Synchronization and Pinning Control in Complex Networks with Interval Time-Varying Delay

Abstract

The problems on synchronization and pinning control for complex dynamical networks with interval time-varying delay are investigated and two less conservative criteria are established based on reciprocal convex technique. Pinning control strategies are designed to make the complex networks synchronized. Moreover, the problem of designing controllers can be converted into solving a series of NMIs (nonlinear matrix inequalities) and LMIs (linear matrix inequalities), which reduces the computation complexity when comparing with those present results. Finally, numerical simulations can verify the effectiveness of the derived methods.

1. Introduction

During the past decades, complex dynamical networks have increasingly become a focal research topic and received much attention in various fields such as physics, chemistry, and computer science [1–4]. Such systems in the real world usually consist of a large number of highly interconnected dynamical units. Transportation networks, coupled engineering systems, artificial neural networks in human brains, and the Internet are only the typically realistic examples. In order to capture their evolving properties, several famous models have been proposed, such as scale-free networks, random networks, and small world networks.

Recently, the research on synchronization and dynamical behaviors of complex networks has become an important direction, and it was mainly based on the techniques including analyzing topological structure and studying the origin of the intrinsic self-organized dynamics; see [5–7] and references therein. Authors in [5–7] have introduced one uniform dynamic network model and studied the phenomena of synchrony of all network states, with some derived conditions that can guarantee the asymptotical synchronization or projective cluster synchronization. Meanwhile, since there inevitably exists communication delay which is the main source of poor dynamical behaviors in various networks, some efforts have been applied to the synchronization of complex systems [8–21]. More recently, some researchers have considered the global or robust or cluster synchronization for various coupled delayed neural networks based on some effective techniques including LMI one in [8–12]. As for complex networks with linear or nonlinear couplings, some delay-dependent criteria are presented to guarantee the synchronization in [13–16], and moreover in [17, 18], some synchronization results are achieved for complex networks based on linear feedback control or adaptive feedback control. Additionally, to avoid the conservatism, some authors employed delay-partitioning idea to tackle the constant delay or time-variant one in [19, 20]. Yet, it has come to our attention that though some delay-dependent methods have appeared, the techniques above still require great improvements owing to ignoring some important terms when estimating the derivative of Lyapunov functional, especially for the case of interval time-varying delay [21].

Presently, the control problem on scale-free dynamical networks was investigated by applying local linear feedback injections to a small fraction of network nodes [22, 23]. Meanwhile, some researchers have considered pinning control of delayed complex networks in [24–31]. In [24], cluster synchronization for stochastic delayed neural networks was studied based on pinning control and LMI approach. By adopting pinning periodically intermittent control, the work [25] studied the exponential synchronization and yet gave some uneasy-to-check results. With existence of various couplings, the synchronization has been analyzed for delayed complex networks using pinning adaptive control in [26]. In [27], the authors have presented linear feedback and adaptive feedback pinning control to synchronize the delayed complex networks. Yet, we notice that as for constructions of Lyapunov functional and analyzed techniques described in [24–29], there still exists much room waiting for the further improvement, which can be fully considered and tackled in this presented work.

In this paper, the problems on synchronization and pinning control for delayed dynamical networks are studied based on pinning control strategies. Then the decentralized feedback controllers are designed to make the addressed networks synchronized. Moreover, since the most improved techniques including reciprocal convex technique have been applied, the design of controllers can be converted into solving optimal solution of an array of NMIs and LMIs. These conditions can be extended to delayed complex networks with different topology and different sizes.

Notations 1. 𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space, and 𝐑^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices. 𝐴^𝑇 stands for the transpose of matrix 𝐴; 𝐼 represents the identity matrix of an appropriate dimension; 𝐴⊗𝐵 indicates the Kronecker product of the 𝑚×𝑛 matrix 𝐴 and 𝑝×𝑞 matrix 𝐵; _𝑌^𝑋𝑌_𝑇𝑍=^𝑋𝑌_∗𝑍 with ∗ denoting the symmetric term in a symmetric matrix.

2. Problem Formulations

Suppose that the nodes are coupled with states 𝑥_𝑖(⋅), 𝑖∈{1,…,𝑁}, then the complex dynamical networks can be described by𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝑔𝑥_𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑗=1𝑙¹_𝑖𝑗𝐸𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑙²_𝑖𝑗𝐹𝑥_𝑗(𝑡−𝜏(𝑡)), (2.1) where 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡)]^𝑇 are the state vectors, and 𝑓(⋅),𝑔(⋅)∶𝐑^𝑛→𝐑^𝑛 are the continuously differentiable functions; here, 𝐸=[𝑒_𝑖𝑗]_𝑛×𝑛 and 𝐹=[𝑓_𝑖𝑗]_𝑛×𝑛 are the inner coupling matrices between the connected nodes 𝑖 and 𝑗 at time 𝑡 and 𝑡−𝜏(𝑡), respectively.

