Pinning Synchronization of Delayed Neural Networks with Nonlinear Inner-Coupling

Abstract

Without assuming the symmetry and irreducibility of the outer-coupling weight configuration matrices, we investigate the pinning synchronization of delayed neural networks with nonlinear inner-coupling. Some delay-dependent controlled stability criteria in terms of linear matrix inequality (LMI) are obtained. An example is presented to show the application of the criteria obtained in this paper.

1. Introduction and Model Description

In the past few decades, the problem of control and synchronization in complex networks has attracted increasing attention. There are attempts to control the dynamics of a complex network and guide it to a desired state, such as an equilibrium point or a periodic orbit of the network. Since a complex network has a large number of nodes, it is difficult to control it by adding controllers to all nodes. To reduce the number of the controllers, Wang and Chen investigated pinning control for complex networks [1]. Pinning control applies local feedback injections to a small fraction of nodes on a large-size network, thereby achieving some intended global performances over the entire network.

In [1], Wang and Chen showed that, due to the extremely inhomogeneous connectivity distribution of scale-free networks, it is much effective to pin some most-highly connected nodes than to pin randomly selected nodes. In [2], Li et al. further investigated the control of complete random networks and scale-free networks via virtual control and showed that the control actions applied to the pinned nodes can be propagated to the rest of network nodes through the couplings in the network and eventually result in the synchronization of the whole network. In [3], Chen et al. proved that, if the coupling strength is large enough, then even one single pinning controller is able to control network. In the sequel, [4–9] also studied the global pinning controllability of complex networks and some sufficient pinning conditions were established. The common feature of the work in [1–9] is that there are no coupling delays in the network. However, due to the limited speeds of transmission and spreading as well as traffic congestion, signals traveling through a network are often associated with time delays, which are very common in biological and physical networks. Therefore, time delays should be modeled in order to simulate more realistic networks. In [10–13], the pinning synchronization of complex networks with homogeneous time delay is studied. In [14], Xiang et al. considered the pinning control of complex networks with heterogeneous delays via linearized method.

The previous researches on pinning control of complex dynamical networks have mainly focused on such networks with some specific coupling schemes. That is, there is a common outer-coupling strength for all connections and the inner-coupling is linear. Moreover, most of the existed studies assume the coupling configuration matrices are symmetric. In a real neural network, however, this is not always the case. Many real neural networks are direct graphs, such as the WWW, whose coupling configuration matrix is not symmetric. Additionally, as pointed in [15–17], synchronization is influenced not only by the topology, but also by the strength of the connections. So, we should consider nonuniform coupling strength while studying complex dynamical networks. Xiang et al. [14] considered the pinning control of complex networks with nonuniform coupling strength. In [18], Zhou et al. investigated the pinning adaptive synchronization of complex network with nonuniform coupling strength as well as time delay under the assumption of the symmetry of the nondelayed and delayed outer-coupling weight configuration matrices, they introduced some specific pinning control technique.

Motivated by the above discussion, we consider a general complex dynamical network described by [18] 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑁_𝑗=1𝑎_𝑖𝑗𝑔𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑥_𝑗(𝑡−𝜏(𝑡)),𝑖=1,2,…,𝑁, (1.1) where 𝑥_𝑖(𝑡) denotes the state vector of the 𝑖th node, 𝑓∶ℝ^𝑛×ℝ⁺→ℝ^𝑛 represents the activity of an individual subsystem, 𝑔∶ℝ^𝑛→ℝ^𝑛 and ℎ∶ℝ^𝑛→ℝ^𝑛 are the inner-coupling and delayed inner-coupling vector functions, and 𝜏(𝑡) is the coupling delay function. 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 are the nondelayed and delayed outer-coupling strength, respectively. If there is a link from node 𝑖 to node 𝑗(𝑗≠𝑖), then 𝑎_𝑖𝑗>0 and 𝑏_𝑖𝑗>0; otherwise, 𝑎_𝑖𝑗=𝑏_𝑖𝑗=0, and 𝑎_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗, 𝑏_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑏_𝑖𝑗, 𝑖=1,2,…,𝑁. We will give some synchronization criteria by adding nonlinear and adaptive feedback controllers to a small fraction of nodes of network (1.1).

