Adaptive Synchronization of Complex Dynamical Networks Governed by Local Lipschitz Nonlinearlity on Switching Topology

Abstract

This paper investigates the adaptive synchronization of complex dynamical networks satisfying the local Lipschitz condition with switching topology. Based on differential inclusion and nonsmooth analysis, it is proved that all nodes can converge to the synchronous state, even though only one node is informed by the synchronous state via introducing decentralized adaptive strategies to the coupling strengths and feedback gains. Finally, some numerical simulations are worked out to illustrate the analytical results.

1. Introduction

In recent years, the research on the synchronization of complex dynamical networks has attracted great attention and emerged a large number of outstanding works (e.g., [1–8] and the references therein).

For a network, the synchronization implies that all nodes will converge to the same state, which can be a homogeneous equilibrium point or a periodic orbit. To enhance the synchronization of such network, a lot of research methods are developed, and one of the most significant methods is to design effective adaptive strategies for the relevant parameters, such as the coupling strengths and the feedback gains [1–4, 8–10]. In references [4, 10, 11], there must have information channel by using a special indicator function; however, in reality, the information channel between any two nodes of a network may be lost or changed. Driven by it, we will investigate the synchronization of the complex dynamical networks with switching topology, which can lead to some information channels that occurred as well as another information channels that disappeared, by introducing adaptive strategies to the coupling strengths and feedback gains. Different from [8], the coupling strengths 𝑐^𝜎(𝑡)_𝑖𝑗(𝑡) are dynamic and variable, which can change according to the switching signal, the neighbor rule, and the switched coupling configuration 𝑎^𝜎(𝑡)_𝑖𝑗. Yet, “fast switching” [5] is hardly to be realized and bears new features and difficulties. Here, we attack the problem by invoking the theory of differential inclusion and nonsmooth analysis [12–18].

So far, most attentions have been focused on nonlinear dynamics satisfying the globally Lipschitz condition. However, many known systems only satisfy the local Lipschitz condition, such as Lorenz system, Chen system, and FitzHugh-Rinzel system [9–11]. In this paper, we will also examine nonlinear dynamics of such complex network satisfying the local Lipschitz condition.

The main contribution of the current work lies in characterizing synchronization of complex dynamical networks. First, the nonlinear dynamics of all nodes and the synchronous goal satisfies the local Lipschitz condition. Second, the adaptive strategies are introduced to the coupling strengths and the feedback gains. Third, the topology of complex dynamical networks can switch even though the information is discontinue, we can also solves the synchronization based on differential inclusion and nonsmooth analysis. In this paper, we will prove that all nodes of the complex dynamical network which is steered by the adaptive strategies can converge to the synchronous goal, even though only one node is informed by the synchronous goal if the neighboring graph remains connected.

This paper is organized as follows. Section 2 describes the model with the nonlinear dynamics satisfying the local Lipschitz condition, and some preliminaries about the nonsmooth analysis and the local Lipschitz condition are given. The main results are shown in Section 3, while Section 4 presents some simulations to illustrate our theoretical results. The conclusion is given in Section 5.

2. Preliminaries and Model Formulation

2.1. Model Formulation

Consider a dynamical network with 𝑁 nodes, and each node updates its state by 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+_{𝑗∈𝑁𝑖(𝑡)}𝑎^𝜎(𝑡)_𝑖𝑗𝑐^𝜎(𝑡)_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+𝑢_𝑖(𝑡),𝑖=1,…,𝑁, (1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈𝑅^𝑛 (𝑖=1,2,…,𝑁) is the state vector of node 𝑖, 𝑡∈[0,+∞), 𝑓∶𝑅^𝑛→𝑅^𝑛 is a continuous map, and 𝑁_𝑖(𝑡) is the neighboring set of node 𝑖 at the instant 𝑡. Notice that the coupling between the nodes may vary in time. Therefore, the interconnection graph 𝒢_𝜎(𝑡) of the model is time varying, where 𝜎(𝑡)∶[0,+∞)→𝑀={1,2,…,𝑚} is the switching signal. For all 𝑘∈𝑀, 𝐴_𝑘=(𝑎^𝑘_𝑖𝑗)∈𝑅^𝑁×𝑁 is the weighted coupling configuration of network (1), which has 𝑎^𝑘_𝑖𝑖=−∑_{𝑗∈𝑁𝑖(𝑡)}𝑎^𝑘_𝑖𝑗. If there exists communication channel between node 𝑖 and node 𝑗 at the switching segment 𝑘, then 𝑎^𝑘_𝑖𝑗=𝑎^𝑘_𝑗𝑖>0; otherwise, 𝑎^𝑘_𝑖𝑗=0. 𝑐^𝜎(𝑡)_𝑖𝑗(𝑡) denotes the coupling strengths between node 𝑖 and node 𝑗. 𝑢_𝑖(𝑡) is the feedback controller designed as 𝑢_𝑖(𝑡)=−ℎ_𝑖𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥₀(𝑡),𝑖=1,…,𝑁, (2) where ℎ_𝑖=1, if the node 𝑖 is controlled; otherwise, ℎ_𝑖=0; 𝑐_𝑖(𝑡) is the feedback gain. 𝑥(𝑡) is a desired synchronous state for network (1) with 𝑥̇₀(𝑡)=𝑓𝑥₀(𝑡). (3)

