𝐹(𝑡,𝑒_𝑖(𝑡),𝑒_𝑖(𝑡−𝜏(𝑡)))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑒_𝑗(𝑡)+𝑑^𝑁_𝑗=1𝑢_𝑖𝑗𝐷𝑒_𝑗+ℎ(𝑡−𝜏(𝑡))]d𝑡_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖(𝑡)+𝑅_𝑖d𝑡,𝑖=1,2,…,𝑙,d𝑒_𝑖(𝑡)=[𝐹(𝑡,𝑒_𝑖(𝑡),𝑒_𝑖(𝑡−𝜏(𝑡)))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑒_𝑗(𝑡)+𝑑^𝑁_𝑗=1𝑢_𝑖𝑗𝐷𝑒_𝑗+ℎ(𝑡−𝜏(𝑡))]d𝑡_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖(𝑡),𝑖=𝑙+1,2,…,𝑁, (2.5) where 𝐹(𝑡,𝑒_𝑖(𝑡),𝑒_𝑖(𝑡−𝜏(𝑡)))=𝑓(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))−𝑓(𝑡,𝑠(𝑡),𝑠(𝑡−𝜏(𝑡))), and ℎ_𝑖(𝑡,𝑒(𝑡))=ℎ_𝑖(𝑡,𝑥(𝑡))−ℎ_𝑖(𝑡,𝑠(𝑡)).

Then the objective here is to find some appropriate adaptive pinning controllers 𝑅_𝑖, 𝑖=1,2,…,𝑙, such that the trivial solution of error system (2.5) is globally asymptotically stable in the mean square, that is, lim_𝑡→∞𝐸[‖𝑒_𝑖(𝑡)‖²]=0,𝑖=1,2,…,𝑁. (2.6)

Definition 2.1. Function class QUAD1(𝑃,Δ,𝜂,𝜃): let 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛) be a positive-definite diagonal matrix, Δ=diag(𝛿₁,𝛿₂,…,𝛿_𝑛) a diagonal matrix, and constants 𝜂>0, 𝜃≥0. QUAD1(𝑃,Δ,𝜂,𝜃) denotes a class of continuous functions 𝑓(𝑡,𝑥,𝑧)∈[0,+∞)×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 satisfying (𝑥−𝑦)^𝑇𝑃[𝑓(𝑡,𝑥,𝑧)−𝑓(𝑡,𝑦,𝑤)−Δ(𝑥−𝑦)]≤−𝜂(𝑥−𝑦)^𝑇(𝑥−𝑦)+𝜃(𝑧−𝑤)^𝑇(𝑧−𝑤) (2.7) for all 𝑡≥0, for all 𝑥,𝑦,𝑧,𝑤∈ℝ^𝑛.

Remark 2.2. The function class QUAD1(𝑃,Δ,𝜂,𝜃) includes almost all the well-known chaotic systems with delays or without delays such as Lorenz system, Rössler system, Chen system, delayed Chua's circuit as well as logistic delay differential system, delayed Hopfield neural networks and delayed CNNs, and so on. In fact, all those systems mentioned above satisfy the following condition [8, 18]: there exist positive 𝑘_𝑖𝑗>0 such that for |𝑓_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))−𝑓_𝑖≤(𝑡,𝑦(𝑡),𝑦(𝑡−𝜏(𝑡)))|_𝑛𝑗=1𝑘_𝑖𝑗(|𝑥_𝑗(𝑡)−𝑦_𝑗(𝑡)|+|𝑥_𝑗(𝑡−𝜏(𝑡))−𝑦_𝑗(𝑡−𝜏(𝑡))|),𝑖=1,2,…,𝑛 (2.8) for any 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈ℝ^𝑛, 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈ℝ^𝑛.

From condition (2.8), we get (𝑥−𝑦)^𝑇≤𝑃(𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))−𝑓(𝑡,𝑦(𝑡),𝑦(𝑡−𝜏(𝑡))))_{𝑛𝑛𝑖=1𝑗=1}𝑝_𝑖𝑘_𝑖𝑗|𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡)|(|𝑥_𝑗(𝑡)−𝑦_𝑗(𝑡)|+|𝑥_𝑗(𝑡−𝜏(𝑡))−𝑦_𝑗=(𝑡−𝜏(𝑡))|)_{𝑛𝑛𝑖=1𝑗=1}𝑝_𝑖[𝑘^1/2_𝑖𝑗|𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡)|𝑘^1/2_𝑖𝑗|𝑥_𝑗(𝑡)−𝑦_𝑗(𝑡)|+𝑘^1/2_𝑖𝑗|𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡)|𝑘^1/2_𝑖𝑗|𝑥_𝑗(𝑡−𝜏(𝑡))−𝑦_𝑗≤(𝑡−𝜏(𝑡))|]_{𝑛𝑛𝑖=1𝑗=1}𝑝_𝑖[𝑘_𝑖𝑗(𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡))²+12𝑘_𝑖𝑗(𝑥_𝑗(𝑡)−𝑦_𝑗(𝑡))²+12𝑘_𝑖𝑗(𝑥_𝑗(𝑡−𝜏(𝑡))−𝑦_𝑗(𝑡−𝜏(𝑡)))²=_{𝑛𝑛𝑖=1𝑗=1}]𝑝_𝑖[𝑘_𝑖𝑗(𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡))²+12𝑘_𝑗𝑖(𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡))²+12𝑘_𝑗𝑖(𝑥_𝑖(𝑡−𝜏(𝑡))−𝑦_𝑖(𝑡−𝜏(𝑡)))²=_{𝑛𝑛𝑖=1𝑗=1}]𝑝_𝑖[(𝑘_𝑖𝑗+12𝑘_𝑗𝑖)(𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡))²+12𝑘_𝑗𝑖(𝑥_𝑖(𝑡−𝜏(𝑡))−𝑦_𝑖(𝑡−𝜏(𝑡)))²=_𝑛𝑖=1]𝑝_𝑖[(−𝛿_𝑖+^𝑛_𝑗=1(𝑘_𝑖𝑗+12𝑘_𝑗𝑖))(𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡))²+12^𝑛_𝑗=1𝑘_𝑗𝑖(𝑥_𝑖(𝑡−𝜏(𝑡))−𝑦_𝑖(𝑡−𝜏(𝑡)))²]+(𝑥(𝑡)−𝑦(𝑡))^𝑇𝑃Δ(𝑥(𝑡)−𝑦(𝑡))≤−𝜂(𝑥(𝑡)−𝑦(𝑡))^𝑇(𝑥(𝑡)−𝑦(𝑡))+(𝑥(𝑡)−𝑦(𝑡))^𝑇𝑃Δ(𝑥(𝑡)−𝑦(𝑡))+𝜃(𝑥(𝑡−𝜏(𝑡))−𝑦(𝑡−𝜏(𝑡)))^𝑇(𝑥(𝑡−𝜏(𝑡))−𝑦(𝑡−𝜏(𝑡))), (2.9)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10) where 𝜂=min_{1≤𝑖≤𝑛}{𝑝_𝑖(𝛿_𝑖−∑^𝑛_𝑗=1(𝑘_𝑖𝑗+(1/2)𝑘_𝑗𝑖))}, 𝜃=max_{1≤𝑖≤𝑛}{(1/2)𝑝_𝑖∑^𝑛_𝑗=1𝑘_𝑗𝑖}, and 2𝑎𝑏≤𝑎²+𝑏² is used in the above inequality. Hence, condition (2.7) is satisfied. Note that function class QUAD1(𝑃,Δ,𝜂,𝜃) is more general than the usual function class QUAD(𝑃,Δ,𝜂) (see [13, 19]) for chaotic systems and includes it as a special class when 𝜃=0.

