Research Article Distributed Adaptive

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 548375, 12 pages http://dx.doi.org/10.1155/2013/548375

Research Article Distributed Adaptive Synchronization for Complex Dynamical Networks with Uncertain Nonlinear Neutral-Type Coupling Shi Miao and Li Junmin Department of Mathematics, Xidian University, Xi’an 710071, China Correspondence should be addressed to Li Junmin; [email protected] Received 10 July 2013; Revised 23 August 2013; Accepted 28 August 2013 Academic Editor: Zhiwei Gao Copyright © 2013 S. Miao and L. Junmin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Distributed adaptive synchronization control for complex dynamical networks with nonlinear derivative coupling is proposed. The distributed adaptive strategies are constituted by directed connections among nodes. By means of the parameters separation, the nonlinear functions can be transformed into the linearly form. Then effective distributed adaptive techniques are designed to eliminate the effect of time-varying parameters and made the considered network synchronize a given trajectory in the sense of square error norm. Furthermore, the coupling matrix is not assumed to be symmetric or irreducible. An example shows the applicability and feasibility of the approach.

1. Introduction A complex network is a large set of interconnected nodes, where the nodes represent individuals in the graph and the edges represent the connections among them, such as climate system [1], biological neural networks [2], human brain system [3]. Many natural and man-made systems can be modeled and characterized by complex networks successfully [4]. Such systems may be characterized by a system with uncertainties, time delays, nonlinearity, neutral properties, hybrid dynamics, distributed dynamics and chaotic dynamics. Synchronization phenomena has been found in different forms in complex networks, such as fireflies in the forest, description of hearts, and routing messages in the internet. Thus synchronization is one of meaningful issues in dynamical characteristics of the complex dynamical networks. A considerable number of papers on this topic have appeared (see [5–7] and references therein.) Recently, various control techniques have been reported to achieve networks synchronization (see [4, 8–16] and references therein) Some control schemes [8–13] were based on a solution of the homogenous system, in which it may be difficult to obtain the state information of an isolated node.

Consequently, utilizing the information from neighborhood to realize the network synchronization is more reasonable. Paper [17] introduced the concept of control topology to describe the whole controller structure. In [18], based on local information of node dynamics, an effective distributed adaptive strategy was designed to tune the coupling weights of a network. A considerable number of controlled synchronization techniques have been derived for complex dynamic networks based on the assumption which is the coupled nodes of CDN with the same dynamics (see the above papers and the references therein). In reality, complex networks are more likely to have different nodes for different dynamics. For example, in a multi-robot system, the robots can have distinct dynamic structures or different parameters. Recently, special attention has been focused on the synchronization of complex dynamical networks with nonidentical nodes [19– 22]. Paper [20] investigated the synchronization problem of a complex network with nonidentical nodes via openloop controllers. The paper [22] considered nonlinearly coupled networks with non-identical nodes and designed pinning control to obtain synchronization criteria. On the other hand, many real-world network systems’ structure will change over time and contain unknown parameters. Very recently, some papers studied complex networks with

2

Journal of Applied Mathematics

unknown time-varying coupling strength [23–26]. In these results, non-identical nodes were not considered. Only in [21], the time-varying complex network with non-identical nodes was investigated, and a criterion of global bounded synchronization of the maximum state deviation between nodes was developed. In other aspects, new complex networks models are proposed to reflect the complexity from the network structure. Thus, the problem of neutral-type couplings has also been widely investigated [27–31]. However, in the above studies, only linear derivative coupling is considered. More recently, [32] studied the synchronization in a class of dynamical networks with distributed delays and nonlinear derivative coupling. Considering the preceding discussion, nonidentical nodes complex dynamic network with nonlinear derivative coupling, and time-varying coupling strength is not concerned yet. Inspired by the aforementioned results, the problem of adaptive synchronization is studied for complex dynamical networks with non-identical nodes, nonlinearly derivative couplings, and unknown time-varying coupling strength. A prominent feature of this network is that its complexity originates not only from the nonlinear dynamics of the nodes, but also from the complex coupling strength. The difficulty in dealing with the nonlinearly derivative couplings with unknown time-varying parameters is solved by using the parameter separation method. The distributed adaptive learning laws of periodically time-varying and constant parameters and the distributed adaptive controllers are constructed to guarantee that the system is asymptotically stable and that all closed loop signals are bounded. The remainder of the paper is organized as follows. The problem statement and preliminaries are given in Section 2. Section 3 gives the main results and proofs. In Section 4, an illustrative example is provided to verify the theoretical results. Finally, conclusion is given.

Remark 1. It should be pointed out that the nonlinear derivative couplings consist of more information in CDN than the linear derivative couplings in [28], the challenge problem in (1) is due to the uncertain nonlinear neutral-type coupling ∑𝑁 𝑗=1 𝑏𝑖𝑗 Γ𝑘(𝑥𝑗̇ (𝑡), 𝛽𝑖 (𝑡)). For the unknown parameter 𝛽𝑖 (𝑡) and the nonlinear function 𝑘(𝑥𝑗̇ (𝑡), 𝛽𝑖 (𝑡)), in this paper, we shall overcome the obstacle though the domination technique and the parameters separation principle. Under later assumption, we can “separate” the parameters from the nonlinear function. Define synchronization error as

2. Problem Statement and Preliminaries

where 𝑢𝑖 (𝑡) = [𝑢𝑖1 (𝑡), 𝑢𝑖2 (𝑡), . . . , 𝑢𝑖𝑛 (𝑡)]𝑇 ∈ 𝑅𝑛 , 𝑖 = 1, 2, . . . , 𝑁, are the adaptive controllers to be designed. Since the row sum of coupling matrix is zero, it is easily obtained that 𝑁 ̇ 𝛽𝑖 (𝑡)) = 0, which ∑𝑁 𝑗=1 𝑎𝑖𝑗 Γ𝑔(𝑠(𝑡), 𝜑𝑖 (𝑡)) = ∑𝑗=1 𝑏𝑖𝑗 Γ𝑘(𝑠(𝑡), implies the dynamical error equations as

The complex dynamical network is described as 𝑁

𝑥𝑖̇ (𝑡) = 𝑓𝑖 (𝑥𝑖 (𝑡)) + ∑ 𝑎𝑖𝑗 Γ𝑔 (𝑥𝑗 (𝑡) , 𝜑𝑖 (𝑡)) 𝐽=1

𝑁

+ ∑𝑏𝑖𝑗 Γ𝑘 (𝑥𝑗̇ (𝑡) , 𝛽𝑖 (𝑡)) ,

(1) 𝑖 = 1, 2, . . . , 𝑁,

𝑗=1

where 𝑥𝑖 (𝑡) = [𝑥𝑖1 (𝑡), 𝑥𝑖2 (𝑡), . . . , 𝑥𝑖𝑛 (𝑡)]𝑇 ∈ 𝑅𝑛 is the state vector of the 𝑖th node, 𝑓𝑖 (⋅) : 𝑅𝑛 → 𝑅𝑛 is a vector-valued continuous function, and 𝑔(⋅, ⋅), 𝑘(⋅, ⋅) : 𝑅𝑛 × 𝑅 → 𝑅𝑛 are unknown continuous nonlinear vector-valued functions, 𝜑𝑖 (𝑡), 𝛽𝑖 (𝑡) represent the unknown time-varying functions, the inner-coupling matrix Γ = diag(𝛾1 , . . . , 𝛾𝑛 ) is positive definite. 𝐴 = (𝑎𝑖𝑗 )𝑁×𝑁 and 𝐵 = (𝑏𝑖𝑗 )𝑁×𝑁 are the coupling 𝑁 configuration matrices, which satisfying ∑𝑁 𝑗=1 𝑎𝑖𝑗 = ∑𝑗=1 𝑏𝑖𝑗 = 0 and defined as follows: if there is a connection between node 𝑖 and node 𝑗 (𝑖 ≠ 𝑗), then 𝑎𝑖𝑗 > 0, 𝑏𝑖𝑗 > 0, otherwise, 𝑎𝑖𝑗 = 𝑏𝑖𝑗 = 0.