The adaptive strategies on the coupling strengths and the feedback gains are designed as 𝑐̇^𝜎(𝑡)_𝑖𝑗(𝑡)=𝑘_𝑖𝑗𝑎^𝜎(𝑡)_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡),𝑐_𝑖̇(𝑡)=𝑘_𝑖ℎ_𝑖𝑥_𝑖(𝑡)−𝑥₀(𝑡)^𝑇𝑥_𝑖(𝑡)−𝑥₀(𝑡), (4) where 𝑐_𝑖𝑗(0)≥0, 𝑐_𝑖(0)≥0, 𝑘_𝑖𝑗>0, and 𝑘_𝑖>0 are the adaptive parameters of the coupling strength and the feedback gains, respectively.

In the 𝑘th (for all𝑘∈𝑀) time period, the weighted coupling configuration matrix of network (1) is defined as 𝐵_𝑘=𝑎^𝑘₁₁𝑐^𝑘₁₁𝑎^𝑘₁₂𝑐^𝑘₁₂⋯𝑎^𝑘_1𝑁𝑐^𝑘_1𝑁𝑎^𝑘₂₁𝑐^𝑘₂₁𝑎^𝑘₂₂𝑐^𝑘₂₂⋯𝑎^𝑘_2𝑁𝑐^𝑘_2𝑁𝑎⋮⋮⋱⋮_𝑘𝑁1𝑐^𝑘_𝑁1𝑎^𝑘_𝑁2𝑐^𝑘_𝑁2⋯𝑎^𝑘_𝑁𝑁𝑐^𝑘_𝑁𝑁∈𝑅^𝑁×𝑁, (5) with 𝑎^𝑘_𝑖𝑖𝑐^𝑘_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎^𝑘_𝑖𝑗𝑐^𝑘_𝑖𝑗.

2.2. Mathematical Preliminaries

In this section, we introduce some useful concepts, assumptions, and lemmas.

2.2.1. Concepts of Nonsmooth Analysis

Definition 1 (see [16], Filippov). A vector function 𝑥(⋅) is called a solution of equation 𝑥̇=𝑓(𝑥,𝑡) on [𝑡₀,𝑡₁] if 𝑥(⋅) is absolutely continuous on [𝑡₀,𝑡₁], and for almost all 𝑡∈[𝑡₀,𝑡₁], ̇𝑓)𝑓𝑥)≜𝑥∈𝐾(𝑥,𝑡≡𝐾(colim𝑓𝑥_𝑖∣𝑥_𝑖⟶𝑥,𝑥_𝑖∉𝑁_𝑓∪𝑁, (6) where 𝑁_𝑓⊂𝑅^𝑛,𝜇(𝑁_𝑓)=0;𝑁⊂𝑅^𝑛,𝜇(𝑁)=0.

Definition 2 (see [16], generalized directional derivative). The generalized directional derivative of 𝑓 at 𝑥 in the direction 𝑣, denoted by 𝑓^∘(𝑥;𝑣), is defined as 𝑓^∘(𝑥;𝑣)∶=lim_𝑦→𝑥,sup_𝑡↓0𝑓𝑦+𝑡𝑣−𝑓𝑦𝑡, (7) where 𝑦 is a vector in the Banach space 𝑋 and 𝑡 is a positive scalar.