Before giving our main results, we present the following lemmas, which are needed in the next section.

Lemma 2.3 (see [12]). If 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁 is an irreducible matrix satisfying 𝑎_𝑖𝑗=𝑎_𝑗𝑖≥0, for 𝑖≠𝑗, and 𝑎_𝑖𝑖=−∑^𝑁_{𝑖=1,𝑖≠𝑗}𝑎_𝑖𝑗, for 1=1,2,…,𝑁, then all eigenvalues of the matrix 𝐴 = 𝐴−diag(𝑘₁,…,𝑘_𝑙,0,…,0) are negative, where 𝑘₁,…,𝑘_𝑙 are positive constants.

Lemma 2.4 (see [19]). Suppose that 𝐺=(𝑔_𝑖𝑗)_𝑁×𝑁 is an irreducible matrix satisfying 𝑔_𝑖𝑗≥0, for 𝑖≠𝑗, and 𝑔_𝑖𝑖=−∑^𝑁_{𝑖=1,𝑖≠𝑗}𝑔_𝑖𝑗, for 1=1,2,…,𝑁. Let 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑁)^𝑇 be the left eigenvector corresponding to the eigenvalue 0, that is, 𝜉^𝑇𝐺=0. Then, 𝜉_𝑖>0, 𝑖=1,2,…,𝑁, and its multiplicity is 1.

Lemma 2.5 (see [20]; (Schur Complement)). The linear matrix inequality (LMI) 𝑆𝑆=¹¹𝑆₁₂𝑆^𝑇₁₂𝑆₂₂<0 (2.10) is equivalent to anyone of the following two conditions:(L₁)𝑆₁₁<0, 𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0,(L₂)𝑆₂₂<0, 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0,where 𝑆₁₁=𝑆^𝑇₁₁ and 𝑆₂₂=𝑆^𝑇₂₂.

Now we list assumptions as follows.(H₁) There exist nonnegative constants 𝜇_𝑖𝑗, 𝑖,𝑗=1,2,…,𝑁, such that trace((ℎ_𝑖(𝑡,𝑒(𝑡)))^𝑇ℎ_𝑖(𝑡,𝑒(𝑡)))≤^𝑁_𝑗=1𝜇_𝑖𝑗‖𝑒_𝑗(𝑡)‖², (2.11) where ℎ_𝑖(𝑡,𝑒(𝑡))=ℎ_𝑖(𝑡,𝑥(𝑡))−ℎ_𝑖(𝑡,𝑠(𝑡)), 𝑖=1,2,…,𝑁.(H₂)𝜏(𝑡) is a continuous differentiable function on ℝ⁺ with 𝜏̇(𝑡)≤ℎ<1.

3. Main Results

In this section, two different adaptive pinning feedback controllers corresponding to two kinds of properties of the inner coupling Γ are designed, and several sufficient conditions for synchronization of the complex networks (2.1) are derived.

For the the inner coupling matrix Γ, there exist the following two cases:(a)𝛾_𝑗≥0, 𝑗=1,2,…,𝑛, and there exists at least one 𝑗₀∈{𝑗=1,2,…,𝑛} such that 𝛾_𝑗0>0.(b)𝛾_𝑗>0, 𝑗=1,2,…,𝑛.

Obviously, case (b) is the special case of case (a). Case (b) means that all the states of connected nodes are coupled, while in case (a) this is not necessary. However, most of existing papers only consider case (b); for example, see references [7, 12, 13, 19]. In this paper, we will consider both of the two cases. We first study the general case (a).

Theorem 3.1. Suppose that the assumptions (H₁)-(H₂) hold and 𝑓∈ QUAD1(𝑃,Δ,𝜂,𝜃). For the case (a), if 2𝑐𝛾𝐺^𝑠_𝑁−𝑙+𝑊_𝑁−𝑙<0, where 𝐺=(𝑔_𝑖𝑗)_𝑁×𝑁, 𝑔_𝑖𝑗=𝑔_𝑖𝑗,𝑖≠𝑗, 𝑔_𝑖𝑖=(𝛾/𝛾)𝑔_𝑖𝑖, 𝛾=max{𝛾_𝑗,𝑗=1,2,…,𝑛}, 𝛾=min{𝛾_𝑗,𝑗=1,2,…,𝑛}, 𝑊=2(−𝜂+𝛿)𝐼_𝑁+2Φ+((5−ℎ)/2(1−ℎ))𝑑𝑑‖𝑉‖𝐼_𝑁, 𝛿=max{𝑝_𝑗𝛿_𝑗,𝑗=1,2,…,𝑛}, Φ=diag(𝜙₁,𝜙₂,…,𝜙_𝑁), 𝜙_𝑗=(1/2)∑^𝑁_𝑖=1𝜇_𝑖𝑗, 𝑑=max{𝑑_𝑗,𝑗=1,2,…,𝑛}, 𝑉=(|𝑢_𝑖𝑗|)_𝑁×𝑁, then the trivial solution of (2.5) is globally asymptotically stable in the mean square with the adaptive pinning controllers 𝑅_𝑖=−𝑐𝜀_𝑖𝑒_𝑖𝜀̇(𝑡),𝑖=1,2,…,𝑙,_𝑖=𝑣_𝑖𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡),𝑖=1,2,…,𝑙. (3.1)