𝑒𝑖 (𝑡) = 𝑥𝑖 (𝑡) − 𝑠 (𝑡) ,

𝑖 = 1, 2, . . . , 𝑁,

(2)

𝑛

where 𝑠(𝑡) ∈ 𝑅 is a solution of the dynamics of the isolated node to which all 𝑥𝑖 (𝑡) are expected to synchronize. Then, (1) is said to be asymptotic synchronization in the 𝐿2𝑇 norm sense if 𝑡

󵄩󵄩 󵄩2 󵄩𝑒 (𝜏)󵄩󵄩󵄩 𝑑𝜏 𝑡 → ∞ 𝑡−𝑇 󵄩 𝑖 lim ∫

= 0,

𝑖 = 1, 2, . . . , 𝑁

(3)

and the system (1) is said to be globally asymptotically synchronized onto state 𝑠(𝑡) if lim 𝑒 𝑡→∞ 𝑖

(𝑡) = 0,

𝑖 = 1, 2, . . . , 𝑁.

(4)

To achieve the control objective in (3) and (4), we need an adaptive control strategy to nodes in network (1). Then, the controlled network is given by 𝑁

𝑥𝑖̇ (𝑡) = 𝑓𝑖 (𝑥𝑖 (𝑡)) + ∑ 𝑎𝑖𝑗 Γ𝑔 (𝑥𝑗 (𝑡) , 𝜑𝑖 (𝑡)) 𝑗=1

𝑁

+ ∑ 𝑏𝑖𝑗 Γ𝑘 (𝑥𝑗̇ (𝑡) , 𝛽𝑖 (𝑡))

(5)

𝑗=1

+ 𝑢𝑖 (𝑡) ,

𝑖 = 1, 2, . . . , 𝑁,

𝑒𝑖̇ (𝑡) = 𝑥𝑖̇ (𝑡) − 𝑠 ̇ (𝑡) = 𝑓𝑖 (𝑥𝑖 (𝑡)) 𝑁

+ ∑ 𝑎𝑖𝑗 Γ (𝑔 (𝑥𝑗 (𝑡) , 𝜑𝑖 (𝑡)) − 𝑔 (𝑠 (𝑡) , 𝜑𝑖 (𝑡))) 𝑗=1 𝑁

(6)

+ ∑ 𝑏𝑖𝑗 Γ (𝑘 (𝑥𝑗̇ (𝑡) , 𝛽𝑖 (𝑡)) − 𝑘 (𝑠 ̇ (𝑡) , 𝛽𝑖 (𝑡))) 𝑗=1

− 𝑠 ̇ (𝑡) + 𝑢𝑖 (𝑡) . Our objective is to design a distributed adaptive control law 𝑢𝑖 (𝑡) so as to obtain the convergence of the synchronization errors. In order to derive the main results, the following assumptions and lemmas are introduced.


3

Assumption 2 (Lipschits condition). For the system (1), there exist positive constants 𝑙𝑖 > 0, 𝑖 = 1, . . . , 𝑁 such that 󵄩󵄩󵄩𝑓𝑖 (𝑥) − 𝑓𝑖 (𝑦)󵄩󵄩󵄩 ≤ 𝑙𝑖 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 , ∀𝑥, 𝑦 ∈ 𝑅𝑛 , 1 ≤ 𝑖 ≤ 𝑁. (7) 󵄩 󵄩 󵄩 󵄩 Remark 3. For many well-known systems, such as Lorenz’s system, Chen’s system, and Lü’s system, the above condition is satisfied.

Lemma 10 (Young’s inequality). For vectors 𝑥, 𝑦 ∈ 𝑅𝑛 , and any positive constant 𝑐, the following matrix inequality holds: 𝑥𝑇 𝑦 ≤ 𝑐𝑥𝑇 𝑥 +

Remark 5. It is well known that the estimation of unknown nonlinear parameters in the systems is a difficult problem. In this paper, we separate the unknown parameters from the nonlinear function in (8), according to the separation principle in [33]; thus, Assumption 4 is reasonable and easily obtained. By using Assumption 4, we are able to solve the synchronization problem for a class of nonlinearly parameterized systems with nonlinear derivative couplings. Assumption 6. In the given networks in (1), 𝜑𝑖 (𝑡), 𝛽𝑖 (𝑡) are unknown time-varying periodic functions with a known period 𝑇. From Assumption 6, it is easy to see that 𝜆(𝜑𝑖 (𝑡)) and each element in 𝑀𝑖 (𝑡) are periodic functions with the same period 𝑇. Suppose 𝜆 (𝜑𝑖 (𝑡)) = 𝜙𝑖 (𝑡) + 𝜃𝑖 ,

(9)

where 𝜙𝑖 (𝑡) is an unknown continuous periodic function with a known period 𝑇 and 𝜃𝑖 is an unknown constant parameter. Assumption 7. In network (1), the inner coupling matrix Γ and ℎ(⋅, ⋅) satisfies 󵄨 󵄨 max (‖Γ‖ , max (󵄨󵄨󵄨𝛾𝑖 󵄨󵄨󵄨)) ≤ 𝛾, ℎ (𝑥𝑖 (𝑡) , 𝑠 (𝑡)) < 𝐻, (10) 𝑖 = 1, 2, . . . , 𝑁, where 𝛾, 𝐻 are positive constants. Assumption 8. Assume that the state and the state derivative of system (1) are measurable. Remark 9. This assumption is necessary to design controller and adaptive laws. Assumption 8 seems to be restrictive. The observer for state derivative will be considered in the near future.

(11)

Lemma 11. For any vectors 𝑥, 𝑦 ∈ 𝑅𝑛 , according to the properties of matrix trace, the following equality holds:

Assumption 4 (see [33]). For the unknown continuous functions 𝑔(⋅, ⋅), 𝑘(⋅, ⋅), the following inequality holds: 󵄩2 󵄩󵄩 󵄩󵄩𝑔(𝑥𝑗 (𝑡), 𝜑𝑖 (𝑡)) − 𝑔 (𝑠 (𝑡) , 𝜑𝑖 (𝑡))󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩2 󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑗 (𝑡) − 𝑠 (𝑡)󵄩󵄩󵄩 ℎ (𝑥𝑗 (𝑡) , 𝑠 (𝑡)) 𝜆 (𝜑𝑖 (𝑡)) , (8) 󵄨 󵄨󵄨 󵄨󵄨𝑘 (𝑥𝑗̇ (𝑡) , 𝛽𝑖 (𝑡)) − 𝑘 (𝑠 ̇ (𝑡) , 𝛽𝑖 (𝑡))󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 ≤ 𝑀 (𝛽𝑖 (𝑡)) 󵄨󵄨󵄨𝑥𝑗̇ (𝑡) − 𝑠 ̇ (𝑡)󵄨󵄨󵄨 ≜ 𝑀𝑖 (𝑡) 󵄨󵄨󵄨󵄨𝑒𝑗̇ (𝑡)󵄨󵄨󵄨󵄨 , where ℎ(⋅, ⋅) is an known nonnegative continuous function 𝜆(⋅) is a unknown nonnegative continuous function, 𝑀(⋅) or 𝑀𝑖 (⋅) is an 𝑛 × 𝑛 unknown nonnegative continuous function matrix.