Definition 3 (see [18]). 𝑓(𝑥)∶𝑅^𝑚×𝑅→𝑅 is called regular if(i)for all 𝑣, the usual one sided directed derivative 𝑓^′(𝑥;𝑣) exists, (ii)for all 𝑣, 𝑓^′(𝑥;𝑣)=𝑓^∘(𝑥;𝑣).

2.2.2. Assumptions

Assumption 1. If each 𝑓_𝑖 of the nonlinear function 𝑓(𝑥_𝑖(𝑡))=(𝑓₁(𝑥_𝑖(𝑡)),𝑓₂(𝑥_𝑖(𝑡)),…,𝑓_𝑛(𝑥_𝑖(𝑡)))^𝑇 in network (1) satisfies the local Lipschitz condition, for any compact set 𝕊∈𝑅^𝑚, there exists a positive constant 𝜂(𝕊), such that 𝑥−𝑦^𝑇𝑓(𝑥)−𝑓𝑦≤𝜂(𝕊)𝑥−𝑦²,∀𝑥,𝑦∈𝕊. (8)

Assumption 2. The synchronous state 𝑥₀ is bounded; that is, there exists a compact set 𝕊=𝕊(𝑥₀(0))∈𝑅^𝑛 such that the trajectory of (3) starting from 𝑥₀(0) is always in the compact set 𝕊.

Remark 4. Note that the nonlinear dynamics of network (1) only satisfies the local Lipschitz condition. If the Jacobian matrix of 𝑓 is continuous, then 𝑓 is at least local Lipschitz. Many famous systems may not be governed by global Lipschitz nonlinearity but by local Lipschitz nonlinearity, such as Lorenz system and Chen system; therefore, it is worthy of discussing the local Lipschitz nonlinearity dynamics.

Assumption 3 (see [13]). The interconnection graph 𝒢 remains connected.

2.2.3. Lemmas

Lemma 5 (see [2]). If 𝐴=(𝑎_𝑖𝑗)∈𝑅^𝑁×𝑁 is a symmetric irreducible matrix with 𝑎_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗 and the matrix 𝐸=diag(𝑒,0,…,0) with 𝑒>0, then all eigenvalues of the matrix 𝐴−𝐸 are negative.

Lemma 6 (see [18], Chain Rule). Let 𝑥(⋅) be a Filippov solution to 𝑥̇=𝑓(𝑥,𝑡) on an interval containing 𝑡 and let 𝑉∶𝑅^𝑛×𝑅→𝑅 be a Lipschitz and regular function. Then 𝑉(𝑥(𝑡),𝑡) is absolutely continuous, (𝑑/𝑑𝑡)𝑉(𝑥(𝑡),𝑡) exists almost everywhere, and 𝑑𝑉(𝑥(𝑡))𝑑𝑡,𝑡∈^𝑎.𝑒.̇𝑉(𝑥,𝑡), (9) where ̇𝑉(𝑥,𝑡)∶=⋂_{𝜉∈𝜕𝑉(𝑥(𝑡),𝑡)}𝜉^𝑇₁^{𝐾𝑓(𝑥(𝑡),𝑡)}.

Remark 7 (see [12]). In Lemma 6, the (global) Lipschitz continuity required to 𝑉(𝑥) can be relaxed to the local condition.

Lemma 8 (see [17], Lyapunov's theorem generalized). If(i)𝑉∶𝑅^𝑛→𝑅,𝑉(0)=0 and 𝑉(𝑥)>0, for all𝑥≠0;(ii)𝑥∶𝑅→𝑅^𝑛 and 𝑉(𝑥(𝑡)) is absolutely continuous on [0,+∞) with 𝑑[𝑉(𝑥(𝑡{(𝑡)},𝑑𝑡))]<−𝜀<0𝑎.𝑒.𝑜𝑛𝑡∣𝑥≠0 (10)then 𝑥 converges to 0 in finite time.

It is worth noting that if the condition 𝑉(0)=0 in (i) of Lemma 8 cannot hold; then we can have the following lemma.

Lemma 9. If(i)𝑉∶𝑅^𝑛→𝑅, 𝑉(𝑥)>0, for all𝑥≠0; (ii)𝑥∶𝑅→𝑅^𝑛 and 𝑉(𝑥(𝑡)) is absolutely continuous on [0,+∞) with 𝑑[𝑉(𝑥(𝑡{(𝑡)},𝑑𝑡))]<−𝜀<0𝑎.𝑒.𝑜𝑛𝑡∣𝑥≠0 (11)then 𝑥 converges to 0 as 𝑡→∞.