For the dynamical system (2.1), the following assumptions are utilized throughout this paper:

(A1) 𝜏(𝑡) denotes the interval time-varying delay satisfying0≤𝜏₀≤𝜏(𝑡)≤𝜏_𝑚,𝜏̇(𝑡)≤𝜇<+∞, (2.2) and we set 𝜏_𝑚=𝜏_𝑚−𝜏₀,

(A2) here, 𝐿^𝑘=[𝑙^𝑘_𝑖𝑗]_𝑁×𝑁(𝑘=1,2) is the configuration matrix that is irreducible and satisfies𝑙^𝑘_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑙^𝑘_𝑖𝑗=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑙^𝑘_𝑗𝑖,𝑘=1,2;𝑖=1,2,…,𝑁, (2.3) with 𝑙^𝑘_𝑖𝑗=𝑙^𝑘_𝑗𝑖>0 if there is a connection between node 𝑖 and the one 𝑗, and otherwise, 𝑙^𝑘_𝑖𝑗=0. Furthermore, suppose that the eigenvalue 𝜆_𝑖(𝜈_𝑖) of the matrix 𝐿¹(𝐿²) can be ordered as follows:0=𝜆₁>𝜆₂≥⋯≥𝜆_𝑁0=𝜈₁>𝜈₂≥⋯≥𝜈_𝑁. (2.4)

Prior to addressing the synchronization results, the following definition and lemmas are introduced.

Definition 2.1 (see [28]). The complex networks described by (2.1) are said to achieve the asymptotical synchronization 𝑥₁(𝑡)=𝑥₂(𝑡)=⋯=𝑥_𝑁(𝑡)=𝑠(𝑡), (2.5) where 𝑠(𝑡) is the solution of the local dynamics of an isolate node satisfying 𝑠̇(𝑡)=𝑓(𝑠(𝑡))+𝑔(𝑠(𝑡−𝜏(𝑡))). (2.6)

Lemma 2.2 (see [32]). For any constant matrix W∈𝐑^𝑛×𝑛, W=W^𝑇>0, scalar functional 0≤𝑟(𝑡)≤𝑟_𝑀, and a vector function 𝑒̇∶[−𝑟_𝑀,0]→𝐑^𝑛 such that the following integration is well defined, then −𝑟_𝑀⁰_{(𝑡)−𝑟}𝑒^𝑇̇(𝑡+𝑠)W𝑒̇(𝑡+𝑠)𝑑𝑠≤[𝑒(𝑡)−𝑒(𝑡−𝑟(𝑡))]^𝑇W[𝑒(𝑡)−𝑒(𝑡−𝑟(𝑡))]. (2.7)

Lemma 2.3 (see [33]). Let the functions 𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑁(𝑡)∶𝐑^m→𝐑 have the positive values in an open subset 𝐃 of 𝐑^m, then for 𝛼_𝑖>0 satisfying ∑_𝑖𝛼_𝑖=1 and 𝑔_𝑖,𝑗(𝑡)∶𝐑^𝑚→𝐑 satisfying ^{𝑓𝑖(𝑡)𝑔𝑖,𝑗}_𝑔^(𝑡)_{𝑗,𝑖(𝑡)𝑓𝑗(𝑡)}≥0, one has that _𝑖1𝛼_𝑖𝑓_𝑖(𝑡)≥_𝑖𝑓_𝑖(𝑡)+_𝑖≠𝑗𝑔_𝑖,𝑗(𝑡). (2.8)

As for the complex networks (2.1) and symmetric matrices 𝐿₁, 𝐿₂, if the following 𝑁−1 systems of 𝑛-dimensional linear delayed differential systems𝜔̇_𝑖(𝑡)=𝐽(𝑡)+𝜆_𝑖𝐸𝜔_𝑖(𝑡)+Γ(𝑡)+𝜈_𝑖𝐹𝜔_𝑖(𝑡−𝜏(𝑡)),𝑖=2,…,𝑁 (2.9) are asymptotically stable with respect to their zero solutions, the complex networks (2.1) can achieve the synchronization at 𝑠(𝑡), where 𝐽(𝑡)Ḃ𝑓^′(𝑠(𝑡)) is the Jacobian matrix of 𝑓(𝑥(𝑡)) at 𝑠(𝑡), and Γ(𝑡)Ḃ𝑔^′(𝑥(𝑡−𝜏(𝑡))) is the Jacobian matrix of 𝑔(𝑥(𝑡−𝜏(𝑡))) at 𝑠(𝑡−𝜏(𝑡)).

3. Synchronization Criterion for Complex Networks

In this section, we will derive one delay-derivative-dependent synchronization criterion for the complex networks in (2.1), which can be much more effective than those present ones based on some most developed techniques in [28–33].