1 𝑇 𝑦 𝑦. 4𝑐

𝑥𝑇 𝑦 = tr (𝑦𝑥𝑇 ) = tr (𝑥𝑦𝑇 ) .

(12)

̇ ∈ Lemma 12 (Barbalat’s Lemma [34]). If 𝑒(𝑡) ∈ 𝐿 2 and 𝑒(𝑡) 𝐿 ∞ , then lim𝑡 → ∞ 𝑒(𝑡) = 0, where 𝐿 2 = {𝑥 | 𝑥 : [0, ∞) → ∞

1/2

< ∞} and 𝐿 ∞ = {𝑥 | 𝑥 : [0, ∞) → 𝑅, (∫0 |𝑥(𝑡)|2 ) 𝑅, sup𝑡≥0 |𝑥(𝑡)| < ∞}. Lemma 13. For a positive constant 𝑇, if ∀𝑡 ≥ 0, 𝑓(𝑡) is a 𝑡 continuous function and ∫𝑡−𝑇 𝑓(𝜏) 𝑑𝜏 exists and is bounded with 𝑀. Then, 𝑓(𝑡) is bounded. Proof. In fact, if 𝑓(𝑡) is unbounded, we can get ∀𝑁 > 0, ∃𝜏 ∈ [𝑡−𝑇, 𝑡], 𝑓(𝜏) > 𝑁. As 𝑓(𝑡) is continuous function, according to the properties of the continuous functions, we have ∃𝜀 > 𝑡 0 ∀𝑘 ∈ [𝜏−𝜀, 𝜏+𝜀] ⊂ [𝑡−𝑇, 𝑡], 𝑓(𝑘) > 𝑁; then ∫𝑡−𝑇 𝑓(𝜏) 𝑑𝜏 ≥ 𝜏+𝜀

∫𝜏−𝜀 𝑓(𝜏) 𝑑𝑡 ≥ 2𝜀𝑁. We can choose 𝑁 = (𝑀/2𝜀) + 1; then 𝑡

∫𝑡−𝑇 𝑓(𝜏) 𝑑𝜏 > 𝑀; this is a contradiction with the assumption. The proof is completed.

3. Adaptive Synchronization of the Complex Dynamical Networks In this section, distributed adaptive controller and distributed adaptive laws are designed to control the given system to synchronize the given trajectory 𝑠(𝑡), and the synchronization process is proved. Considering the controlled networks, we design the controller by 𝑢𝑖 (𝑡) = 𝑠 ̇ (𝑡) − 𝑓𝑖 (𝑠 (𝑡)) 𝑁

− 𝛾𝐻𝑒𝑖 (𝑡) ∑ 𝑎𝑗𝑖2 (𝜙̂𝑗 (𝑡) + 𝜃̂𝑗 (𝑡)) 𝑗=1

(13)

𝑁 󵄨 󵄨 ̂𝑖 (𝑡) 󵄨󵄨󵄨󵄨𝑒𝑗̇ (𝑡)󵄨󵄨󵄨󵄨 , − 𝛾 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 diag (sign (𝑒𝑖𝑗 (𝑡))) 𝑀 󵄨 󵄨 𝑗=1

1≤𝑗≤𝑁

where 𝛾, 𝐻 satisfy condition (10), 𝑁 is the node number, and ̂𝑖 (𝑡) used to estimate 𝜙𝑖 (𝑡), 𝜃𝑖 , 𝑀𝑖 (𝑡). 𝜙̂𝑖 (𝑡), and 𝜃̂𝑖 (𝑡), are 𝑀 ̇ (𝑡)|, . . . , |𝑒𝑗𝑛 ̇ (𝑡)|)𝑇 . |𝑒𝑗̇ (𝑡)| = (|𝑒𝑗1 The constant parameter distributed update law is designed as follows: 𝑁

̇ 𝜃̂𝑖 (𝑡) = 𝑟𝑖 ∑ 𝑎𝑖𝑗2 𝑒𝑗𝑇 (𝑡) 𝑒𝑗 (𝑡) 𝑗=1

(14)

4


and the time-varying distributed periodic adaptive learning laws as 𝜙̂𝑖 (𝑡) 0, 𝑡 ∈ [−𝑇, 0) { { { 𝑁 { { { 2 𝑇 { 𝑡 ∈ [0, 𝑇) {𝑞𝑖0 (𝑡) ∑𝑎𝑖𝑗 𝑒𝑗 (𝑡) 𝑒𝑗 (𝑡) , ={ 𝑗=1 { { 𝑁 { { { ̂ { − 𝑇) + 𝑞 𝜙 𝑎𝑖𝑗2 𝑒𝑗𝑇 (𝑡) 𝑒𝑗 (𝑡) , 𝑡 ∈ [𝑇, +∞) , ∑ (𝑡 𝑖 { 𝑖 𝑗=1 {

It should be noted that we define 𝜙𝑖 (𝑡) = 𝜙𝑖 (0), 𝑀𝑖 (𝑡) = 𝑀𝑖 (0) 𝑡 ∈ [−𝑇, 0); then, from the adaptive laws (15) and (16), we get 𝜙̃𝑖2 (𝑡 − 𝑇) = 𝜙𝑖2 (𝑡 − 𝑇) = 𝜙𝑖2 (0) , ̃𝑖 (𝑡 − 𝑇) 𝑀 ̃𝑇 (𝑡 − 𝑇) = 𝑀𝑖 (𝑡 − 𝑇) 𝑀𝑇 (𝑡 − 𝑇) 𝑀 𝑖 𝑖

(15)

̂𝑖 (𝑡) 𝑀 0, 𝑡 ∈ [−𝑇, 0) { { { 𝑁 { { 󵄨 󵄨󵄨 󵄨󵄨 𝑇 󵄨 { { 𝑡 ∈ [0, 𝑇) {𝑓𝑖0 (𝑡) ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗̇ (𝑡)󵄨󵄨󵄨󵄨 , ={ 𝑗=1 { { 𝑁 { 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 { { ̂ 󵄨󵄨𝑏𝑖𝑗 󵄨󵄨 󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑗𝑇̇ (𝑡)󵄨󵄨󵄨 , 𝑡 ∈ [𝑇, +∞) , { 𝑀 − 𝑇) + 𝑓 ∑ (𝑡 𝑖 { 𝑖 󵄨 󵄨 󵄨 󵄨 𝑗=1 { (16)

=

Proof. Define a Lyapunov-Krasovskii-like function as follows:

(18)

𝑡 ∈ [0, 𝑇] .