Let 𝐶([−𝜏,0],ℝ^𝑛) be the Banach space of continuous functions mapping [−𝜏,0] into ℝ^𝑛 with the norm ‖𝜙‖=sup_{−𝜏≤𝜃≤0}‖𝜙(𝜃)‖, where 𝜏=sup_𝑡∈ℝ+{𝜏(𝑡)}. For the complex network (1.1), its initial conditions are given by 𝑥_𝑖(𝑡)=𝜙_𝑖(𝑡)∈𝐶([−𝜏,0],ℝ^𝑛). We always assume that (1.1) has a unique solution with respect to initial conditions.

2. Preliminaries

In this section, we present some lemmas and assumptions required throughout this paper.

Lemma 2.1 (see [19] (Schur Complement)). The following linear matrix inequality (LMI) 𝑆₁₁𝑆₁₂𝑆^𝑇₁₂𝑆₂₂<0, (2.1) where 𝑆₁₁=𝑆^𝑇₁₁,𝑆₂₂=𝑆^𝑇₂₂, is equivalent to the following condition: 𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0. (2.2)

Lemma 2.2 (see [18]). Assume that 𝑀 is a diagonal matrix whose 𝑖_𝑘th (1≤𝑖_𝑘≤𝑁,1≤𝑘≤𝑙,1≤𝑙≤𝑁) diagonal elements are 𝑚 and the others are 0, where 𝑚>0 is a constant. Then, for a symmetric matrix 𝐺 which has the same dimension with 𝑀, 𝐺−𝑀<0 is equivalent to 𝐺_𝑙<0 when 𝑚 is large enough, where 𝐺_𝑙 denotes the minor matrix of the matrix 𝐺 by removing all the 𝑖_𝑘th row-column pairs of 𝐺.

Assumption 1. Suppose that the delay function 𝜏(𝑡) is differentiable and satisfies 𝜏̇(𝑡)≤𝛼, where 0≤𝛼<1 is a constant.

Assumption 2 (see [3]). Assume that there is a positive definite diagonal matrix 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛) and a diagonal matrix Δ=diag(𝛿₁,𝛿₂,…,𝛿_𝑛), such that 𝑓 satisfies the following inequality: 𝑥−𝑦^𝑇𝑃𝑓(𝑥)−𝑓𝑦−Δ𝑥−𝑦≤−𝜂𝑥−𝑦^𝑇𝑥−𝑦 (2.3) for some 𝜂>0, all 𝑥, 𝑦∈ℝ^𝑛, and 𝑡>0.

Assumption 3. Assume that there exist positive constants 𝛽₁ and 𝛽₂ such that 𝛽₁𝑥−𝑦²≤𝑥−𝑦^𝑇𝑔(𝑥)−𝑔𝑦≤𝑥−𝑦𝑔(𝑥)−𝑔𝑦≤𝛽₂𝑥−𝑦² (2.4) for all 𝑥,𝑦∈ℝ^𝑛.

Assumption 4. Assume that there exists a positive constant 𝜇>0 satisfying ℎ(𝑥)−ℎ𝑦≤𝜇𝑥−𝑦 (2.5) for all 𝑥,𝑦∈ℝ^𝑛.

Remark 2.3. In [18], the delayed inner-coupling vector function ℎ was required to satisfy 𝑒₁𝑥−𝑦²≤𝑥−𝑦^𝑇ℎ(𝑥)−ℎ𝑦≤𝑥−𝑦ℎ(𝑥)−ℎ𝑦≤𝑒₂𝑥−𝑦² (2.6) for all 𝑥, 𝑦∈ℝ^𝑛, where 𝑒₁ and 𝑒₂ are positive constants. Obviously, our assumption is weaker than that.