Proof. If Lemma 9 is not true, then we have 𝑥(𝑡)≠0, for all𝑡∈[𝑡₀,+∞) and (𝑑/𝑑𝑡)[𝑉(𝑥(𝑡))]<−𝜀<0 a.e. on {𝑡∣𝑥(𝑡)≠0}. Therefore, lim_𝑡→∞𝑉(𝑡)=𝑉(𝑥(0))+^∞₀𝑉̇(𝑡)𝑑𝑡=−∞, (12) since 𝑉(𝑥(0)) is bounded, which contradicts with 𝑉(𝑥)>0, for all𝑥≠0.
This completes the proof.

3. Main Results

Denote 𝑑₀≜12^𝑁_𝑖=1𝑥_𝑖(0)−𝑥₀², (13) and construct a closed space 𝔹𝜎𝑑₀,𝑥₀=𝑥∈𝑅^𝑁𝑛∣12^𝑁_𝑖=1𝑥_𝑖−𝑥₀²≤𝜎𝑑₀, (14) where 𝑥=(𝑥^𝑇₁,𝑥^𝑇₂,…,𝑥^𝑇_𝑁)^𝑇 and 𝜎>1 is a constant. Under Assumption 1, for all 𝑥∈𝔹(𝜎𝑑₀,𝑥₀), there exists a constant 𝜂(𝜎,𝑥₀) such that ^𝑁_𝑖=1𝑥_𝑖−𝑥₀^𝑇𝑓𝑥_𝑖−𝑓𝑥₀≤𝜂𝜎,𝑥₀𝑥_𝑖−𝑥₀². (15)

In order to get the main result, we construct the following potential function: 𝑉(𝑡)=12^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑁_{𝑖=1𝑗∈𝑁𝑖(𝑡)}𝑐_𝑖𝑗(𝑡)−𝑚²4𝑘_𝑖𝑗+^𝑁_𝑖=1𝑐_𝑖(𝑡)−𝑚²2𝑘_𝑖, (16) where 𝑚>0 is sufficiently large. For convenience, we denote 𝑐=(𝑐_𝑖𝑗)_𝑁×𝑁 and 𝑐̃=(𝑐^𝑇₁,𝑐^𝑇₂,…,𝑐^𝑇_𝑁)^𝑇. Then, we can have the following results.

Theorem 10. Supposing that Assumptions 1–3 hold, then the combined trajectories of all nodes and the parameters (𝑥,𝑐,𝑐)̃ in (1) and (2) are geared to a compact hyper-ellipsoid Ω𝜎𝑑₀,𝑚,𝑥₀=(𝑥,𝑐,𝑐)̃∈𝑅^𝑁𝑛×𝑅^𝑁𝑛×𝑅^𝑁∣𝑉<𝜎𝑑₀, (17) if the initial value is selected from Ω₀𝜎𝑑₀,𝑚,𝑥₀=(𝑥,𝑐,𝑐)̃∈𝑅^𝑁𝑛×𝑅^𝑁𝑛×𝑅^𝑁∣𝑉(0)<𝜎𝑑₀. (18)