Theorem 3.1. Considering the time delay in (2.2), the complex networks (2.1) can achieve the asymptotical synchronization if there exist the appropriately dimensional matrices 𝑃_𝑖>0, 𝑄_𝑖>0, 𝑅_𝑖>0, 𝑇_𝑖>0, 𝑈_𝑖>0, 𝑉_𝑖, 𝑆_𝑖, 𝐻_𝑖, 𝑀_𝑖, and 𝑁_𝑖  for 𝑖=2,…,𝑁, satisfying the matrix inequalities in (3.1), 𝑉_𝑖𝑆_𝑖𝑆^𝑇_𝑖𝑉_𝑖Ω≥0,𝑖=2,…,𝑁,_𝑖11𝑈_𝑖Ω_𝑖130Ω_𝑖15∗Ω_𝑖22𝑉_𝑖−𝑆_𝑖𝑆_𝑖0∗∗Ω_𝑖33𝑉_𝑖−𝑆_𝑖Ω_𝑖35∗∗∗−𝑅_𝑖−𝑉_𝑖0∗∗∗∗Ω_𝑖55<0, (3.1) with Ω_𝑖11=−𝑈_𝑖+𝑄_𝑖+𝐻^𝑇_𝑖𝐽(𝑡)+𝜆_𝑖𝐸+𝐽(𝑡)+𝜆_𝑖𝐸^𝑇𝐻_𝑖,Ω_𝑖13=𝐻^𝑇_𝑖Γ(𝑡)+𝜈_𝑖𝐹+𝐽(𝑡)+𝜆_𝑖𝐸^𝑇𝑁_𝑖,Ω_𝑖15=𝑃_𝑖−𝐻^𝑇_𝑖+𝐽(𝑡)+𝜆_𝑖𝐸^𝑇𝑀_𝑖,Ω_𝑖22=−𝑄_𝑖+𝑅_𝑖+𝑇_𝑖−𝑉_𝑖−𝑈_𝑖,Ω_𝑖33=−1−𝜇𝑇_𝑖−𝑁^𝑇_𝑖Γ(𝑡)+𝜈_𝑖𝐹+Γ(𝑡)+𝜈_𝑖𝐹^𝑇𝑁_𝑖−2𝑉_𝑖+𝑆_𝑖+𝑆^𝑇_𝑖,Ω_𝑖35=Γ(𝑡)+𝜈_𝑖𝐹^𝑇𝑀_𝑖−𝑁^𝑇_𝑖,Ω_𝑖55=−𝑀^𝑇_𝑖−𝑀_𝑖+𝜏²₀𝑈_𝑖+𝜏²_𝑚𝑉_𝑖. (3.2)