Proof. According to (H₁) and (H₂), the origin is an equilibrium point of the error system (2.5). We define the Lyapunov-Krasovskii functional as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡), (3.2) where 𝑉₁1(𝑡)=2^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖1(𝑡)+2^𝑙_𝑖=1𝑐(𝜀_𝑖−𝑘_𝑖)²𝑣_𝑖,𝑉₂(𝑡)=^𝑡_{𝑡−𝜏(𝑡)}𝜁^𝑇(𝑠)𝑄𝜁(𝑠)d𝑠, (3.3)𝜁(𝑡)=(‖𝑒₁(𝑡)‖,‖𝑒₂(𝑡)‖,…,‖𝑒_𝑁(𝑡)‖)^𝑇, 𝑄 is a positive definite matrix, 𝑘_𝑖, 𝑖=1,2,…,𝑙, are positive constants, and 𝑄 and 𝑘_𝑖, 𝑖=1,2,…,𝑙, are to be determined.
Differentiating 𝑉₁(𝑡) along the solution of (2.5) and using the properties of vector-form Weiner process [16], we get d𝑉₁(𝑡)=^𝑁_𝑖=1[𝑒^𝑇_𝑖(𝑡)𝑃d𝑒_𝑖1(𝑡)+2(d𝑒_𝑖(𝑡))^𝑇𝑃d𝑒_𝑖(𝑡)]+𝑐^𝑙_𝑖=1(𝜀_𝑖−𝑘_𝑖)𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖=(𝑡)d𝑡_𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃[𝐹(𝑡,𝑒_𝑖(𝑡),𝑒_𝑖(𝑡−𝜏(𝑡)))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑒_𝑗(𝑡)+𝑑^𝑁_𝑗=1𝑢_𝑖𝑗𝐷𝑒_𝑗(𝑡−𝜏(𝑡))]d𝑡−𝑐^𝑙_𝑖=1𝑘_𝑖𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡)d𝑡+^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)ℎ_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖+1(𝑡)2^𝑁_𝑖=1trace[(ℎ_𝑖(𝑡,𝑒(𝑡)))^𝑇ℎ_𝑖(𝑡,𝑒(𝑡))]d𝑡={^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃(𝐹(𝑡,𝑒_𝑖(𝑡),𝑒_𝑖(𝑡−𝜏(𝑡)))−Δ𝑒_𝑖(𝑡))+^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃Δ𝑒_𝑖(𝑡)−𝑐^𝑙_𝑖=1𝑘_𝑖𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃(𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑒_𝑗(𝑡)+𝑑^𝑁_𝑗=1𝑢_𝑖𝑗𝐷𝑒_𝑗+1(𝑡−𝜏(𝑡)))2^𝑁_𝑖=1trace[(ℎ_𝑖(𝑡,𝑒(𝑡))^𝑇ℎ_𝑖(𝑡,𝑒(𝑡))]}d𝑡+^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)ℎ_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖(𝑡)≤{−𝜂^𝑁_𝑖=1‖𝑒_𝑖(𝑡)‖²+𝜃^𝑁_𝑖=1‖𝑒_𝑖(𝑡−𝜏(𝑡))‖²+𝛿^𝑁_𝑖=1‖𝑒_𝑖(𝑡)‖²−𝑐𝑝^𝑙_𝑖=1𝑘_𝑖‖𝑒_𝑖(𝑡)‖²+𝑐^𝑁𝑁_{𝑖=1𝑗=1,𝑗≠𝑖}𝑔_𝑖𝑗𝛾‖𝑒_𝑖(𝑡)‖‖𝑒_𝑗(𝑡)‖+𝑐^𝑁_𝑖=1𝑔_𝑖𝑖𝛾‖𝑒_𝑖(𝑡)‖²+12^𝑁𝑁_𝑖=1𝑗=1𝜇_𝑖𝑗‖𝑒_𝑗(𝑡)‖²+𝑑^𝑁𝑁_{𝑖=1𝑗=1,𝑗≠𝑖}|𝑢_𝑖𝑗|𝑑‖𝑒_𝑖(𝑡)‖‖𝑒_𝑗(𝑡−𝜏(𝑡))‖}d𝑡+^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)ℎ_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖(𝑡), (3.4) where 𝑝=max{𝑝_𝑗,𝑗=1,2,…,𝑛}.