Proof. Taking the derivative of 𝑉(𝑡) in [𝑡₀,𝑡₁), we can get 𝑉̇(𝑡)=^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑓𝑥_𝑖(𝑡)−𝑓𝑥₀(𝑡)+_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+12^𝑁_{𝑖=1𝑗∈𝑁𝑖}𝑐_𝑖𝑗(𝑡)−𝑚𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇×𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+^𝑁_𝑖=1𝑐_𝑖(𝑡)−𝑚ℎ_𝑖𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)=^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝜂𝜎,𝑥₀𝐼_𝑁𝑥_𝑖(𝑡)−12𝑚^𝑁_{𝑖=1𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑚^𝑁_𝑖=1ℎ_𝑖𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)=𝑥^𝑇(𝑡)𝜂𝜎,𝑥₀𝐼_𝑁+𝑚(𝐴−𝐻)⊗𝐼_𝑛𝑥(𝑡). (19) Under Assumption 3, Lemma 5 implies that 𝐴−𝐻<0 in [𝑡₀,𝑡₁). Since 𝑚 is a sufficiently large positive constant, then 𝑉̇≤0, (20) which implies 𝑉(𝑡)≤𝑉(0),𝑡∈𝑡₀,𝑡₁. (21) Similarly, 𝑉(𝑡)≤𝑉𝑡_𝑘−1,𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘=1,2,…,𝑚. (22) Thus, we have 𝑉(𝑡)≤𝑉(0)<𝜎𝑑₀. (23) Therefore, we can conclude that the combined trajectories of all nodes with the parameters (𝑥,𝑐,𝑐)̃ are geared to a compact hyper-ellipsoid Ω(𝜎𝑑₀,𝑚,𝑥₀)={(𝑥,𝑐,𝑐)̃∈𝑅^𝑁𝑛×𝑅^𝑁𝑛×𝑅^𝑁∣𝑉<𝜎𝑑₀} if the initial value is selected from Ω₀(𝜎𝑑₀,𝑚,𝑥₀)={(𝑥,𝑐,𝑐)̃∈𝑅^𝑁𝑛×𝑅^𝑁𝑛×𝑅^𝑁∣𝑉(0)<𝜎𝑑₀}.
This completes the proof.

Theorem 11. If network (1) steered by adaptive laws (3) and the initial value of (𝑥,𝑐,𝑐)̃ is defined as (18), under Assumptions 1–3, then all nodes will converge to the synchronous state even when only one node is controlled by (2).

Proof. Let 𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥(𝑡). Construct the Lyapunov function, one has 𝑉(𝑡)=12^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑁_{𝑖=1𝑗∈𝑁𝑖(𝑡)}𝑐_𝑖𝑗(𝑡)−𝑚²4𝑘_𝑖𝑗+∑^N_i=1c_i(t)−m²2k_i, (24) where 𝑚>0 is a sufficiently large constant.
By Definition 1, we can obtain 𝑥̇_𝑖(𝑡)∈^a.e.𝐾𝑓𝑥_𝑖(𝑡)+_{𝑗∈𝑁𝑖(𝑡)}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−ℎ_𝑖𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥(𝑡),𝑐̇_𝑖𝑗(𝑡)∈^a.e.𝐾𝑘_𝑖𝑗𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡). (25) Because the topology of network (1) is switching, we have 𝑥̇_𝑖(𝑡)⊆𝐾𝑓𝑥_𝑖(𝑡)+_{𝑗∈𝑁𝑖(𝑡)}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−ℎ_𝑖𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥(𝑡). (26)
Also by Definition 3, we know that 𝑉(𝑡) is regular, then 𝜕𝑉⊆𝑥^𝑇₁(𝑡),…,𝑥^𝑇_𝑁(𝑡),^𝑁_𝑗=2𝑐_1𝑗(𝑡)−𝑚2𝑘_1𝑗,…,_{𝑁−1𝑗=1}𝑐_𝑁−1𝑗(𝑡)−𝑚2𝑘_𝑁−1𝑗,𝑐₁(𝑡)−𝑚𝑘₁𝑐,⋯,^𝑁(𝑡)−𝑚𝑘_𝑁^𝑇. (27) Hence, ̇𝑉⊂𝐾^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑓𝑥_𝑖(𝑡)−𝑓𝑥₀(𝑡)+_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)1+𝐾2^𝑁_{𝑖=1𝑗∈𝑁𝑖}𝑐_𝑖𝑗(𝑡)−𝑚𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇×𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+^𝑁_𝑖=1𝑐_𝑖(𝑡)−𝑚ℎ_𝑖𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)=𝐾^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝜂𝜎,𝑥₀𝐼_𝑁𝑥_𝑖(𝑡)−12𝑚^𝑁_{𝑖=1𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑚^𝑁_𝑖=1ℎ_𝑖𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡). (28)
Consequently, ̇𝑉⊂𝐾𝑥^𝑇(𝑡)𝜂𝜎,𝑥₀𝐼_𝑁+𝑚(𝐴−𝐻)⊗𝐼_𝑛𝑥(𝑡). (29) Lemma 5 indicates that all the eigenvalues of 𝐴−𝐻 are negative, and there exists 𝜀>0 such that 𝑑[𝑉(𝑥(𝑡{(𝑡)},𝑑𝑡))]<−𝜀<0a.e.on𝑡∣𝑥≠0 (30) since 𝑚 is a sufficiently large positive constant.
From Lemma 9, we can get that all the nodes of network (1) can converge to the synchronous state 𝑥(𝑡). This completes the proof.