Proof. As for the 𝑖th node, we choose the following Lyapunov-Krasovskii functional: 𝑉_𝑖𝜔_𝑖𝑡=𝜔^𝑇_𝑖(𝑡)𝑃_𝑖𝜔_𝑖(𝑡)+^𝑡_𝑡−𝜏0𝜔^𝑇_𝑖(𝑠)𝑄_𝑖𝜔_𝑖(𝑠)𝑑𝑠+^𝑡−𝜏0_{𝑡−𝜏𝑚}𝜔^𝑇_𝑖(𝑠)𝑅_𝑖𝜔_𝑖(𝑠)𝑑𝑠+^𝑡−𝜏0_{(𝑡)𝑡−𝜏}𝜔^𝑇_𝑖(𝑠)𝑇_𝑖𝜔_𝑖(𝑠)+𝑑𝑠_{0−𝜏0𝑡𝑡+𝜃}𝜏₀𝜔̇^𝑇_𝑖(𝑠)𝑈_𝑖𝜔̇_𝑖(𝑠)𝑑𝑠𝑑𝜃+^−𝜏0_−𝜏𝑚^𝑡_𝑡+𝜃𝜏_𝑚𝜔̇^𝑇_𝑖(𝑠)𝑉_𝑖𝜔̇_i(𝑠)𝑑𝑠𝑑𝜃, (3.3) with symmetric matrices 𝑃_𝑖, 𝑄_𝑖, 𝑅_𝑖, 𝑇_𝑖, 𝑈_𝑖, and 𝑉_𝑖 waiting to be determined. Then for any 𝑛×𝑛 matrices 𝐻_𝑖, 𝑀_𝑖, and 𝑁_𝑖, the time derivative of 𝑉_𝑖(𝜔_𝑖(𝑡)) along the system (2.9) can be derived as 𝑉̇_𝑖𝜔_𝑖𝑡=2𝜔^𝑇_𝑖(𝑡)𝑃_𝑖𝜔̇_𝑖(𝑡)+𝜔^𝑇_𝑖(𝑡)𝑄_𝑖𝜔_𝑖(𝑡)−𝜔^𝑇_𝑖𝑡−𝜏₀𝑄_𝑖−𝑅_𝑖−𝑇_𝑖𝜔_𝑖𝑡−𝜏₀−𝜔^𝑇_𝑖𝑡−𝜏_𝑚𝑅_𝑖𝜔_𝑖𝑡−𝜏_𝑚−(1−𝜏̇(𝑡))𝜔^𝑇_𝑖(𝑡−𝜏(𝑡))𝑇_𝑖𝜔_𝑖(𝑡−𝜏(𝑡))+𝜏²₀𝜔̇^𝑇_𝑖(𝑡)𝑈_𝑖𝜔̇_𝑖(𝑡)−^𝑡_𝑡−𝜏0𝜏₀𝜔̇^𝑇_𝑖(𝑠)𝑈_𝑖𝜔̇_𝑖(𝑠)+𝑑𝑠𝜏²_𝑚𝜔̇^𝑇_𝑖(𝑡)𝑉_𝑖𝜔̇_𝑖(𝑡)−^𝑡−𝜏0_{(𝑡)𝑡−𝜏}𝜏_𝑚𝜔̇^𝑇_𝑖(𝑠)𝑉_𝑖𝜔̇_𝑖(𝑠)𝑑𝑠−^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑚}𝜏_𝑚𝜔̇^𝑇_𝑖(𝑠)𝑉_𝑖𝜔̇_𝑖(𝑠)+𝜔𝑑𝑠_𝑇𝑖(𝑡)𝐻^𝑇_𝑖+𝜔̇^𝑇_𝑖(𝑡)𝑀^𝑇_𝑖+𝜔^𝑇_𝑖(𝑡−𝜏(𝑡))𝑁^𝑇_𝑖×−𝜔̇_𝑖(𝑡)+𝐽(𝑡)+𝜆_𝑖𝐸𝜔_𝑖(𝑡)+Γ(𝑡)+𝜈_𝑖𝐹𝜔_𝑖(𝑡−𝜏(𝑡)). (3.4) Through employing Lemmas 2.2 and 2.3, we can yield that −^𝑡_𝑡−𝜏0𝜏₀𝜔̇^𝑇_𝑖(𝑠)𝑈_𝑖𝜔̇_𝑖(𝑠)𝑑𝑠≤𝜔_𝑖(𝑡)−𝜔_𝑖𝑡−𝜏₀^𝑇𝑈_𝑖𝜔_𝑖(𝑡)−𝜔_𝑖𝑡−𝜏₀,−^𝑡−𝜏0_{(𝑡)𝑡−𝜏}𝜏_𝑚𝜔̇^𝑇_𝑖(𝑠)𝑉_𝑖𝜔̇_𝑖(𝑠)𝑑𝑠−^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑚}𝜏_𝑚𝜔̇^𝑇_𝑖(𝑠)𝑉_𝑖𝜔̇_𝑖(𝑠)𝑑𝑠≤−𝜏_𝑚𝜏(𝑡)−𝜏₀𝜔_𝑖𝑡−𝜏₀−𝜔_𝑖(𝑡−𝜏(𝑡))^𝑇𝑉_𝑖𝜔_𝑖𝑡−𝜏₀−𝜔_𝑖((𝑡−𝑡−𝜏))𝜏_𝑚𝜏_𝑚(𝑡)𝜔−𝜏^𝑖(𝑡−𝜏(𝑡))−𝜔_𝑖𝑡−𝜏_𝑚^𝑇𝑉_𝑖𝜔_𝑖(𝑡−𝜏(𝑡))−𝜔_𝑖𝑡−𝜏_𝑚𝜔≤−^𝑖𝑡−𝜏₀−𝜔_𝑖((𝑡𝜔𝑡−𝜏))_𝑖(𝑡−𝜏(𝑡))−𝜔_𝑖𝑡−𝜏_𝑚^𝑇𝑉_𝑖𝑆_𝑖𝑆^𝑇_𝑖𝑉_𝑖𝜔_𝑖𝑡−𝜏₀−𝜔_𝑖((𝑡𝜔𝑡−𝜏))_𝑖(𝑡−𝜏(𝑡))−𝜔_𝑖𝑡−𝜏_𝑚. (3.5) Then together with the terms (3.4)-(3.5), it can be deduced that 𝑉̇_𝑖𝜔_𝑖𝑡≤𝜔_𝑖(𝑡)𝜔_𝑖𝑡−𝜏₀𝜔_𝑖((𝑡𝜔𝑡−𝜏))_𝑖𝑡−𝜏_𝑚𝜔̇_𝑖(𝑡)^𝑇Ω_𝑖11𝑈_𝑖Ω_𝑖130Ω_𝑖15∗Ω_𝑖22𝑉_𝑖−𝑆_𝑖𝑆_𝑖0∗∗Ω_𝑖33𝑉_𝑖−𝑆_𝑖Ω_𝑖35∗∗∗−𝑅_𝑖−𝑉_𝑖0∗∗∗∗Ω_𝑖55𝜔_𝑖(𝑡)𝜔_𝑖𝑡−𝜏₀𝜔_𝑖((𝑡𝜔𝑡−𝜏))_𝑖𝑡−𝜏_𝑚𝜔̇_𝑖(𝑡). (3.6) Then there must exist one scalar 𝜀>0 such that 𝑉̇_𝑖(𝜔_𝑖𝑡)≤−𝜀‖P𝜔_𝑖(𝑡)P‖<0 for any 𝜔_𝑖(𝑡)≠0. Based on the Lyapunov-Krasovskii stability theory, all nodes in complex networks (2.1) are asymptotically synchronized, and it completes the proof.