Firstly, the finiteness property of 𝑉(𝑡) for the period [0, 𝑇) is given. Consider the system (6) and the proposed control laws (14)–(16); it can be seen that the right-hand side of (6) is continuous with respect to all arguments. According to the existence theorem of differential equation, (6) has unique solution in the interval [0, 𝑇1 ) ⊂ [0, 𝑇) with 0 < 𝑇1 ≤ 𝑇. This can guarantee the boundedness of 𝑉(𝑡) over [0, 𝑇1 ). Therefore, we need only focus in the interval [𝑇1 , 𝑇). The derivative of 𝑉(𝑡) with respect to time is given by 𝑁

𝑉̇ (𝑡) = ∑𝑒𝑖𝑇 (𝑡) 𝑒𝑖̇ (𝑡) 𝑖=1

+

where 𝑟𝑖 , 𝑞𝑖 , and 𝑓𝑖 are positive constants and 𝑞𝑖0 (𝑡), 𝑓𝑖0 (𝑡) are continuous and strictly increasing nonnegative functions satisfying 𝑞𝑖0 (0) = 0, 𝑞𝑖0 (𝑇) = 𝑞𝑖 , and 𝑓𝑖0 (0) = 0, 𝑓𝑖0 (𝑇) = 𝑓𝑖 ̂𝑖 (𝑡) are continuous in 𝑡 = 𝑖𝑇, which ensure that 𝜙̂𝑖 (𝑡), 𝑀 𝑖 = 1, 2, . . . , 𝑁, |𝑒𝑗 (𝑡)| = (|𝑒𝑗1 (𝑡)|, . . . , |𝑒𝑗𝑛 (𝑡)|)𝑇 . The following theorem will give a sufficient condition for the controlled network in (5) to be asymptotical synchronization. Theorem 14. Under Assumptions 2–7, the control law (13) with the adaptive learning laws (14)–(16) guarantees the controlled networks in (5) achieve asymptotic synchronization in the 𝐿2𝑇 norm sense. All closed-loop signals are bounded. Furthermore, if 𝑘(⋅, ⋅) is bounded function, we can obtain globally asymptotical synchronization.

𝑀𝑖 (0) 𝑀𝑖𝑇 (0) ,

𝛾𝐻 𝑁 −1 ̃2 ∑ [𝑞 𝜙 (𝑡) − 𝑞𝑖−1 𝜙̃𝑖2 (𝑡 − 𝑇)] 2 𝑖=1 𝑖 𝑖 𝑁

̇ + 𝛾𝐻∑𝑟𝑖−1 (𝜃̃𝑖 (𝑡) + 𝐿) 𝜃̃𝑖 (𝑡)

(19)

𝑖=1

𝑁

𝛾 ̃𝑇 (𝑡)) ̃𝑖 (𝑡) 𝑀 + ∑ [𝑓𝑖−1 tr (𝑀 𝑖 2 𝑖=1 ̃𝑇 (𝑡 − 𝑇))] . ̃𝑖 (𝑡 − 𝑇) 𝑀 − 𝑓𝑖−1 tr (𝑀 𝑖 Let us introduce some notations as 𝑓𝑖 (𝑥𝑖 (𝑡)) − 𝑓𝑖 (𝑠 (𝑡)) = Φ, 𝑔 (𝑥𝑗 (𝑡) , 𝜑𝑖 (𝑡)) − 𝑔 (𝑠 (𝑡) , 𝜑𝑖 (𝑡)) = Λ,

(20)

𝑘 (𝑥𝑗̇ (𝑡) , 𝛽𝑖 (𝑡)) − 𝑘 (𝑠𝑗̇ (𝑡) , 𝛽𝑖 (𝑡)) = 𝐾. From (6) and (13), the first term on the right hand side of (19) satisfies 𝑁

∑𝑒𝑖𝑇 (𝑡) 𝑒𝑖̇ (𝑡) 𝑖=1

𝑁

𝑁

𝑡

𝛾𝐻 1 𝑉 (𝑡) = ∑𝑒𝑖𝑇 𝑒𝑖 + ∑ ∫ 𝑞−1 𝜙̃2 (𝜏) 𝑑𝜏 2 𝑖=1 2 𝑖=1 𝑡−𝑇 𝑖 𝑖 +

2 𝛾𝐻 𝑁 −1 ̃ ∑𝑟𝑖 (𝜃𝑖 (𝑡) + 𝐿) 2 𝑖=1

𝑁

𝑁

= ∑𝑒𝑖𝑇 (𝑡) [𝑓𝑖 (𝑥𝑖 (𝑡)) + ∑ 𝑎𝑖𝑗 ΓΛ 𝑖=1 𝑗=1 [ 𝑁

+ ∑𝑏𝑖𝑗 Γ𝐾 − 𝑠 ̇ (𝑡) + 𝑢𝑖 (𝑡)] 𝑗=1 ]

(17)

𝛾𝑁 𝑡 ̃𝑇 (𝜏)) 𝑑𝜏, ̃𝑖 (𝜏) 𝑀 + ∑ ∫ 𝑓𝑖−1 tr (𝑀 𝑖 2 𝑖=1 𝑡−𝑇 𝑡 ∈ [0, ∞) , ̃𝑖 (𝑡) = 𝑀𝑖 (𝑡) − where 𝜙̃𝑖 (𝑡) = 𝜙𝑖 (𝑡) − 𝜙̂𝑖 (𝑡), 𝜃̃𝑖 (𝑡) = 𝜃𝑖 − 𝜃̂𝑖 (𝑡), 𝑀 ̂ 𝑀𝑖 (𝑡), and 𝐿 is a sufficiently large positive constant which will be determined later.

𝑁

𝑁

𝑖=1

𝑗=1

= ∑ (𝑒𝑖𝑇 Φ + ∑ 𝑎𝑖𝑗 𝑒𝑖𝑇 ΓΛ 𝑁

𝑁

𝑗=1

𝑗=1

+ ∑𝑏𝑖𝑗 𝑒𝑖𝑇 Γ𝐾 − 𝛾𝐻∑ 𝑎𝑗𝑖2 (𝜙̂𝑗 (𝑡) + 𝜃̂𝑗 (𝑡)) 𝑒𝑖𝑇 𝑒𝑖 𝑁

󵄨̂ 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 −𝛾 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑖𝑇 (𝑡)󵄨󵄨󵄨󵄨 𝑀 𝑖 (𝑡) 󵄨󵄨󵄨𝑒𝑗̇ (𝑡)󵄨󵄨󵄨) . 𝑗=1

(21)


5

According to Assumptions 2–7 and Lemma 10, from the above equation, we get

𝑁 𝑙 1 𝑁𝛾 = ∑[( 𝑖 + + 2 2 4 𝑖=1 [ 𝑁

𝑁

∑𝑒𝑖𝑇 (𝑡) 𝑒𝑖̇ 𝑖=1

+𝛾𝐻∑𝑎𝑗𝑖2 (𝜙̃𝑗 (𝑡) + 𝜃̃𝑗 (𝑡))) 𝑒𝑖𝑇 𝑒𝑖

(𝑡)

𝑗=1

𝑁 󵄨󵄨 󵄨 󵄨̃ 󵄨󵄨 󵄨󵄨 𝑇 󵄨󵄨 ] +𝛾 tr ( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝑀 𝑖 (𝑡) 󵄨󵄨󵄨𝑒𝑗̇ (𝑡)󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨) 𝑗=1 ]