Remark 12. Note also that Theorem 11 is still established when 𝐴^𝑘 is asymmetric.

Similar to the proof of Theorem 11, we also have ̇𝑉⊂𝐾𝑥^𝑇(𝑡)𝜂𝜎,𝑥₀𝐼_𝑁+𝑚𝐴+𝐴^𝑇2−𝐻⊗𝐼_𝑛𝑥(𝑡), (31) since (𝐴+𝐴^𝑇)/2 is symmetric.

4. Simulations

In this section, number simulations are given to illustrate our theoretical results. All nodes of network (1) and the synchronous goal share the same nonlinear dynamics described as the Lorenz system as follows: 𝑓(𝑥(𝑡))=𝑓𝑥¹,𝑥²,𝑥³=𝑥̇¹=10𝑥²−𝑥¹,𝑥̇²=28𝑥¹−𝑥¹𝑥³−𝑥²,𝑥̇³=𝑥¹𝑥²−83𝑥³, (32) as shown in Figure 1. Then the network can be shown as follows: 𝑥̇¹_𝑖𝑥̇²_𝑖𝑥̇³_𝑖=10𝑥²_𝑖−𝑥¹_𝑖28𝑥¹_𝑖−𝑥¹_𝑖𝑥³_𝑖−𝑥²_𝑖𝑥¹_𝑖𝑥²_𝑖−83𝑥³_𝑖+_{𝑗∈𝑁𝑖(𝑡)}𝑎^𝜎(𝑡)_𝑖𝑗𝑐^𝜎(𝑡)_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−ℎ_𝑖𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥₀(𝑡). (33) Figure 2 shows the topology of network (1) with subgraphs (a), (b), and (c) during the time intervals [0,4], [4,12], [12,50], respectively.

Figure 1 

Trajectories of the synchronous state.

(a)

(b)

(c)

Figure 2 

The topologies of network (1).

The initial values of the 10 nodes are chosen randomly, and the initial value of the synchronous goal is 𝑥₀(0)=[10,10,10]^𝑇. There is only one node informed by the synchronous goal, and the 10th node is given, that is, ℎ_𝑖(𝑖=1,2,…,9)=0 and ℎ₁₀=1. The initial values of the adaptive parameters are 𝑐_𝑖𝑗(0)=0 with the weight of 𝑘_𝑖𝑗(0)=0 for all 𝑖 and 𝑗, while 𝑐_𝑖(0)=0 with the weight of 𝑘_𝑖(0)=0 for all 𝑖.

Figure 3 describes the convergence of the state errors on the 𝑥-axis, 𝑦-axis, and 𝑧-axis, respectively. From this figure, we can see that all nodes of network (1) can synchronize to the synchronous state when the neighboring graph 𝔾 remains connected with switching topology even though only one node is informed by the synchronous state. Figure 4 shows the change trends of the adaptive coupling strengths and the adaptive feedback gains, respectively, and all these parameters converge to the constants.

Figure 3 

Trajectories convergence of 10 nodes with only one node informed by controllers (2).

(a) Coupling strengths

(b) Feedback gain

Figure 4 

Convergence of the adaptive coupling strengths and the feedback gain.

5. Conclusion

In this paper, we have investigated the synchronization of complex dynamical networks with switching topology via differential inclusion method. Different from the most previous work, all nodes and the synchronous state in this paper share the same intrinsic nonlinear dynamics governed by the local Lispchitz condition. By adding decentralized adaptive strategies to the coupling strengths and the feedback gains, all nodes can converge to the synchronous state even when only one node is pinning controlled by the synchronous state if the neighboring graph 𝒢 of the switching topology remains connected.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants (61304049, 61104140, 61203150, 61170113, 61174116), Beijing Natural Science Foundation Program 4122019, Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the jurisdiction of Beijing Municipality (PHR201108055), Science and Technology Development Plan Project of Beijing Education Commission (no. KM201310009011), the Fundamental Research Funds for the Central Universities (HUST: Grant no. 2011JC055), the Research Fund for the Doctoral Program of Higher Education (RFDP) under Grant no. 20100142120023, and the Natural Science Foundation of Hubei Province of China under Grant no. 2011CDB042.