Remark 3.2. Presently, the convex combination technique has been widely employed to tackle time-varying delay owing to that it could reduce the conservatism more effectively than the previous methods; see [21, 32]. In [33], the authors put forward the reciprocal convex approach, which can reduce the conservatism more effectively than convex combination ones. Yet, it has come to our attention that no researchers have utilized reciprocal convex idea to tackle the synchronization for delayed complex networks.

4. Pinning Control of Delayed Complex Networks

Suppose that we want to stabilize the system (2.1) onto a homogeneous stationary state. In order to achieve the goal, we will apply the pinning control strategy on a small fraction 𝛿(0<𝛿≪1) of the nodes in network (2.1). Suppose that the nodes 𝑖₁,𝑖₂,…,𝑖_𝑙 are selected to be under pinning control, and 𝑖_𝑙+1,𝑖_𝑙+2,…,𝑖_𝑁 are the unselected ones, where 𝑙=[𝛿𝑁] stands for the smaller but nearest to the real number 𝛿𝑁, then 𝑙 is no less than 0 for pinning control, and the controlled networks can be described by𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝑔𝑥_𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑗=1𝑙¹_𝑖𝑗𝐸𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑙²_𝑖𝑗𝐹𝑥_𝑗(𝑡−𝜏(𝑡))+𝑢_𝑖(𝑡),𝑖=1,2,…,𝑙,𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝑔𝑥_𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑗=1𝑙¹_𝑖𝑗𝐸𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑙²_𝑖𝑗𝐹𝑥_𝑗(𝑡−𝜏(𝑡)),𝑖=𝑙+1,𝑙+2,…,𝑁. (4.1) For a special case that 𝜏(𝑡) is available, we adopt the following local linear feedback control law:𝑢_𝑖(𝑡)=−𝑘_𝑖𝐸𝑥_𝑖(𝑡)−𝑠(𝑡)−𝑙_𝑖𝐹𝑥_𝑖(𝑡−𝜏(𝑡))−𝑠(𝑡−𝜏(𝑡)), (4.2) with the feedback gains 𝑘_𝑖>0,𝑙_𝑖>0.

The aim of this section is to stabilize the network (4.1) to be of homogenous stationary state defined by𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑁(𝑡)⟶𝑠(𝑡),𝑠̇(𝑡)=𝑓(𝑠(𝑡))+𝑔(𝑠(𝑡−𝜏(𝑡))),as𝑡⟶+∞. (4.3) Denoting 𝑘_𝑖=𝑙_𝑖=0(𝑖=𝑙+1,𝑙+2,…,𝑁), then we can derive that𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝑔𝑥_𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑗=1𝑙¹_𝑖𝑗𝐸𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑙²_𝑖𝑗𝐹𝑥_𝑗(𝑡−𝜏(𝑡))−𝑘_𝑖𝐸𝑥_𝑖(𝑡)−𝑠(𝑡)−𝑙_𝑖𝐹𝑥_𝑖(𝑡−𝜏(𝑡))−𝑠(𝑡−𝜏(𝑡)),𝑖=1,2,…,𝑁. (4.4) If we let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡) and 𝑒(𝑡)=[𝑒₁(𝑡),𝑒₂(𝑡),…,𝑒_𝑁(𝑡)]^𝑇, then we can linearize the network (4.4) as𝑒_𝑖̇(𝑡)=𝐽(𝑡)𝑒_𝑖(𝑡)+Γ(𝑡)𝑒_𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑗=1𝑙¹_𝑖𝑗𝐸𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑙²_𝑖𝑗𝐹𝑒_𝑗(𝑡−𝜏(𝑡))−𝑘_𝑖𝐸𝑒_𝑖(𝑡)−𝑙_𝑖𝐹𝑒_𝑖(𝑡−𝜏(𝑡)), (4.5) where 𝐽(𝑡) and Γ(𝑡) are the Jacobian matrices of the functions 𝑓,𝑔 at 𝑠(𝑡) and 𝑠(𝑡−𝜏(𝑡)), respectively.