𝑁

1 ≤ ∑ [𝑐𝑒𝑖𝑇 𝑒𝑖 + 𝜙𝑇 𝜙 4𝑐 𝑖=1 [ 𝑁

+ ∑ (𝑑𝑒𝑖𝑇 𝑒𝑖 + 𝑗=1

𝑁 𝑙 1 𝑁𝛾 = ∑[( 𝑖 + + 2 2 4 𝑖=1 [

1 2 𝑇 𝑇 𝑎 Λ Γ ΓΛ) 4𝑑 𝑖𝑗

𝑁

𝑁

󵄨 󵄨󵄨 󵄨 + 𝛾 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑖𝑇 󵄨󵄨󵄨󵄨 |𝐾|

+ ∑𝑎𝑗𝑖2 𝛾𝐻 (𝜙̃𝑗 (𝑡) + 𝜃̃𝑗 (𝑡))) 𝑒𝑖𝑇 𝑒𝑖

𝑗=1

𝑗=1

𝑁

𝑁 󵄨 ̃𝑇 󵄨 󵄨󵄨 󵄨󵄨 ] +𝛾 tr ( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ (𝑡)󵄨󵄨󵄨󵄨 𝑀 𝑖 (𝑡)) . 𝑗=1 ] (23)

− 𝛾𝐻∑ 𝑎𝑗𝑖2 (𝜙̂𝑗 (𝑡) + 𝜃̂𝑗 (𝑡)) 𝑒𝑖𝑇 𝑒𝑖 𝑗=1

𝑁 󵄨̂ 󵄨󵄨 󵄨󵄨] 󵄨 󵄨󵄨 −𝛾 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑖𝑇 (𝑡)󵄨󵄨󵄨󵄨 𝑀 𝑖 (𝑡) 󵄨󵄨󵄨𝑒𝑗̇ (𝑡)󵄨󵄨󵄨 𝑗=1 ]

Applying (14), the third term on the right-hand side of (19) satisfies

𝑁 𝑙2 ≤ ∑ [ (𝑐 + 𝑖 ) 𝑒𝑖𝑇 𝑒𝑖 + 𝑁𝑑𝑒𝑖𝑇 𝑒𝑖 4𝑐 𝑖=1 [

+

𝑁

𝑁

̇ 𝛾𝐻∑𝑟𝑖−1 (𝜃̃𝑖 (𝑡) + 𝐿) 𝜃̃𝑖 (𝑡) 𝑖=1

𝐻𝛾2 𝑇 (𝜙𝑖 (𝑡) + 𝜃𝑖 ) 𝑒 𝑒 4𝑑 𝑗 𝑗

∑𝑎𝑖𝑗2 𝑗=1

𝑁

= −𝛾𝐻∑ (𝜃̃𝑖 (𝑡) + 𝑖=1

𝑁

󵄨 󵄨 󵄨 󵄨 󵄨󵄨 + 𝛾 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑖𝑇 (𝑡)󵄨󵄨󵄨󵄨 𝑀𝑖 (𝑡) 󵄨󵄨󵄨󵄨𝑒𝑗̇ (𝑡)󵄨󵄨󵄨󵄨

𝑁

Let us focus on the second and fourth terms on the righthand side of (19). In the interval [𝑇1 , 𝑇), since 𝑞𝑖0 (𝑡), 𝑓𝑖0 (𝑡) are −1 continuous and strictly increasing functions, 𝑞𝑖−1 ≤ 𝑞𝑖0 (𝑡) < −1 −1 ∞, 𝑓𝑖 ≤ 𝑓𝑖0 (𝑡) < ∞ are ensured, we obtain

𝑗=1

𝑁

− 𝛾𝐻∑ 𝑎𝑗𝑖2 (𝜙̂𝑗 (𝑡) + 𝜃̂𝑗 (𝑡)) 𝑒𝑖𝑇 𝑒𝑖 𝑗=1

𝛾𝐻 𝑁 −1 ̃2 ∑ [𝑞 𝜙 (𝑡) − 𝑞𝑖−1 𝜙̃𝑖2 (𝑡 − 𝑇)] 2 𝑖=1 𝑖 𝑖

𝑁

󵄨̂ 󵄨󵄨 󵄨󵄨] 󵄨 󵄨󵄨 −𝛾 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑖𝑇 (𝑡)󵄨󵄨󵄨󵄨 𝑀 𝑖 (𝑡) 󵄨󵄨󵄨𝑒𝑗̇ (𝑡)󵄨󵄨󵄨 , 𝑗=1 ] (22)

=

𝛾𝐻 𝑁 −1 ̃2 ∑ [𝑞 𝜙 (𝑡) − 𝑞𝑖−1 𝜙̃𝑖2 (0)] 2 𝑖=1 𝑖 𝑖

where 𝑐, 𝑑 are positive constants. Choosing 𝑐 = 1/2, 𝑑 = 𝛾/4, and according to Lemma 11, we have

≤

𝛾𝐻 𝑁 −1 ̃2 ∑𝑞 𝜙 (𝑡) 2 𝑖=1 𝑖0 𝑖

=

𝛾𝐻 𝑁 −1 2 ∑𝑞 [𝜙 (𝑡) + 2𝜙̂𝑖2 (𝑡) 2 𝑖=1 𝑖0 𝑖

𝑁

∑𝑒𝑖𝑇 (𝑡) 𝑒𝑖̇ 𝑖=1

(𝑡)

−2𝜙𝑖 (𝑡) 𝜙̂𝑖 (𝑡) − 𝜙̂𝑖2 (𝑡)]

𝑁

𝑙 1 𝑁𝛾 ≤ ∑ [( 𝑖 + + 2 2 4 𝑖=1 [ 𝑁

≤

+𝛾𝐻∑𝑎𝑗𝑖2 (𝜙̃𝑗 (𝑡) + 𝜃̃𝑗 (𝑡))) 𝑒𝑖𝑇 𝑒𝑖

𝛾𝐻 𝑁 −1 2 ∑𝑞 [𝜙 (𝑡) + 2𝜙̂𝑖2 (𝑡) 2 𝑖=1 𝑖0 𝑖 −2𝜙𝑖 (𝑡) 𝜙̂𝑖 (𝑡)]

𝑗=1

𝑁 󵄨̃ 󵄨󵄨 󵄨󵄨 ] 󵄨 󵄨󵄨 +𝛾 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑖𝑇 (𝑡)󵄨󵄨󵄨󵄨 𝑀 𝑖 (𝑡) 󵄨󵄨󵄨𝑒𝑗̇ (𝑡)󵄨󵄨󵄨 𝑗=1 ]

(24)

𝐿) ∑ 𝑎𝑖𝑗2 𝑒𝑗𝑇 𝑒𝑗 . 𝑗=1

=

𝑁 𝑁 𝛾𝐻 𝑁 −1 2 ∑𝑞𝑖0 𝜙𝑖 (𝑡) − 𝛾𝐻 ∑ ∑ 𝑎𝑖𝑗2 𝜙̃𝑖 (𝑡) 𝑒𝑗𝑇 𝑒𝑗 . 2 𝑖=1 𝑖=1 𝑗=1

(25)

6

Journal of Applied Mathematics According to Lemma 11, we can get

According to (28) we have 𝛾𝑁 𝛾𝐻 𝑁 −1 𝑉̇ (𝑡) ≤ ∑𝑞𝑖0 (𝑡) 𝜙𝑖2 (𝑡) + ∑𝑓𝑖0−1 tr (𝑀𝑖 𝑀𝑖𝑇 ) , 2 𝑖=1 2 𝑖=1

̃𝑇 (𝑡)) ̃𝑖 (𝑡) 𝑀 tr (𝑀 𝑖 ̂𝑖 (𝑡)) (𝑀𝑖 − 𝑀 ̂𝑖 (𝑡))𝑇 ] = tr [(𝑀𝑖 − 𝑀

𝑡 ∈ [𝑇1 , 𝑇) .