3. Synchronization Criteria of Directed Networks

The objective in this section is to stabilize network (1.1) onto a homogeneous trajectory satisfying 𝑠(̇𝑡)=𝑓(𝑠(𝑡),𝑡), where 𝑠(𝑡)∈ℝ^𝑛 is a solution of an isolate node. To achieve this goal, we first add feedback pinning controllers to a small fraction of nodes in the network. The pinning controlled network is described as follows: 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑁_𝑗=1𝑎_𝑖𝑗𝑔𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑥_𝑗(𝑡−𝜏(𝑡))+𝑢_𝑖(𝑡), (3.1) with nonlinear feedback controllers given by𝑢_𝑖(𝑡)=−𝑑_𝑖𝑔𝑥_𝑖(𝑡)−𝑔(𝑠(𝑡)),𝑖=1,2,…,𝑁, (3.2) where the feedback gain satisfies 𝑑_𝑖>0 for 𝑖=1,2,…,𝑙 and 𝑑_𝑖=0 for 𝑖=𝑙+1,…,𝑁.

For the convenience of later use, we introduce some notations employed through this section. We let 𝑝=min_{1≤𝑗≤𝑛}{𝑝_𝑗};|𝑥|=(|𝑥₁|,|𝑥₂|,…,|𝑥_𝑛|)^𝑇 for all 𝑥∈ℝ^𝑛; 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑙,0,…,0); 𝐴=diag(𝑎₁₁,𝑎₂₂,…,𝑎_𝑁𝑁), 𝐴=𝐴−𝐴;𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡)∈ℝ^𝑛, 𝑔(𝑒_𝑖(𝑡))=𝑔(𝑒_𝑖(𝑡)+𝑠(𝑡))−𝑔(𝑠(𝑡)), ℎ(𝑒_𝑖(𝑡−𝜏(𝑡)))=ℎ(𝑒_𝑖(𝑡−𝜏(𝑡))+𝑠(𝑡−𝜏(𝑡)))−ℎ(𝑠(𝑡−𝜏(𝑡))), 𝑖=1,2,…,𝑁; 𝑒_𝑗̃(𝑡)=(𝑒_1𝑗(𝑡), 𝑒_2𝑗(𝑡),…,𝑒_𝑁𝑗(𝑡))^𝑇∈ℝ^𝑁, 𝑔_𝑗(𝑒(𝑡))=(𝑔_𝑗(𝑒₁(𝑡)), 𝑔_𝑗(𝑒₂(𝑡)),…,𝑔_𝑗(𝑒_𝑁(𝑡)))^𝑇∈ℝ^𝑁, ℎ_𝑗(𝑒(𝑡−𝜏(𝑡)))=(ℎ_𝑗(𝑒₁(𝑡−𝜏(𝑡))), ℎ_𝑗(𝑒₂(𝑡−𝜏(𝑡))),…,ℎ_𝑗(𝑒_𝑁(𝑡−𝜏(𝑡))))^𝑇∈ℝ^𝑁, 𝑗=1,2,…,𝑛;‖𝐴‖₂=𝜆_max(𝐴^𝑇𝐴) for all 𝐴∈ℝ^𝑁×𝑁.

Theorem 3.1. Under the Assumptions 1–4, the pinning controlled directed network (3.1) is globally asymptotically stable at the homogenous trajectory 𝑠(𝑡) if there exist positive diagonal matrices 𝑅 and 𝑄, such that 𝑝_𝑗𝛿_𝑗𝐼_𝑁+𝛽₁𝑝𝐴−𝛽₁𝑝𝑝𝐷+^2𝑗4𝐴𝑅⁻¹𝐴^𝑇<0,−𝜂𝐼_𝑁+𝜇²𝑄+𝛽²₂()𝑝𝑅+1−𝛼²_𝑗4𝐵𝑄⁻¹𝐵^𝑇<0 (3.3) for 𝑗=1,2,…,𝑛.