In what follows, we define 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑁), 𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑁), and𝐿¹=𝑙¹_{𝑖𝑗𝑁×𝑁},𝐿²=𝑙²_{𝑖𝑗𝑁×𝑁},𝑒(𝑡)=𝑒^𝑇₁(𝑡),𝑒^𝑇₂(𝑡),…,𝑒^𝑇_𝑁(𝑡)^𝑇. (4.6) Then based on Kronecker product, we can deduce𝑒̇(𝑡)=𝐼_𝑁⊗𝐽(𝑡)𝑒(𝑡)+𝐼_𝑁⊗Γ(𝑡)𝑒(𝑡−𝜏(𝑡))+𝐿¹−𝐾⊗𝐸𝑒(𝑡)+𝐿²−𝐿⊗𝐹𝑒(𝑡−𝜏(𝑡)). (4.7)

Assumption 4.1. The matrices 𝐿¹−𝐾 and 𝐿²−𝐿 satisfy [(𝐿¹−𝐾)(𝐿²−𝐿)]^𝑇=(𝐿¹−𝐾)(𝐿²−𝐿).

Then there does exist an orthogonal matrix 𝑈 such that 𝑈(𝐿¹−𝐾)𝑈^𝑇=Λ and 𝑈(𝐿²−𝐿)𝑈^𝑇=Π, in which Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑁) and Π=diag(𝜈₁,𝜈₂,…,𝜈_𝑁) satisfy 0>𝜆₁≥𝜆₂≥⋯≥𝜆_𝑁 and 0>𝜈₁≥𝜈₂≥⋯≥𝜈_𝑁.

Denoting 𝜂(𝑡)=(𝑈^𝑇⊗𝐼_𝑛)𝑒(𝑡)=[𝜂^𝑇₁(𝑡),𝜂^𝑇₂(𝑡),…,𝜂^𝑇_𝑁(𝑡)]^𝑇 and employing the property of Kronecker product (𝐴₁⊗𝐵₁)(𝐴₂⊗𝐵₂)=(𝐴₁𝐴₂)⊗(𝐵₁𝐵₂), one can derive𝜂̇(𝑡)=𝐼_𝑁⊗𝐽(𝑡)𝜂(𝑡)+𝐼_𝑁⊗Γ(𝑡)𝜂(𝑡−𝜏(𝑡))+Λ⊗𝐸𝜂(𝑡)+Π⊗𝐹𝜂(𝑡−𝜏(𝑡)), (4.8) that is,𝜂̇_𝑖(𝑡)=𝐽(𝑡)+𝜆_𝑖𝐸𝜂_𝑖(𝑡)+Γ(𝑡)+𝜈_𝑖𝐹𝜂_𝑖(𝑡−𝜏(𝑡)),𝑖=1,2,…,𝑁. (4.9)

Theorem 4.2. Considering the time delay in (2.2), the controlled networks (4.9) can achieve local asymptotical stability at 𝑠(𝑡) if there exist the appropriately dimensional matrices 𝑃_𝑖>0, 𝑄_𝑖>0,𝑅_𝑖>0,𝑇_𝑖>0,𝑈_𝑖>0, 𝑉_𝑖>0,𝑆_𝑖,𝐻_𝑖, 𝑀_𝑖, and 𝑁_𝑖 for 𝑖=1,2,…,𝑁, satisfying the following inequalities in (4.10): 𝑉_𝑖𝑆_𝑖𝑆^𝑇_𝑖𝑉_𝑖Ξ≥0,𝑖=1,2,…,𝑁,_𝑖11𝑈_𝑖Ξ_𝑖130Ξ_𝑖15∗Ξ_𝑖22𝑉_𝑖−𝑆_𝑖𝑆_𝑖0∗∗Ξ_𝑖33𝑉_𝑖−𝑆_𝑖Ξ_𝑖35∗∗∗−𝑅_𝑖−𝑉_𝑖0∗∗∗∗Ξ_𝑖55<0, (4.10) with Ξ_𝑖11=−𝑈_𝑖+𝑄_𝑖+𝐻^𝑇_𝑖𝐽(𝑡)+𝜆_𝑖𝐸+𝐽(𝑡)+𝜆_𝑖𝐸^𝑇𝐻_𝑖,Ξ_𝑖13=𝐻^𝑇_𝑖Γ(𝑡)+𝜈_𝑖𝐹+𝐽(𝑡)+𝜆_𝑖𝐸^𝑇𝑁_𝑖,Ξ_𝑖15=𝑃_𝑖−𝐻^𝑇_𝑖+𝐽(𝑡)+𝜆_𝑖𝐸^𝑇𝑀_𝑖,Ξ_𝑖22=−𝑄_𝑖+𝑅_𝑖+𝑇_𝑖−𝑉_𝑖−𝑈_𝑖,Ξ_𝑖33=−1−𝜇𝑇_𝑖−𝑁^𝑇_𝑖Γ(𝑡)+𝜈_𝑖𝐹+Γ(𝑡)+𝜈_𝑖𝐹^𝑇𝑁_𝑖−2𝑉_𝑖+𝑆_𝑖+𝑆^𝑇_𝑖,Ξ_𝑖35=Γ(𝑡)+𝜈_𝑖𝐹^𝑇𝑀_𝑖−𝑁^𝑇_𝑖,Ξ_𝑖55=−𝑀^𝑇_𝑖−𝑀_𝑖+𝜏²₀𝑈_𝑖+𝜏²_𝑚𝑉_𝑖, (4.11) where 𝜆_𝑖 and 𝜈_𝑖 are the 𝑖th eigenvalues of the matrices 𝐿¹−𝐾 and 𝐿²−𝐿, respectively.