Similarly, from 𝑉₂(𝑡) we have d𝑉₂(𝑡)=[𝜁^𝑇(𝑡)𝑄𝜁(𝑡)−(1−𝜏̇(𝑡))𝜁^𝑇(𝑡−𝜏(𝑡))𝑄𝜁(𝑡−𝜏(𝑡))]≤[𝜁^𝑇(𝑡)𝑄𝜁(𝑡)−(1−ℎ)𝜁^𝑇(𝑡−𝜏(𝑡))𝑄𝜁(𝑡−𝜏(𝑡))]d𝑡. (3.5) Hence, d𝑉(𝑡)≤[(𝛿−𝜂)𝜁^𝑇(𝑡)𝜁(𝑡)+𝜃𝜁^𝑇(𝑡−𝜏(𝑡))𝜁(𝑡−𝜏(𝑡))−𝑐𝑝𝜁^𝑇(𝑡)𝐾𝜁(𝑡)+𝑐𝛾𝜁^𝑇(𝑡)𝐺^𝑠𝜁(𝑡)+𝑑𝑑𝜁^𝑇(𝑡)𝑉𝜁(𝑡−𝜏(𝑡))+𝜁^𝑇(𝑡)(𝑅+𝑄)𝜁(𝑡)−(1−ℎ)𝜂^𝑇+(𝑡−𝜏(𝑡))𝑄𝜂(𝑡−𝜏(𝑡))]d𝑡_𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)ℎ_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖=1(𝑡)2𝜛^𝑇(𝑡)𝚷𝜛(𝑡)d(𝑡)+^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)ℎ_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖(𝑡), (3.6) where 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑙,0,…,0), 𝜛^𝑇(𝑡)=(𝜁^𝑇(𝑡),𝜁^𝑇(𝑡−𝜏(𝑡)))^𝑇, and 𝚷=2((−𝜂+𝛿)𝐼_𝑁−𝑐𝑝𝐾+𝑐𝛾𝐺^𝑠+Φ+𝑄)𝑑𝑑𝑉^𝑇𝑑𝑑𝑉2𝜃𝐼_𝑁−2(1−ℎ)𝑄. (3.7)
In view of Lemma 2.5, 𝚷<0 is equivalent to Υ=2((−𝜂+𝛿)𝐼_𝑁−𝑐𝑝𝐾+𝑐𝛾𝐺^𝑠1+Φ+𝑄)+2𝑑²𝑑²𝑉^𝑇((1−ℎ)𝑄−𝜃𝐼_𝑁)⁻¹𝑉<0,−(1−ℎ)𝑄+𝜃𝐼_𝑁<0. (3.8)
Let 𝑄=(1/(1−ℎ))(𝑑𝑑‖𝑉‖+𝜃)𝐼_𝑁, we obtain Υ≤2(−𝜂+𝛿)𝐼_𝑁−2𝑐𝑝𝐾+2𝑐𝛾𝐺^𝑠+2Φ+5−ℎ𝑑2(1−ℎ)𝑑‖𝑉‖𝐼_𝑁=−2𝑐𝑝𝐾_𝑙+2𝑐𝛾𝐺^𝑠_𝑙+𝑊_𝑙2𝑐𝛾(𝐺^𝑠_∗)^𝑇2𝑐𝛾𝐺^𝑠_∗2𝑐𝛾𝐺^𝑠_𝑁−𝑙+𝑊_𝑁−𝑙, (3.9) where 𝐺^𝑠_∗ is matrix with appropriate dimension.
Since 2𝑐𝛾𝐺^𝑠_𝑁−𝑙+𝑊_𝑁−𝑙<0, obviously, there exist positive constants 𝑘₁,𝑘₂,…,𝑘_𝑙 such that −2𝑐𝑝𝐾_𝑙+2𝑐𝛾𝐺^𝑠_𝑙+𝑊_𝑙−2𝑐²𝛾²(𝐺^𝑠_∗)^𝑇(2𝑐𝛾𝐺^𝑠_𝑁−𝑙+𝑊_𝑁−𝑙)⁻¹𝐺^𝑠_∗<0. (3.10) Again, from Lemma 2.5 we obtain Υ<0. Hence, (3.8) holds, which means that 𝚷<0. Taking the mathematical expectation of both sides of (3.6), we have d𝐸𝑉(𝑡)1d𝑡≤𝐸[2𝜛^𝑇1(𝑡)𝚷𝜛(𝑡)]≤−2𝜆_min(−𝚷)𝐸[‖𝜁(𝑡)‖²+‖𝜁(𝑡−𝜏(𝑡))‖²]1≤−2𝜆_min(−𝚷)𝐸[‖𝜁(𝑡)‖²1]=−2𝜆_min(−𝚷)𝐸[^𝑁_𝑖=1‖𝑒_𝑖(𝑡)‖²], (3.11) where 𝜆_min(−𝚷) is positive. In view of the LaSalle invariance principle of stochastic differential equation, which was developed in [21], we have lim_𝑡→∞𝐸[∑^𝑁_𝑖=1‖𝑒_𝑖(𝑡)‖²]=0, which in turn illustrates that lim_𝑡→∞𝐸[‖𝑒_𝑖(𝑡)‖²]=0, and at the same time, lim_𝑡→∞𝜀_𝑖(𝑡)=𝜀_𝑖 (constants). This completes the proof.