Proof. From Lemma 2.1, conditions (3.3) are equivalent to Ω₁=𝑝_𝑗𝛿_𝑗𝐼_𝑁+𝛽₁𝑝𝐴−𝛽₁𝑝𝐷𝑝_𝑗2𝐴𝑝_𝑗2𝐴^𝑇Ω−𝑅<0,₂=−𝜂𝐼_𝑁+𝜇²𝑄+𝛽²₂𝑅𝑝_𝑗2𝐵𝑝_𝑗2𝐵^𝑇−(1−𝛼)𝑄<0. (3.4) We consider a Lyapunov function as 𝑉(𝑡)=12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡)+𝜇^2𝑛_𝑗=1^𝑡_{𝑡−𝜏(𝑡)}𝑒^𝑇_𝑗̃(𝑠)𝑄𝑒_𝑗̃(𝑠)𝑑𝑠. (3.5) Differentiating the function 𝑉(𝑡) along the trajectories of (3.1), one obtains 𝑑𝑉(𝑡)𝑑𝑡=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))−Δ𝑒_𝑖(𝑡)+Δ𝑒_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗𝑔𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑒_𝑗(𝑡−𝜏(𝑡))−𝑑_𝑖𝑔𝑒_𝑖(𝑡)+𝜇^2𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝑄𝑒_𝑗̃(𝑡)−𝜇^2𝑛_𝑗=1(1−𝜏̇(𝑡))𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄𝑒_𝑗̃(𝑡−𝜏(𝑡)). (3.6) From Assumption 3, one can obtain ^𝑁_𝑖=1𝑒^𝑇_𝑖𝑃𝑎_𝑖𝑖𝑔𝑒_𝑖(𝑡)=^𝑁_𝑖=1𝑎_𝑖𝑖^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)𝑝_𝑗𝑔_𝑗𝑒_𝑖(𝑡)≤𝑝^𝑁_𝑖=1𝑎_𝑖𝑖^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)𝑔_𝑗𝑒_𝑖(𝑡)=𝑝^𝑁_𝑖=1𝑎_𝑖𝑖𝑒^𝑇_𝑖(𝑡)𝑔𝑒_𝑖(𝑡)≤𝛽₁𝑝^𝑁_𝑖=1𝑎_𝑖𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡). (3.7) Similar to (3.7), we can get ^𝑁_𝑖=1𝑒^𝑇_𝑖𝑃𝑑_𝑖𝑔𝑒_𝑖(𝑡)≥𝛽₁𝑝^𝑁_𝑖=1𝑑_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡). (3.8) It follows from (3.7) and (3.8), and combining with Assumptions 1-2, we can derive 𝑑𝑉(𝑡)𝑑𝑡≤−𝜂^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃Δ𝑒_𝑖(𝑡)+𝛽₁𝑝^𝑁_𝑖=1𝑎_𝑖𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+^𝑁_𝑖=1^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑃𝑔𝑒_𝑗(𝑡)+^𝑁_𝑖=1^𝑁_𝑗=1𝑏_𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑃ℎ𝑒_𝑗(𝑡−𝜏(𝑡))−𝛽₁𝑝^𝑁_𝑖=1𝑑_