̂𝑇 − 𝑀 ̂𝑇 ) ̂𝑖 𝑀𝑇 + 𝑀 ̂𝑖 𝑀 = tr (𝑀𝑖 𝑀𝑖𝑇 − 𝑀𝑖 𝑀 𝑖 𝑖 𝑖 =

tr (𝑀𝑖 𝑀𝑖𝑇

+

̂𝑇 ̂𝑖 𝑀 2𝑀 𝑖

−

̂𝑇 𝑀𝑖 𝑀 𝑖

−

̂𝑖 𝑀𝑇 𝑀 𝑖

−

̂𝑇 ) ̂𝑖 𝑀 𝑀 𝑖

̂𝑇 − 𝑀𝑖 𝑀 ̂𝑇 − 𝑀 ̂𝑖 𝑀 ̂𝑖 𝑀𝑇 ) ≤ tr (𝑀𝑖 𝑀𝑖𝑇 + 2𝑀 𝑖 𝑖 𝑖 ̂𝑇 − 𝑀𝑖 𝑀 ̂𝑇 ) ̂𝑖 𝑀 = tr (𝑀𝑖 𝑀𝑖𝑇 ) + 2 tr (𝑀 𝑖 𝑖 ̂𝑇 ) . ̃𝑖 𝑀 = tr (𝑀𝑖 𝑀𝑖𝑇 ) − 2 tr (𝑀 𝑖

For ∀𝑡 ∈ [𝑇1 , 𝑇), since 𝜙𝑖 (𝑡) is continuous and periodic and every element in matrix 𝑀𝑖 (⋅) is continuous function, the boundedness of them can be obtained. The boundedness of ̇ That is, 𝜙𝑖 (𝑡) and tr(𝑀𝑖 𝑀𝑖𝑇 ) leads to the boundedness of 𝑉(𝑡). 𝑉(𝑡) is bounded in [0, 𝑇). For 𝑉(𝑇1 ) is bounded, the finiteness of 𝑉(𝑡) is obvious by using integral technique, ∀𝑡 ∈ [0, 𝑇). Next step, the asymptotical convergence of 𝑒(𝑡) is proved. According to (17), ∀𝑡 ≥ 𝑇, we can get

(26)

Δ𝑉 (𝑡) = 𝑉 (𝑡) − 𝑉 (𝑡 − 𝑇)

Then from (26), the last term on the right-hand side of (19) satisfies

1𝑁 = − ∑𝑒𝑖𝑇 (𝑡) 𝑒𝑖 (𝑡) 2 𝑖=1 1𝑁 − ∑𝑒𝑖𝑇 (𝑡 − 𝑇) 𝑒𝑖 (𝑡 − 𝑇) 2 𝑖=1

𝛾 𝑁 −1 ̃𝑇 (𝑡)) ̃𝑖 (𝑡) 𝑀 ∑ [𝑓 tr (𝑀 𝑖 2 𝑖=1 𝑖 ̃𝑇 (𝑡 − 𝑇))] ̃𝑖 (𝑡 − 𝑇) 𝑀 −𝑓𝑖−1 tr (𝑀 𝑖

+

2 𝛾𝐻 𝑁 −1 ̃ ∑𝑟𝑖 (𝜃𝑖 (𝑡) + 𝐿) 2 𝑖=1

≤

𝛾 𝑁 −1 ̃𝑇 (𝑡)) ̃𝑖 (𝑡) 𝑀 ∑𝑓 tr (𝑀 𝑖 2 𝑖=1 𝑖

−

2 𝛾𝐻 𝑁 −1 ̃ ∑𝑟 (𝜃𝑖 (𝑡 − 𝑇) + 𝐿) 2 𝑖=1 𝑖

≤

𝛾 𝑁 −1 ̂𝑇 )] ̃𝑖 𝑀 ∑𝑓 [tr (𝑀𝑖 𝑀𝑖𝑇 ) − 2 tr (𝑀 𝑖 2 𝑖=1 𝑖0

+

𝛾𝐻 𝑁 𝑡 ∑ ∫ [𝑞−1 𝜙̃2 (𝜏) − 𝑞𝑖−1 𝜙̃𝑖2 (𝜏 − 𝑇)] 𝑑𝜏 2 𝑖=1 𝑡−𝑇 𝑖 𝑖

=

𝑁 𝛾 𝑁 −1 ̃𝑖 𝑀 ̂𝑇 ) ∑𝑓𝑖0 tr (𝑀𝑖 𝑀𝑖𝑇 ) − 𝛾∑ tr (𝑓𝑖0−1 𝑀 𝑖 2 𝑖=1 𝑖=1

(27)

𝛾𝑁 𝑡 ̃𝑇 (𝜏)) ̃𝑖 (𝜏) 𝑀 + ∑ ∫ 𝑓𝑖−1 [tr (𝑀 𝑖 2 𝑖=1 𝑡−𝑇 ̃𝑇 (𝜏−𝑇))] 𝑑𝜏. ̃𝑖 (𝜏−𝑇) 𝑀 − tr (𝑀 𝑖 (31)

𝛾𝑁 = ∑𝑓𝑖0−1 tr (𝑀𝑖 𝑀𝑖𝑇 ) 2 𝑖=1 𝑁

𝑁

𝑖=1

𝑗=1

With Newton-Leibniz’s formula, the first two terms on the right-hand side of (31) satisfied

󵄨 ̃𝑇 󵄨 󵄨󵄨 󵄨󵄨 − 𝛾∑ tr ( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ (𝑡)󵄨󵄨󵄨󵄨 𝑀 𝑖 (𝑡)) .

1𝑁 𝑇 1𝑁 ∑𝑒𝑖 (𝑡) 𝑒𝑖 (𝑡) − ∑𝑒𝑖𝑇 (𝑡 − 𝑇) 𝑒𝑖 (𝑡 − 𝑇) 2 𝑖=1 2 𝑖=1

Substituting (23)–(27) into (19) we obtain 𝑁 𝑙 1 𝑁𝛾 𝑉̇ (𝑡) ≤ ∑ ( 𝑖 + + − 𝐿∑ 𝑎𝑗𝑖2 ) 𝑒𝑖𝑇 𝑒𝑖 2 2 4 𝑖=1 𝑗=1

𝑁

𝑁

𝑁

𝑁

= ∑∫ (28)

𝑖 = 1, . . . , 𝑁.