Next, we consider the case (b). Obviously, the adaptive pinning controllers (3.1) can also synchronize (2.1) for this case. Here we use another kind of controllers, which is related to the inner matrix Γ, to synchronize the complex networks (2.1).

Assume that 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑁)^𝑇 is defined in Lemma 2.4; we have 𝜉_𝑖>0, 𝑖=1,2,…,𝑁. Define Ξ=diag(𝜉₁,𝜉₂,…,𝜉_𝑁). It is easy to verify that Ξ𝐺+𝐺^𝑇Ξ is a symmetric and zero row sum matrix. Moreover, if 𝐺 is irreducible, then Ξ𝐺 is also irreducible; hence, Ξ𝐺+𝐺^𝑇Ξ is irreducible. Therefore, by Lemma 2.3, (Ξ𝐺)^𝑠=(1/2)(Ξ𝐺+𝐺^𝑇Ξ)=(1/2)(Ξ𝐺+𝐺^𝑇Ξ)−Ξ𝐾 is negative definite, where 𝐺=𝐺−𝐾, 𝐾=diag(𝑘₁,…,𝑘_𝑙,0,…,0), and 𝑘_𝑖, 𝑖=1,…,𝑙, are positive constants.

Theorem 3.2. Suppose that the assumptions (H₁)-(H₂) hold and 𝑓∈ QUAD1(𝑃,Δ,𝜂,𝜃). If 𝛾_𝑗>0 and 2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐺)^𝑠_𝑁−𝑙+(𝑊_𝑗)_𝑁−𝑙<0, 𝑗=1,2,…,𝑛, where 𝑊_𝑗=−2𝜂Ξ+2𝑝_𝑗𝛿_𝑗Ξ+Φ+((2−ℎ)/(1−ℎ))𝑑𝑝_𝑗𝑑_𝑗‖Ξ𝑈‖𝐼_𝑁+(2𝜃/(1−ℎ))Ξ, Φ=diag(𝜙₁,𝜙₂,…,𝜙_𝑁), and 𝜙_𝑗=∑^𝑁_𝑖=1𝜉_𝑖𝜇_𝑖𝑗, then the trivial solution of (2.5) is globally asymptotically stable in the mean square with the adaptive pinning controllers 𝑅_𝑖=−𝑐𝜀_𝑖Γ𝑒_𝑖𝜀̇(𝑡),𝑖=1,2,…,𝑙,_𝑖=𝑣_𝑖𝑒^𝑇_𝑖(𝑡)𝑃Γ𝑒_𝑖(𝑡),𝑖=1,2,…,𝑙. (3.12)

Proof. Let 𝑒_𝑗̃(𝑡)=(𝑒_1𝑗,𝑒_2𝑗,…,𝑒_𝑁𝑗)^𝑇, 𝑗=1,2,…,𝑛. We define the Lyapunov-Krasovskii functional as 1𝑉(𝑡)=2^𝑁_𝑖=1𝜉_𝑖𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖1(𝑡)+2^𝑛_𝑗=1^𝑡_{𝑡−𝜏(𝑡)}𝑒^𝑇_𝑗̃(𝑠)𝑄_𝑗𝑒_𝑗̃1(𝑠)d𝑠+2^𝑙_𝑖=1𝑐𝜉_𝑖(𝜀_𝑖−𝑘_𝑖)²𝑣_𝑖, (3.13) where 𝑄_𝑗, 𝑗=1,2,…,𝑛, are positive definite matrices, 𝑘_𝑖, 𝑖=1,2,…,𝑙, are positive constants, and 𝑄_𝑗 and 𝑘_𝑖,𝑖=1,2,…,𝑙, are to be determined.
Differentiating both sides of (3.13) along the solution of (2.5) yields d𝑉(𝑡)=^𝑁_𝑖=1𝜉_𝑖[𝑒^𝑇_𝑖(𝑡)𝑃d𝑒_𝑖1(𝑡)+2(d𝑒_𝑖(𝑡))^𝑇𝑃d𝑒_𝑖1(𝑡)]+{2^𝑛_𝑗=1[𝑒^𝑇_𝑗̃(𝑡)𝑄_𝑗𝑒_𝑗̃(𝑡)−(1−𝜏̇(𝑡))𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄_𝑗𝑒_𝑗̃(𝑡−𝜏(𝑡))]+𝑐^𝑙_𝑖=1𝜉_𝑖(𝜀_𝑖−𝑘_𝑖)𝑒^𝑇_𝑖(𝑡)𝑃Γ𝑒_𝑖=(𝑡)d𝑡}_𝑁𝑖=1𝜉_𝑖𝑒^𝑇_𝑖(𝑡)𝑃[𝐹(𝑡,𝑒_𝑖(𝑡),𝑒_𝑖(𝑡−𝜏(𝑡)))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑒_𝑗(𝑡)+𝑑^𝑁_𝑗=1𝑢_𝑖𝑗𝐷𝑒_𝑗(𝑡−𝜏(𝑡))]d𝑡−𝑐^𝑙_𝑖=1𝜉_𝑖𝑘_𝑖𝑒^𝑇_𝑖(𝑡)𝑃Γ𝑒_𝑖(𝑡)d𝑡+^𝑁_𝑖=1𝜉_𝑖𝑒^𝑇_𝑖(𝑡)ℎ_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖+1(𝑡)2^𝑁_𝑖=1𝜉_𝑖trace[(ℎ_𝑖(𝑡,𝑒(𝑡)))^𝑇ℎ_𝑖+1(𝑡,𝑒(𝑡))]d𝑡2^𝑛_𝑗=1[𝑒^𝑇_𝑗̃(𝑡)𝑄_𝑗𝑒_𝑗̃(𝑡)−(1−𝜏̇(𝑡))𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄_𝑗𝑒_𝑗̃(𝑡−𝜏(𝑡))]d𝑡={^𝑁_𝑖=1𝜉_𝑖𝑒^𝑇_𝑖(𝑡)𝑃(𝐹(𝑡,𝑒_𝑖(𝑡),𝑒_𝑖(𝑡−𝜏(𝑡)))−Δ𝑒_𝑖(𝑡))+^𝑁_𝑖=1𝜉_𝑖𝑒^𝑇_𝑖(𝑡)𝑃Δ𝑒_𝑖(𝑡)−𝑐^𝑙_𝑖=1𝜉_𝑖𝑘_𝑖𝑒^𝑇_𝑖(𝑡)𝑃Γ𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝜉_𝑖𝑒^𝑇_𝑖(𝑡)𝑃(𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑒_𝑗(𝑡)+𝑑^𝑁_𝑗=1𝑢_𝑖𝑗𝐷𝑒_𝑗+1(𝑡−𝜏(𝑡)))2^𝑁_𝑖=1𝜉_𝑖trace[(ℎ_𝑖(𝑡,𝑒(𝑡))^𝑇ℎ_𝑖+1(𝑡,𝑒(𝑡))]2^𝑛_𝑗=1[𝑒^𝑇_𝑗̃(𝑡)𝑄_𝑗𝑒_𝑗̃(𝑡)−(1−𝜏̇(𝑡))𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄_𝑗𝑒_𝑗̃+(𝑡−𝜏(𝑡))]}d𝑡_𝑁𝑖=1𝜉_𝑖𝑒^𝑇_𝑖(𝑡)ℎ_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖(𝑡)≤{−𝜂^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)Ξ𝑒_𝑗̃(𝑡)+𝜃^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))Ξ𝑒_𝑗̃(𝑡−𝜏(𝑡))+^𝑛_𝑗=1𝑝_𝑗𝛿_𝑗𝑒^𝑇_𝑗̃(𝑡)Ξ𝑒_𝑗̃(𝑡)+𝑐^𝑛_𝑗=1𝑝_𝑗𝛾_𝑗𝑒^𝑇_𝑗̃(𝑡)(Ξ𝐺)^𝑠𝑒_𝑗̃(𝑡)+𝑑^𝑛_𝑗=1𝑝_𝑗𝑑_𝑗𝑒^𝑇_𝑗̃(𝑡)Ξ𝑈𝑒_𝑗̃1(𝑡−𝜏(𝑡))+2^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)Φ𝑒_𝑗̃+1(𝑡)2^𝑛_𝑗=1[𝑒^𝑇_𝑗̃(𝑡)𝑄_𝑗𝑒_𝑗̃(𝑡)−(1−ℎ)𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄_𝑗𝑒_𝑗̃+(𝑡−𝜏(𝑡))]}d𝑡_𝑁𝑖=1𝜉_𝑖𝑒^𝑇_𝑖(𝑡)ℎ_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖=1(𝑡)2^𝑛_𝑗=1𝜁^𝑇_𝑗𝚷_𝑗𝜁_𝑗d𝑡+^𝑁_𝑖=1𝜉_𝑖𝑒^𝑇_𝑖(𝑡)ℎ_𝑖(𝑡,𝑒(𝑡))d𝜔_𝑖(𝑡), (3.14)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26) where 𝜁_𝑗=(𝑒^𝑇_𝑗̃(𝑡),𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡)))^𝑇 and 𝚷_𝑗=−2𝜂Ξ+2𝑝_𝑗𝛿_𝑗Ξ+2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐺)^𝑠+Φ+𝑄_𝑗𝑑𝑝_𝑗𝑑_𝑗Ξ𝑈𝑑𝑝_𝑗𝑑_𝑗(Ξ𝑈)^𝑇−(1−ℎ)𝑄_𝑗+2𝜃Ξ,𝑗=1,2,…,𝑛. (3.15) In view of Lemma 2.5, 𝚷_𝑗<0 is equivalent to Υ_𝑗=−2𝜂Ξ+2𝑝_𝑗𝛿_𝑗Ξ+2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐺)^𝑠+Φ+𝑄_𝑗+(𝑑𝑝_𝑗𝑑_𝑗)²(Ξ𝑈)[(1−ℎ)𝑄_𝑗−2𝜃Ξ]⁻¹(Ξ𝑈)^𝑇<0−(1−ℎ)𝑄_𝑗+2𝜃Ξ<0. (3.16) Let 𝑄_𝑗=(1/(1−ℎ))[𝑑𝑝_𝑗𝑑_𝑗‖Ξ𝑈‖𝐼_𝑁+2𝜃Ξ]; we have Υ_𝑗≤−2𝜂Ξ+2𝑝_𝑗𝛿_𝑗Ξ+2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐺)^𝑠+Φ+2−ℎ1−ℎ𝑑𝑝_𝑗𝑑_𝑗‖Ξ𝑈‖𝐼_𝑁+2𝜃Ξ=1−ℎ2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐺)^𝑠_𝑙−2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐾)_𝑙+(𝑊_𝑗)_𝑙2𝑐𝑝_𝑗𝛾_𝑗((Ξ𝐺)^𝑠_∗)^𝑇2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐺)^𝑠_∗2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐺)^𝑠_𝑁−𝑙+(𝑊_𝑗)_𝑁−𝑙, (3.17) where (Ξ𝐺)^𝑠_∗ is the matrix with appropriate dimension.
Since 2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐺)^𝑠_𝑁−𝑙+(𝑊_𝑗)_𝑁−𝑙<0, (Ξ𝐺)^𝑠_𝑁−𝑙=(Ξ𝐺)^𝑠_𝑁−𝑙, there exist positive constants 𝑘₁,𝑘₂,…,𝑘_𝑙 such that 2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐺)^𝑠_𝑙−2𝑐𝑝_𝑗𝛾_𝑗(Ξ𝐾)_𝑙+(𝑊_𝑗)_𝑙−4𝑐²𝑝²_𝑗𝛾²_𝑗((Ξ𝐺)^𝑠_∗)^𝑇[2𝑐𝛾_𝑗(Ξ𝐺)^𝑠_𝑁−𝑙+(𝑊_𝑗)_𝑁−𝑙]⁻¹(Ξ𝐺)^𝑠_∗<0. (3.18) Again, from Lemma 2.5 we obtain Υ_𝑗<0. Hence, (3.16) holds, which implies 𝚷_𝑗<0. The rest steps are the same as those in the proof of Theorem 3.1. This completes the proof.