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+𝜇^2𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝑄𝑒_𝑗̃(𝑡)−𝜇^2𝑛_𝑗=1(1−𝛼)𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄𝑒_𝑗̃(𝑡−𝜏(𝑡))=−𝜂^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝑒_𝑗̃(𝑡)+^𝑛_𝑗=1𝑝_𝑗𝛿_𝑗𝑒^𝑇_𝑗̃(𝑡)𝑒_𝑗̃(𝑡)+𝛽₁𝑝^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝐴𝑒_𝑗̃(𝑡)+^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝐴𝑔_𝑗(𝑒(𝑡))−^𝑛_𝑗=1𝑔^𝑇_𝑗(𝑒(𝑡))𝑅𝑔_𝑗(𝑒(𝑡))+^𝑛_𝑗=1𝑔^𝑇_𝑗(𝑒(𝑡))𝑅𝑔_𝑗(𝑒(𝑡))+^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝐵ℎ_𝑗(𝑒(𝑡−𝜏(𝑡)))+𝜇^2𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝑄𝑒_𝑗̃(𝑡)−𝛽₁𝑝^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝐷𝑒_𝑗̃(𝑡)−𝜇^2𝑛_𝑗=1(1−𝛼)𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄𝑒_𝑗̃(𝑡−𝜏(𝑡)). (3.9) From Assumption 4, one can obtain ^𝑛_𝑗=1ℎ^𝑇_𝑗(𝑒(𝑡−𝜏(𝑡)))𝑄ℎ_𝑗(𝑒(𝑡−𝜏(𝑡)))=^𝑛_𝑗=1^𝑁_𝑖=1𝑞_𝑖ℎ²_𝑗𝑒_𝑖(𝑡−𝜏(𝑡))=^𝑁_𝑖=1𝑞_𝑖ℎ𝑒_𝑖(𝑡−𝜏(𝑡))²≤𝜇^2𝑁_𝑖=1𝑞_𝑖𝑒^𝑇_𝑖(𝑡−𝜏(𝑡))𝑒_𝑖(𝑡−𝜏(𝑡))=𝜇^2𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄𝑒_𝑗̃(𝑡−𝜏(𝑡)). (3.10) Similar to (3.10), from Assumption 3, we can derive ^𝑛_𝑗=1𝑔^𝑇_𝑗(𝑒(𝑡))𝑅𝑔_𝑗(𝑒(𝑡))=^𝑛_𝑗=1^𝑁_𝑖=1𝑟_𝑖𝑔²_𝑗𝑒_𝑖(𝑡)=^𝑁_𝑖=1𝑟_𝑖𝑔𝑒_𝑖(𝑡)²≤𝛽²₂^𝑁_𝑖=1𝑟_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)=𝛽²₂^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝑅𝑒_𝑗̃(𝑡). (3.11) From (3.7)–(3.11), we can get 𝑑𝑉(𝑡)=𝑑𝑡^𝑛𝑗=1𝑒^𝑇_𝑗̃(𝑡),𝑔^𝑇_𝑗(𝑒(𝑡))Ω₁𝑒_𝑗̃(𝑡)𝑔_𝑗(𝑒(𝑡))+^𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡),ℎ^𝑇_𝑗(𝑒(𝑡−𝜏(𝑡)))Ω₂𝑒_𝑗̃(𝑡)ℎ_𝑗(𝑒(𝑡−𝜏(𝑡)))<0. (3.12) From Lyapunov stability theory, the controlled system (3.1) is globally asymptotically stable at 𝑠(𝑡). This completes the proof.