Proof. As for the 𝑖th node, we choose the following Lyapunov-Krasovskii functional: 𝑉_𝑖𝜂_𝑖𝑡=𝜂^𝑇_𝑖(𝑡)𝑃_𝑖𝜂_𝑖(𝑡)+^𝑡_𝑡−𝜏0𝜂^𝑇_𝑖(𝑠)𝑄_𝑖𝜂_𝑖(𝑠)𝑑𝑠+^𝑡−𝜏0_{𝑡−𝜏𝑚}𝜂^𝑇_𝑖(𝑠)𝑅_𝑖𝜂_𝑖(𝑠)𝑑𝑠+^𝑡−𝜏0_{(𝑡)𝑡−𝜏}𝜂^𝑇_𝑖(𝑠)𝑇_𝑖𝜂_𝑖(𝑠)+𝑑𝑠_{0−𝜏0𝑡𝑡+𝜃}𝜏₀𝜂̇^𝑇_𝑖(𝑠)𝑈_𝑖𝜂̇_𝑖(𝑠)𝑑𝑠𝑑𝜃+^−𝜏0_−𝜏𝑚^𝑡_𝑡+𝜃𝜏_𝑚𝜂̇^𝑇_𝑖(𝑠)𝑉_𝑖𝜂̇_𝑖(𝑠)𝑑𝑠𝑑𝜃, (4.12) with appropriately dimensional matrices 𝑃_𝑖>0,𝑄_𝑖>0, 𝑅_𝑖>0, 𝑇_𝑖>0,𝑈_𝑖>0,𝑉_𝑖>0 waiting to be determined.
Together with similar proof of Theorem 3.1, one can yield 𝑉̇_𝑖𝜂_𝑖𝑡≤𝜂_𝑖(𝑡)𝜂_𝑖𝑡−𝜏₀𝜂_𝑖((𝑡𝜂𝑡−𝜏))_𝑖𝑡−𝜏_𝑚𝜂̇_𝑖(𝑡)^𝑇Ξ_𝑖11𝑈_𝑖Ξ_𝑖130Ξ_𝑖15∗Ξ_𝑖22𝑉_𝑖−𝑆_𝑖𝑆_𝑖0∗∗Ξ_𝑖33𝑉_𝑖−𝑆_𝑖Ξ_𝑖35∗∗∗−𝑅_𝑖−𝑉_𝑖0∗∗∗∗Ξ_𝑖55𝜂_𝑖(𝑡)𝜂_𝑖𝑡−𝜏₀𝜂_𝑖((𝑡𝜂𝑡−𝜏))_𝑖𝑡−𝜏_𝑚𝜂̇_𝑖(𝑡). (4.13) Then there does exist one scalar 𝜀>0 such that 𝑉̇_𝑖(𝜂_𝑖(𝑡))≤−𝜀‖P𝜂_𝑖(𝑡)P‖<0 for any 𝜂_𝑖(𝑡)≠0. Based on the theory of Lyapunov stability, the controlled network (4.9) is locally stable for any time-delay satisfying (2.2). The proof is completed.

Remark 4.3. Based on Theorem 4.2, for the controlled dynamical network with fixed network structure and coupling strength as in (4.1), it is easy to determine the number of pinned nodes for network synchronization and select nodes to pin since there are no restrictions for the positions of the pinned nodes. Moreover, the reciprocal convex technique can reduce the conservatism more effectively than those earlier methods.

5. Numerical Examples

Now, two numerical examples with simulations will be presented to illustrate efficiency of our results.