𝑁

≤ ∑∫

𝑡

𝑖=1 𝑡−𝑇

𝑒𝑖𝑇 (𝜏) 𝑒𝑖̇ (𝜏) 𝑑𝜏 𝑁𝛾 𝑇 1 1 ( + 𝑙𝑖2 + ) 𝑒𝑖 (𝜏) 𝑒𝑖 (𝜏) 𝑑𝜏 2 2 4

𝑁 𝑁

It is obvious that there exist sufficiently large positive constants 𝐿 such that 1 1 2 𝑁𝛾 + 𝑙 + − 𝐿∑ 𝑎𝑗𝑖2 < 0, 2 2𝑖 4 𝑗=1

𝑡

𝑖=1 𝑡−𝑇

𝛾 𝛾𝐻 −1 + ∑𝑞𝑖0 (𝑡) 𝜙𝑖2 (𝑡) + ∑𝑓𝑖0−1 tr (𝑀𝑖 𝑀𝑖𝑇 ) 2 𝑖=1 2 𝑖=1

𝑁

(30)

(29)

𝑡

+ ∑∑ ∫

𝑖=1 𝑗=1 𝑡−𝑇 𝑁

𝑎𝑗𝑖2 𝛾𝐻𝑒𝑖𝑇 (𝜏) 𝑒𝑖 (𝜏) (𝜙̃𝑗 (𝜏) + 𝜃̃𝑗 (𝜏)) 𝑑𝜏

𝑡

𝑁 󵄨 ̃𝑇 󵄨 󵄨 󵄨󵄨 󵄨 + ∑ ∫ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛾 tr ( ∑ 󵄨󵄨󵄨𝑒𝑖 (𝜏)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ (𝜏)󵄨󵄨󵄨󵄨 𝑀 𝑖 (𝜏)) 𝑑𝜏. 𝑖=1

𝑡−𝑇

𝑗=1

(32)


7 𝛾𝑁 𝑡 ∑ ∫ 𝑓−1 2 𝑖=1 𝑡−𝑇 𝑖

Using the algebraic relation (𝑎 − 𝑏)𝑇 𝐺 (𝑎 − 𝑏) − (𝑎 − 𝑐)𝑇 𝐺 (𝑎 − 𝑐) = (𝑐 − 𝑏)𝑇 𝐺 [2 (𝑎 − 𝑏) + (𝑏 − 𝑐)] ,

̃𝑇 (𝜏)) ̃𝑖 (𝜏) 𝑀 × [tr (𝑀 𝑖

(33)

where 𝑎, 𝑏, 𝑐 ∈ 𝑅𝑝 , 𝐺 ∈ 𝑅𝑝×𝑝 . ̂ 𝑎 − 𝑏 = 𝜙̃𝑖 (𝜏), After choosing 𝐺 = 1, 𝑎 = 𝜙𝑖 (𝜏), 𝑏 = 𝜙(𝜏), 𝜙𝑖 (𝜏) = 𝜙𝑖 (𝜏 − 𝑇), 𝑐 = 𝜙̂𝑖 (𝜏 − 𝑇), and 𝑎 − 𝑐 = 𝜙̃𝑖 (𝜏 − 𝑇), we can get

̃𝑇 (𝜏 − 𝑇))] 𝑑𝜏 ̃𝑖 (𝜏 − 𝑇) 𝑀 − tr (𝑀 𝑖 =

𝛾𝑁 𝑡 ∑ ∫ 𝑓−1 2 𝑖=1 𝑡−𝑇 𝑖 𝑁 󵄨 󵄨󵄨 󵄨󵄨 󵄨 × tr [ (𝑓𝑖 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ 󵄨󵄨󵄨󵄨) 𝑗=1 [ 𝑇

𝑁

𝑁 ̃𝑇 (𝜏)−(𝑓𝑖 ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑇̇ 󵄨󵄨󵄨󵄨) )] 𝑑𝜏 × (2𝑀 𝑖 󵄨 󵄨󵄨 󵄨󵄨 𝑗󵄨 𝑗=1 ]

𝑁

𝛾𝐻 𝛾𝐻 ∑𝑞𝑖−1 𝜙̃𝑖2 (𝜏) − ∑𝑞−1 𝜙̃2 (𝜏 − 𝑇) 2 𝑖=1 2 𝑖=1 𝑖 𝑖 =

𝛾𝐻 𝑁 −1 ̃ ∑𝑞 [2𝜙𝑖 (𝜏) + 𝜙̂𝑖 (𝜏) − 𝜙̂𝑖 (𝜏 − 𝑇)] 2 𝑖=1 𝑖

𝑁

𝑡

= −𝛾∑ ∫ (34)

× [𝜙̂𝑖 (𝜏 − 𝑇) − 𝜙̂𝑖 (𝜏)]

𝑖=1 𝑡−𝑇

𝑁 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̃𝑇 tr ( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ 󵄨󵄨󵄨󵄨 𝑀 𝑖 (𝜏)) 𝑑𝜏 𝑗=1

𝑁 𝛾𝑁 𝑡 𝑁 󵄨 󵄨󵄨 󵄨󵄨 󵄨 − ∑ ∫ ∑𝑓𝑖 tr [( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ 󵄨󵄨󵄨󵄨) 2 𝑖=1 𝑡−𝑇 𝑖=1 [ 𝑗=1

𝑁 𝑁 1 𝑁 2 𝑇 𝑒𝑘 𝑒𝑘 . = −𝛾𝐻∑ (𝜙̃𝑖 (𝜏) + 𝑞𝑖 ∑ 𝑎𝑖𝑗2 𝑒𝑗𝑇 𝑒𝑗 ) ∑ 𝑎𝑖𝑘 2 𝑖=1 𝑗=1 𝑘=1

𝑁

From Lemma 11, we have

The other terms of right-hand side of (31) can be simplified as follows:

̃𝑇 (𝜏)) − tr (𝑀 ̃𝑖 (𝜏 − 𝑇) 𝑀 ̃𝑇 (𝜏 − 𝑇)) ̃𝑖 (𝜏) 𝑀 tr (𝑀 𝑖 𝑖 ̃𝑇 (𝜏) − 𝑀 ̃𝑇 (𝜏) ̃𝑖 (𝜏) 𝑀 ̃𝑖 (𝜏 − 𝑇) 𝑀 = tr (𝑀 𝑖 𝑖 ̃𝑇 (𝜏) − 𝑀 ̃𝑇 (𝜏 − 𝑇)) ̃𝑖 (𝜏 − 𝑇) 𝑀 ̃𝑖 (𝜏 − 𝑇) 𝑀 +𝑀 𝑖 𝑖

2 2 𝛾𝐻 𝑁 −1 ̃ 𝛾𝐻 𝑁 −1 ̃ ∑𝑟𝑖 (𝜃𝑖 (𝑡) + 𝐿) − ∑𝑟𝑖 (𝜃𝑖 (𝑡 − 𝑇) + 𝐿) 2 𝑖=1 2 𝑖=1 𝑁

̃𝑖 (𝜏 − 𝑇)] 𝑀 ̃𝑇 (𝜏) ̃𝑖 (𝜏) − 𝑀 = tr ([𝑀 𝑖

= 𝛾𝐻∑ ∫

𝑡

𝑖=1 𝑡−𝑇

̃𝑖 (𝜏 − 𝑇) [𝑀 ̃𝑇 (𝜏) − 𝑀 ̃𝑇 (𝜏 − 𝑇)]) +𝑀 𝑖 𝑖

𝑁

̃𝑖 (𝜏 − 𝑇)] [𝑀 ̃𝑇 (𝜏) + 𝑀 ̃𝑇 (𝜏 − 𝑇)]) ̃𝑖 (𝜏) − 𝑀 = tr ([𝑀 𝑖 𝑖

̇ 𝑟𝑖−1 (𝜃̃𝑖 (𝜏) + 𝐿) 𝜃̃𝑖 (𝜏) 𝑑𝜏

𝑡

= −𝛾𝐻∑ ∫

𝑖=1 𝑡−𝑇

̃𝑖 (𝜏 − 𝑇)] ̃𝑖 (𝜏) − 𝑀 = tr ( [𝑀 (35)