Remark 3.3. If there is no noise perturbation in (2.1), then Φ=0 in Theorem 3.1 and Φ=0 in Theorem 3.2, respectively. Hence, if (2.1) is synchronized with the first 𝑙 nodes to be controlled by adding controllers (3.1) or (3.12), then (2.1) with no noise perturbation can sure be synchronized by using adapting pinning controllers (3.1) or (3.12) with no more than 𝑙 nodes to be controlled.

Remark 3.4. Theorems 3.1 and 3.2 are applicable to directed networks and undirected networks. Although they do not point out which nodes should be controlled first; however, according to [22, 23], for the graph corresponding to 𝐺, nodes whose out-degrees are larger than their in-degrees [24] should be controlled first for directed networks, while the controlled nodes can be randomly selected and it is better to successively pin the most-highly connected nodes for undirected networks.

4. Numerical Examples

In this section, we provide three examples to illustrate the effectiveness of the results obtained above.

Example 4.1. Consider the following chaotic delayed neural networks: d𝑠(𝑡)=[−𝐶𝑠(𝑡)+𝐴𝑓(𝑠(𝑡))+𝐵𝑓(𝑠(𝑡−𝜏))]d𝑡, (4.1) where 𝑓(𝑠)=tanh(𝑠), 𝜏=1, 𝐶=1001,𝐴=2−0.1−54.5,𝐵=−1.5−0.1−0.2−4. (4.2) In the case that the initial condition is chosen as 𝑠₁(𝑡)=0.4, 𝑠₂(𝑡)=0.6, for all 𝑡∈[−1,0], the chaotic attractor can be seen in Figure 1.
Taking 𝑃=diag(1,1) and Δ=diag(5,11.5) and we have 𝜂=0.15, 𝜃=3.25. Hence the condition (2.7) is satisfied.
In order to verify our new results, consider the following coupled networks: d𝑥_𝑖(𝑡)=[−𝐶𝑥_𝑖(𝑡)+𝐴𝑓(𝑥_𝑖(𝑡))+𝐵𝑓(𝑥_𝑖(𝑡−𝜏))+𝑐³_𝑗=1𝑔_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑑³_𝑗=1𝑢_𝑖𝑗𝐷𝑥_𝑗(𝑡−𝜏)]d𝑡+ℎ_𝑖(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))d𝜔_𝑖(𝑡),𝑖=1,2,3, (4.3) where Γ and 𝐷 are 2-dimensional identity matrices, 𝑐=4, 𝑑=2, and 𝐺 and 𝑈 are asymmetric and zero-row sum matrices as the following: 𝐺=−1010−1146−10,𝑈=−2110−1120−2. (4.4)
Through simple computation, we get that the left eigenvector of 𝐺 corresponding to eigenvalue 0 is (0.5494,0.8242,0.1374)^𝑇 and ‖Ξ𝑈‖=1.3901.
Let ℎ_𝑖(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)) = 0.2diag(𝑥_𝑖,1(𝑡)−𝑥_𝑖+1,1(𝑡),𝑥_𝑖,2(𝑡)−𝑥_𝑖+1,2(𝑡)), 𝑖=1,2,3, where 𝑥₄(𝑡)=𝑥₁(𝑡). Obviously, trace((ℎ_𝑖(𝑒(𝑡)))^𝑇ℎ_𝑖(𝑒(𝑡)))=0.04[(𝑥_𝑖1(𝑡)−𝑥_𝑖+1,1(𝑡))²+(𝑥_𝑖2−𝑥_𝑖+1,2(𝑡))²]=0.04[(𝑒_𝑖1(𝑡)−𝑒_𝑖+1,1(𝑡))²+(𝑒_𝑖2(𝑡)−𝑒_𝑖+1,2(𝑡))²]≤0.08(‖𝑒_𝑖(𝑡)‖²+‖𝑒_𝑖+1(𝑡)‖²),𝑖=1,2,3, (4.5) where 𝑒₄(𝑡)=𝑒₁(𝑡). Hence, the assumption conditions (H₁) and (H₂) are satisfied. Selecting the first two nodes to be controlled, we have 2𝑐𝑝₁𝛾₁(Ξ𝐺)^𝑠₃₋₂+(𝑊₁)₃₋₂<2𝑐𝑝₂𝛾₂(Ξ𝐺)^𝑠₃₋₂+(𝑊₂)₃₋₂=−1.2986<0. According to Theorem 3.2, the complex networks (4.3) can be controlled to the state of (4.1) under the adaptive pinning controllers (3.12) with the first two nodes to be controlled.
The initial conditions of the numerical simulations are as follows: 𝑥_𝑖(𝑠)=(−4+12𝑖),−10+12𝑖)^𝑇, 1≤𝑖≤3, for all 𝑠∈[−1,0], 𝜀₁(0)=𝜀₂(0)=1, 𝑣₁=𝑣₂=0.04. Figure 2 shows the time evolutions of synchronization errors 𝑒_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)−𝑠_𝑗(𝑡), 𝑖=1,2,3, 𝑗=1,2, with the first two nodes being controlled, which verify Theorem 3.2 perfectly. The trajectories of the gains are shown in Figure 3.

Figure 1 

Chaotic trajectory of system (4.1).

(a)

(b)

Figure 2 

Time evolutions of synchronization errors with the first two nodes being controlled.

Figure 3 

Trajectories of 𝜀1(𝑡) and 𝜀2(𝑡) for (4.3) with adaptive pinning control.

Example 4.2. Consider the following coupled neural networks: d𝑥_𝑖(𝑡)=[−𝐶𝑥_𝑖(𝑡)+𝐴𝑓(𝑥_𝑖(𝑡))+𝐵𝑓(𝑥_𝑖(𝑡−𝜏))+𝑐¹⁰_𝑗=1𝑔_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑑¹⁰_𝑗=1𝑢_𝑖𝑗𝐷𝑥_𝑗(𝑡−𝜏)]d𝑡+ℎ_𝑖(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥₁₀(𝑡))d𝜔_𝑖(𝑡),𝑖=1,2,3,…,10, (4.6) where 𝑐=1.7, 𝑑=0.3, ℎ_𝑖(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥₁₀(𝑡)) = 0.1diag(𝑥_𝑖,1(𝑡)−𝑥_𝑖+1,1(𝑡),𝑥_𝑖,2(𝑡)−𝑥_𝑖+1,2(𝑡)), 𝑖=1,2,…,10, 𝑥₁₁(𝑡)=𝑥₁(𝑡), and 𝐺=𝑈=−11000000001−10000000010−10000000010−10000001100−20000011000−20000110000−20001100000−20011000000−20110000000−2,Γ=𝐷=1000. (4.7) Take the first two nodes to be pinned. Through simple computation, we obtain that all the conditions of Theorem 3.1 are satisfied. Hence, system (4.6) is synchronized with adaptive pinning controllers (3.1).
The initial conditions of the numerical simulations are as follows: 𝑥_𝑖(𝑠)=(−4+12𝑖),−10+12𝑖)^𝑇, 1≤𝑖≤10, for all 𝑠∈[−1,0], 𝜀₁(0)=𝜀₂(0)=1, 𝑣₁=𝑣₂=1. Figure 4 shows the time evolutions of synchronization errors with adaptive pinning control. The trajectories of the adaptive pinning control gains are shown in Figure 5.

(a)

(b)

Figure 4 

Trajectories of 𝜀1(𝑡) and 𝜀2(𝑡) for (4.6) with adaptive pinning control.

Figure 5 

Trajectories of 𝜀1(𝑡) and 𝜀2(𝑡) for (4.6) with adaptive pinning control.