Remark 3.2. It is hard to compare our results with existing ones, because the issues are different. However, from the aspect of the network model, network (1.1) contains the models studied in [1–13], moreover, because the coupling configuration matrices 𝐴 and 𝐵 are assumed to be asymmetric, which are more consistent with the real-world network.
In the coupled system (3.1), if 𝑑_𝑖, 𝑖=1,2,…,𝑙 are defined as 𝑑̇_𝑖(𝑡)=𝜀_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡),𝑖=1,2,…,𝑙, (3.13) where 𝜀_𝑖 are positive constants, then the controller 𝑢_𝑖(𝑡)=−𝑑_𝑖(𝑡)(𝑔(𝑥_𝑖(𝑡))−𝑔(𝑠(𝑡))) is said to be adaptive pinning controller. In the following, we will prove that under some conditions the system (3.1) would get global synchronization with nonlinear adaptive pinning controllers.

Theorem 3.3. Under the Assumptions 1–4, the controlled directed network (3.1) is globally asymptotically stable with adaptive pinning controllers (3.13) if there exist positive diagonal matrices 𝑅 and 𝑄, such that 𝑝_𝑗𝛿_𝑗𝐼_𝑁+𝛽₁𝑝𝑝𝐴+^2𝑗4𝐴𝑅⁻¹𝐴^𝑇_𝑙<0,−𝜂𝐼_𝑁+𝜇²𝑄+𝛽²₂()𝑝𝑅+1−𝛼²_𝑗4𝐵𝑄⁻¹𝐵^𝑇<0, (3.14) for 𝑗=1,2,…,𝑛.

Proof. Construct the Lyapunov function as 𝑉(𝑡)=12^𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡)+𝜇^2𝑛_𝑗=1^𝑡_{(𝑡)𝑡−𝜏}𝑒^𝑇_𝑗̃(𝑠)𝑄𝑒_𝑗̃(𝑠)𝑑𝑠+𝛽₁𝑝^𝑙_𝑖=1𝑑_𝑖(𝑡)−𝑑²2𝜀_𝑖, (3.15) where 𝑑 is a sufficiently large positive constant to be determined. From Lemma 2.2, when 𝑑 is large enough, 𝑝_𝑗𝛿_𝑗𝐼_𝑁+𝛽₁𝑝𝑝𝐴+^2𝑗4𝐴𝑅⁻¹𝐴^𝑇_𝑙<0 (3.16) is equivalent to 𝑝_𝑗𝛿_𝑗𝐼_𝑁+𝛽₁𝑝𝐴−𝛽₁𝑝𝑝𝐷+^2𝑗4𝐴𝑅⁻¹𝐴^𝑇<0, (3.17) where 𝐷=diag(𝑑,…,_𝑙𝑑,0…,0_𝑁−𝑙).
Differentiating the function 𝑉(𝑡) along the trajectories of (3.1), and combining with (3.8), one can obtain 𝑑𝑉(𝑡)𝑑𝑡≤^𝑁_𝑖=1𝑒^𝑇_𝑖𝑃𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))+^𝑁_𝑗=1𝑎_𝑖𝑗𝑔𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑒_𝑗(𝑡−𝜏(𝑡))−𝛽₁𝑝^𝑁_𝑖=1𝑑_𝑖(𝑡)𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+𝜇^2𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝑄𝑒_𝑗̃(𝑡)−𝜇^2𝑛_𝑗=1(1−𝜏̇(𝑡))𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄𝑒_𝑗̃(𝑡−𝜏(𝑡))+𝛽₁𝑝^𝑙_𝑖=1𝑑_𝑖(𝑡)−𝑑𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖𝑃𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))+^𝑁_𝑗=1𝑎_𝑖𝑗𝑔𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑒_𝑗(𝑡−𝜏(𝑡))−𝛽₁𝑝^𝑙_𝑖=1𝑑𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+𝜇^2𝑛_𝑗=1𝑒^𝑇_𝑗̃(𝑡)𝑄𝑒_𝑗̃(𝑡)−𝜇^2𝑛_𝑗=1(1−𝜏̇(𝑡))𝑒^𝑇_𝑗̃(𝑡−𝜏(𝑡))𝑄𝑒_𝑗̃(𝑡−𝜏(𝑡)). (3.18) The remaining part of the proof is similar to that of Theorem 3.1, hence we omit it.

To make Theorem 3.3 more applicable, we let 𝑃=𝐼_𝑁,𝑅=‖𝐴‖₂𝐼_𝑁,𝑄=‖𝐵‖₂𝐼_𝑁, then we can easily obtain the following corollary.

Corollary 3.4. Under the Assumptions 1–4, the pinning controlled directed network (3.1) is globally asymptotically stable at the homogenous trajectory 𝑠(𝑡) if the following conditions are satisfied: max_{𝑙+1≤𝑖≤𝑁}𝑎_𝑖𝑖<−4𝛿^∗+𝐴₂4𝛽₁,𝜇²+1−𝛼4‖𝐵‖₂+𝛽²₂𝐴₂<𝜂, (3.19) where 𝛿^∗=max_{1≤𝑗≤𝑛}{𝛿_𝑗}.