Example 5.1. Consider one 3-node complex network, in which each node is a simple 3-dimensional nonlinear system [28], 𝑥̇_𝑖1(𝑡)𝑥̇_𝑖2(𝑡)𝑥̇_𝑖3(𝑡)=−𝑥_𝑖1(𝑡)+𝑥²_𝑖2(𝑡)−2𝑥_𝑖2(𝑡)−3𝑥_𝑖3(𝑡)+𝑥_𝑖2(𝑡)𝑥_𝑖3(𝑡),𝑖=1,2,3, (5.1) which is asymptotically stable at 𝑠(𝑡)=0, and its Jacobian matrix is 𝐽=diag(−1,−2,−3). Assuming that the inner coupling matrices and the coupling configuration ones can be represented as 𝐸=𝐹=100010001,𝐿¹=𝐿²=−10.50.50.5−10.50.50.5−1, (5.2) then the eigenvalues of matrices 𝐿¹, 𝐿² can be computed as 0,−1.5,−1.5, respectively.

In the case of 𝜏₀=𝜇=0, using [28, Theorem 1] in gives 𝜏_max=0.615, while applying theorem in this paper can verify that the maximum delay upper bound guaranteeing the synchronized state to be asymptotically stable is 𝜏_max=0.867. This means that our results can be less conservative than some existing ones.

When 𝜏(𝑡)=sin²(2𝑡), it follows from Theorem 3.1 that the feasible solution to the condition in (3.1) can be existent. Defining the total error of the array of the system (5.1) as follows:error(𝑡)=³_𝑖=1𝑥_1𝑖(𝑡)−𝑥_2𝑖(𝑡)²+𝑥_2𝑖(𝑡)−𝑥_3𝑖(𝑡)², (5.3) then the synchronization error can be seen in Figure 1, and it shows that system (5.1) can achieve the asymptotical synchronization. During the process of simulation, the initial conditions of nodes are selected as 𝑥₁=[−0.5,−0.3,0.3]^𝑇, 𝑥₂=[0.7,−0.5,−0.6]^𝑇, and 𝑥₃=[1,0.5,0.3]^𝑇.

Figure 1 

The total synchronous error of system (5.1).

Example 5.2. Together with local linear feedback pinning control in (4.2) to the first node and setting feedback gains 𝑘₁=𝑙₁=1, we consider the following nonlinear system, 𝑥̇_𝑖1(𝑡)𝑥̇_𝑖2(𝑡)=−𝑥_𝑖1(𝑡)+𝑥²_𝑖2(𝑡)−3𝑥_𝑖1(𝑡)+𝑥_𝑖1(𝑡)𝑥_𝑖2(𝑡)+2𝑥_𝑖1(𝑡−𝜏(𝑡))−𝑥_𝑖2((𝑡𝑥𝑡−𝜏))_𝑖1(𝑡−𝜏(𝑡))+3𝑥_𝑖2(𝑡−𝜏(𝑡)),𝑖=1,2,3, (5.4) with 𝜏(t)=cos²(2𝑡). Its Jacobians are 𝐽=diag(−1,−3) and Γ=²⁻¹₁₃, respectively. The inner coupling matrices and the coupling configuration ones can be represented as 𝐸=𝐹=1001,𝐿¹=𝐿²=−10.50.50.5−10.50.50.5−1. (5.5) Then we can stabilize the network onto the original equilibrium point 𝑠=[0,0,0]^𝑇 based on Theorem 4.2, and the feasible solution to (4.10) can be existent with the eigenvalues of matrices 𝐿¹−𝐾 and 𝐿²−𝐿 being {−2.9808,−1.5000,−0.2192}. We also define the total error of the array of system (5.4) as follows: error(𝑡)=²_𝑖=1𝑥_1𝑖(𝑡)−𝑥_2𝑖(𝑡)²+𝑥_2𝑖(𝑡)−𝑥_3𝑖(𝑡)², (5.6) then the synchronization error can be seen in Figure 2, and it shows that system (5.4) can achieve the asymptotical synchronization. During the process of simulation, the initial conditions of nodes are selected as 𝑥₁=[−0.5,−0.3]^𝑇, 𝑥₂=[0.7,−0.5]^𝑇, and 𝑥₃=[0.4,0.3]^𝑇.

Figure 2 

The total synchronous error of system (5.4).

6. Conclusions

In this paper, the asymptotical synchronization and pinning control for delayed dynamical networks have been studied based on Lyapunov-Krasovskii functional theory, and the decentralized feedback controller has been designed to make the networks synchronized. Moreover, since the most improved techniques including reciprocal convex one have been applied, the established results can be more applicable than those present ones, and the design of controllers can be achieved by solving an array of NMIs and LMIs. Finally, two illustrative examples can show the efficiency of the proposed methods.

Acknowledgment

This work is supported by National Natural Science Foundation of China under Grant 61004032.