Example 4.3. Consider the delayed Chua's circuit: d𝑠(𝑡)=[𝐶𝑠(𝑡)+𝑔₁(𝑠(𝑡))+𝑔₂(𝑠(𝑡−𝜏))]d𝑡, (4.8) where 𝑠(𝑡)=(𝑠₁(𝑡),𝑠₂(𝑡),𝑠₃(𝑡))^𝑇, 𝑔₁(𝑠(𝑡))=(−(1/2)𝛿(𝑎−𝑏)(|𝑠₁(𝑡)+1|−|𝑠₁(𝑡)−1|),0,0)^𝑇, 𝑔₂(𝑠(𝑡−𝜏))=(0,0,−𝜉𝜖sin(𝜈𝑠₁(𝑡−𝜏)))^𝑇, 𝐶=−𝛿(1+𝑏)𝛿01−110−𝜉−𝜔, (4.9) and 𝛿=10,𝜉=19.53,𝜔=0.1636,𝑎=−1.4325,𝑏=−0.8731,𝜈=0.5,𝜖=0.2,𝜏=0.02.
In the case that the initial condition is chosen as 𝑠_𝑡(𝑡)=2; 𝑠₂(𝑡)=0.2; 𝑠₃(𝑡)=0.3, for all 𝑡∈[−0.02,0], the chaotic attractor can be seen in Figure 6.
Take 𝑘₁₁=(1/2)𝛿(𝑏−1)+𝛿(1+𝑏)=5.416, 𝑘₁₂=𝛿=10, 𝑘₁₃=0; 𝑘₂₁=𝑘₂₂=𝑘₂₃=1, 𝑘₃₁=𝜉𝜖𝜈=1.953, 𝑘₃₂=𝜉=19.53; 𝑘₃₃=𝜔=0.1636, 𝑃 = diag(1,1,1), Δ = diag(19,18,23), we have 𝜂=0.035, 𝜃=15.265. Hence the condition (2.7) is satisfied.
Now we construct a complex network, which obeys the scale-free distribution of the Barabási-Albert model [17]. The parameters in the process of constructing are the following: initial graph is complete with 𝑚₀=5 nodes, 𝑚=5 edges are added to the network when a new node is introduced, and the final number of nodes is 𝑁=100. See Figure 7 for the BA scare-free network.
Consider the following complex networks: d𝑥_𝑖(𝑡)=[𝐶𝑥_𝑖(𝑡)+𝑔₁(𝑥_𝑖(𝑡))+𝑔₂(𝑥_𝑖(𝑡−𝜏))+6¹⁰⁰_𝑗=1𝑔_𝑖𝑗𝑥_𝑗(𝑡)+0.35¹⁰⁰_𝑗=1𝑔_𝑖𝑗𝑥_𝑗(𝑡−𝜏)]d𝑡+0.4ℎ_𝑖(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥₁₀₀(𝑡))d𝜔_𝑖(𝑡),𝑖=1,2,…,100, (4.10) where 𝐺=𝑈=(𝑔_𝑖𝑗)_100×100 is the graph laplacian of the BA scale-free network, ℎ_𝑖(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥₁₀₀(𝑡))=diag(𝑥_𝑖,1(𝑡)−𝑥_𝑖+1,1(𝑡),𝑥_𝑖,2(𝑡)−𝑥_𝑖+1,2(𝑡),𝑥_𝑖,3(𝑡)−𝑥_𝑖+1,3(𝑡)) with 𝑥₁₀₁(𝑡)=𝑥₁(𝑡), 𝑖=1,2,…,100.
Without loss of generality, we take the first 10 nodes to be controlled. The initial conditions of the numerical simulations are as follows: 𝑥_𝑖(𝑠)=(−8+𝑖,−5+𝑖,−10+𝑖)^𝑇, 𝑠∈[−0.02,0], 1≤𝑖≤100, 𝜀_𝑗(0)=1, 𝑣_𝑗=0.04. For the synchronization errors 𝑒_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)−𝑠_𝑗(𝑡), 1≤𝑖≤100, 𝑗=1,2,3 and trajectories of the control gains; see Figures 8 and 9.

Figure 6 

Chaotic trajectory of system (4.8).

Figure 7 

BA scale-free network initial graph is complete with 𝑚0=5 nodes; 𝑚=5 edges are added in the network when a new node is introduced and the final number of nodes is 𝑁=100.

(a)

(b)

(c)

Figure 8 

Synchronization errors 𝑒𝑖1(𝑡) (a), 𝑒𝑖2(𝑡) (b), and 𝑒𝑖3(𝑡) (c) (1≤𝑖≤100) of (4.10) with the first 10 nodes being controlled.

Figure 9 

Trajectories of control gains 𝜀𝑖(𝑡), 1≤𝑖≤10 for (4.10).

Remark 4.4. In Example 4.2, only the first state of all nodes is coupled; however, the complex network (4.6) is synchronized by adaptive pinning of the first two nodes, which verifies the effectiveness of our theoretical results. In [7, 12, 13, 19], however, all states of the nodes should be coupled. Therefore, our results extend some of existing results. On the other hand, Examples 4.1 and 4.2 are directed networks, while Example 4.3 is undirected network with 100 nodes. Examples with numerical simulations show that our theoretical results are applicable to directed and undirected networks, even networks of large size.

5. Conclusions

In this paper, we have studied the adaptive pinning synchronization of the complex networks with delays and vector-form stochastic perturbations, in which dynamical behaviors are more realistic and complicated. Since the new condition QUAD1, our proof is very simple. Via two simple adaptive pinning feedback control schemes, several sufficient conditions guaranteeing the synchronization of the proposed complex networks are obtained. Our results are valid even if only partial states of nodes are coupled. Numerical simulations verified the effectiveness of our results. Our results improve and extend some existing results. Models and results in this paper provide possible new applications for network designers.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant no. 60874088, and the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2009271, the Scientific Research Fund of Yunnan Provincial Education Department under Grant no. 07Y10085, and the Scientific Research Fund of Yunnan Province under Grant no. 2008CD186.