Remark 3.5. According to this corollary, we can rearrange network nodes in ascending order based on their in-degrees −𝑎_𝑖𝑖 [7] and choose the first 𝑙 network nodes as pinned candidates to satisfy the first pinning condition of this corollary.

4. Numerical Simulation

In this section, a simple example is used to explain the effectiveness of the proposed network synchronization criteria.

Example 4.1. We consider a directed network consisting of three identical Hindmarsh-Rose (HR) neuron systems [19], which is described by 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑁_𝑗=1𝑎_𝑖𝑗𝑔𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑥_𝑗(𝑡−𝜏(𝑡)), (4.1) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))^𝑇 is the state variable of the 𝑖th node, 𝑓(𝑥(𝑡))=𝑓,𝑡¹(𝑥(𝑡𝑓))₂(𝑥(𝑡𝑓))₃(𝑥(𝑡=𝑎))¹𝑥²₁(𝑡)−𝑥³₁(𝑡)−𝑥₂(𝑡)−𝑥₃(𝑡)𝑎₁+𝑏₁𝑥²₁(𝑡)−𝑥₂(𝑡)𝑏₂𝑎₂𝑥₁(𝑡)−𝑥₃(𝑡)+𝑎₃=𝑏0−1−10−10₂𝑎₂0−𝑏₂𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡)+𝑎₁𝑥²₁(𝑡)−𝑥³₁(𝑡)𝑎₁+𝑏₁𝑥²₁(𝑡)𝑏₂𝑎₃. (4.2)
According to the discussion in [18], 𝑔(𝑥_𝑗(𝑡)) and ℎ(𝑥_𝑗(𝑡−𝜏(𝑡))) are realistically specified as 𝑔(𝑥_𝑗(𝑡))=(𝑔₁(𝑥_𝑗1(𝑡)),0,0))^𝑇 and ℎ(𝑥_𝑗(𝑡−𝜏(𝑡)))=(ℎ₁(𝑥_𝑗1(𝑡−𝜏(𝑡))),0,0)^𝑇, so we can select 𝑔(𝑥_𝑗(𝑡))=(𝑥_𝑗1(𝑡),0,0)^𝑇 and ℎ(𝑥_𝑗(𝑡−𝜏(𝑡)))=((1/10)sin𝑥_𝑗1(𝑡−𝜏(𝑡)),0,0)^𝑇, then we can easily drive that 𝛽₁=𝛽₂=1, 𝜇=1/10. We let 𝐴=𝑎_𝑖𝑗=𝑏−2111−1022−4,𝐵=_𝑖𝑗=−0.10.100.03−0.0300.10−0.1, (4.3) and 𝜏(𝑡)=1, 𝑎₁=0.1, 𝑎₂=1.5, 𝑎₃=0.1, 𝑏₁=1.1, 𝑏₂=0.1. In this case, 𝛼=0 and the bound 𝑀 of the first variable in the HR equation is 0.5. If we let 𝑃=𝐼₃, Δ=3𝐼₃, then we have 𝑥−𝑦^𝑇𝑃𝑓(𝑥)𝑦=−𝑓−Δ𝑥−𝑦𝑥−𝑦^𝑇+−3−1−10−400.150−3.1𝑥−𝑦𝑥−𝑦^𝑇0.1𝑥²₁−𝑦²₁−𝑥³₁−𝑦³₁1.2𝑥²₁−𝑦²₁0. (4.4) Note that 𝑥−𝑦^𝑇0.1𝑥²₁−𝑦²₁−𝑥³₁−𝑦³₁1.2𝑥²₁−𝑦²₁0=𝑥₁−𝑦₁²0.1𝑥₁+𝑦₁−𝑥²₁+𝑥₁𝑦₁+𝑦²₁+1.2𝑥₁+𝑦₁𝑥₁−𝑦₁𝑥₂−𝑦₂≤0.2𝑀𝑥₁−𝑦₁²+2.4𝑀𝑥₁−𝑦₁𝑥₂−𝑦₂=𝑥−𝑦^𝑇.0.10.600.600000𝑥−𝑦 (4.5) It follows that 𝑥−𝑦^𝑇𝑃𝑓(𝑥)𝑦≤−𝑓−Δ𝑥−𝑦𝑥−𝑦^𝑇+−2.9110−400.150−3.1𝑥−𝑦𝑥−𝑦^𝑇=−0.08750.600.600000𝑥−𝑦𝑥−𝑦^𝑇−2.98751.10.5751.1−400.5750−3.1𝑥−𝑦≤−2.0513𝑥−𝑦^𝑇𝑥−𝑦. (4.6) So, we can select 𝜂=2.0513. Moreover, if we let 𝑅=2.01𝐼₃, 𝑄=𝐼₃,𝑙=2, then, by simple computation, we can get 𝑝_𝑗𝛿_𝑗𝐼_𝑁+𝛽₁𝑝𝑝𝐴+^2𝑗4𝐴𝑅⁻¹𝐴^𝑇_𝑙=1.248800.248802.12440.24880.24880.2488−0.0048₂=−0.0048<0,−𝜂𝐼_𝑁+𝜇²𝑄+𝛽²₂()𝑝𝑅+1−𝛼²_𝑗4𝐵𝑄⁻¹𝐵^𝑇=−0.0263−0.0015−0.0025−0.0015−0.030850.00075−0.00250.00075−0.0263<0 (4.7) for 𝑗=1,2,3. So, according to Theorem 3.3, when the first two neurons are pinned, the directed network (4.1) with adaptive pinning controllers is globally asymptotically stable at the homogenous trajectory 𝑠(𝑡). Figure 1 shows the state variables 𝑥_𝑖𝑟(𝑡)(1≤𝑖≤3,1≤𝑟≤3) of network (4.1) with initial values as 𝑥_𝑖(0)=(0.1+0.1𝑖,0.2+0.1𝑖,0.3+0.1𝑖)^𝑇, 1≤𝑖≤3 without control. Figure 2 shows the synchronization errors 𝑒_𝑖𝑟(𝑡)(1≤𝑖≤3,1≤𝑟≤3) for pinning the first two neurons. From Figure 2, it is easy to see that the errors between the synchronized states converge to zero under the given conditions.

(a)

(b)

(c)

Figure 1 

State trajectories 𝑥𝑖𝑟(𝑡)(1≤𝑖≤3,1≤𝑟≤3) of network (4.1) without control.

(a)

(b)

(c)

Figure 2 

Synchronization errors 𝑥𝑖𝑟−𝑠𝑟(1≤𝑖≤3,1≤𝑟≤3) of network (4.1) for pinning the first two neurons.

Remark 4.2. Since the nondelayed and delayed coupling matrices 𝐴 and 𝐵 are asymmetric, the theorem in [18] fails to conclude whether the dynamical system (4.1) can be synchronized. However, one can obtain the global synchronization using our results.

5. Conclusion

In this paper, on the basis of the previous work, we investigated the stabilization problem of delayed complex dynamical networks with nonlinear inner-coupling by pinning a small fraction of nodes. By using Lyapunov stability theory and linear matrix inequality (LMI) approach, some sufficient conditions ensuring the pinning synchronization are obtained. A comparison between our results and the previous results implies that our results establish a new set of pinning synchronization criteria for delayed complex dynamical networks with nonlinear inner-coupling. As an example, a network consisting of 3 identical Hindmarsh-Rose neuron systems is studied.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant nos. 60874088 and 11072059 and the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2009271.