𝑁

(𝜃̃𝑖 (𝜏) + 𝐿) ∑ 𝑎𝑖𝑗2 𝑒𝑗𝑇 (𝜏) 𝑒𝑗 (𝜏) 𝑑𝜏. 𝑗=1

𝑁

𝑁 𝑁𝛾 1 1 ( + 𝑙𝑖2 + − 𝐿∑ 𝑎𝑗𝑖2 ) 𝑒𝑖𝑇 (𝜏) 𝑒𝑖 (𝜏) 𝑑𝜏. 2 2 4 𝑡−𝑇 𝑗=1

Δ𝑉 (𝑡) ≤ ∑ ∫ 𝑖=1

𝑡

According to (34) and (35), the last two terms on the right-hand side of (31) satisfy

𝑁 [𝜙̃𝑖 (𝜏) + 1 𝑞𝑖 ∑ 𝑎2 𝑒𝑇 𝑒𝑗 ] 2 𝑗=1 𝑖𝑗 𝑗 𝑡−𝑇 [ ]

𝑖=1

×

𝑁

Δ𝑉 (𝑡) < 0.

𝑡

2 𝑇 𝑒𝑘 𝑒𝑘 𝑑𝜏, ∑ 𝑎𝑖𝑘 𝑘=1

(39) 2 Choosing 1/2 + (1/2)𝑙𝑖2 + 𝑁𝛾/4 − 𝐿 ∑𝑁 𝑗=1 𝑎𝑗𝑖 < 0, 𝑖 = 1, . . . , 𝑁, we can obtain

𝛾𝐻 𝑁 𝑡 ∑ ∫ [𝑞−1 𝜙̃2 (𝜏) − 𝑞𝑖−1 𝜙̃𝑖2 (𝜏 − 𝑇)] 𝑑𝜏 2 𝑖=1 𝑡−𝑇 𝑖 𝑖 𝑁

(38)

Substituting (32) and (36)–(38) into (31), we can attain

̃𝑇 (𝜏 − 𝑇) − 𝑀 ̃𝑇 (𝜏)]) . ̃𝑇 (𝜏) + 𝑀 × [2𝑀 𝑖 𝑖 𝑖

= −𝛾𝐻∑ ∫

𝑇

󵄨 󵄨󵄨 󵄨󵄨 󵄨 × ( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ 󵄨󵄨󵄨󵄨) ] 𝑑𝜏. 𝑗=1 ] (37)

(36)

(40)

For any 𝑡 ∈ [𝑙𝑇, (𝑙 + 1)𝑇], 𝑙 = 1, 2, . . ., and denoting 𝑡0 = 𝑡 − 𝑙𝑇, 𝑡0 ∈ [0, 𝑇), we have 𝑙−1

𝑉 (𝑡) = 𝑉 (𝑡0 ) + ∑ Δ𝑉 (𝑡 − 𝑗𝑇) . 𝑗=0

(41)

8


Considering 𝑡0 ∈ [0, 𝑇) and the positive of 𝑉(𝑡), according to (41), we obtain

𝑁 𝑡 𝛾𝑁 󵄨 󵄨󵄨 󵄨󵄨 󵄨 − ∑𝑓𝑖 ∫ tr [( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ 󵄨󵄨󵄨󵄨) 2 𝑖=1 𝑡−𝑇 [ 𝑗=1

𝑉 (𝑡) < max 𝑉 (𝑡0 ) 𝑡0 ∈[0,𝑇)

𝑙−1 𝑁

− ∑∑ ∫

𝑁

𝑡−𝑗𝑇

𝑗=0 𝑖=1 𝑡−(𝑗+1)𝑇

×

(𝐿∑ 𝑎𝑗𝑖2 − 𝑗=1

𝑒𝑖𝑇 (𝜏) 𝑒𝑖

(𝜏) 𝑑𝜏.

𝑡

lim ∫ 𝑒𝑇 (𝜏) 𝑒𝑖 𝑡 → ∞ 𝑡−𝑇 𝑖

(𝜏) 𝑑𝜏 = 0.

(43)

Finally, we prove that all the closed-loop signals are bounded. Considering ∀𝑡 ∈ [𝑇, ∞), the derivative of 𝑉(𝑡) is 𝑁 𝛾𝐻 𝑁 −1 ̃2 𝑉̇ (𝑡) = ∑𝑒𝑖𝑇 (𝑡) 𝑒𝑖̇ (𝑡) + ∑𝑞 𝜙 (𝑡) 2 𝑖=1 𝑖 𝑖 𝑖=1

𝛾𝐻 𝑁 −1 ̃2 ∑𝑞 𝜙 (𝑡 − 𝑇) 2 𝑖=1 𝑖 𝑖

+

𝑁

𝛾𝐻∑𝑟𝑖−1 𝑖=1

𝛾 ̃𝑇 (𝑡)) ̃𝑖 (𝑡) 𝑀 + 𝑓𝑖−1 [ tr (𝑀 𝑖 2 ̃𝑇 (𝑡 − 𝑇))] ̃𝑖 (𝑡 − 𝑇) 𝑀 − tr (𝑀 𝑖

𝑁

𝑁

𝑖=1

𝑗=1

− ∑ (𝐿∑ 𝑎𝑗𝑖2 − 𝑁

2

2 Choosing (𝐿 > 1/2 + ((1/2)𝑙𝑖2 ) + 𝑁𝛾/4)/ ∑𝑁 𝑗=1 𝑎𝑗𝑖 , 𝑖 = 1, . . . , 𝑁, we have

𝑉 (𝑡) < 𝑉 (𝑇) .

From the boundedness of 𝑉(𝑡) and (17), we can con𝑡 clude that 𝑒𝑖 , ∫𝑡−𝑇 𝜙̃𝑖2 (𝜏)𝑑𝜏, 𝜃̃𝑖 (𝑡) For 𝜃𝑖 is constant and 𝑡 ̃𝑖 (𝜏)𝑀 ̃𝑇 (𝜏))𝑑𝜏 are all bounded., it implies the ∫𝑡−𝑇 tr(𝑀 𝑖 boundedness of 𝜃̂𝑖 (𝑡). According to Lemma 13 and the coñ𝑖 (𝑡), the boundedness of 𝜙̃𝑖 (𝑡) and 𝑀 ̃𝑖 (𝑡) tinuity of 𝜙̃𝑖 (𝑡) and 𝑀 are obviously obtained. As 𝜙𝑖 (𝑡) is a continuous periodic function and 𝑀𝑖 (𝑡) is continuous periodic matrix, we can ̂𝑖 (𝑡) are bounded. According to (1), the get that 𝜙̂𝑖 (𝑡), 𝑀 boundedness of the control input 𝑢𝑖 (𝑡) is obtained. Since 𝑒𝑖 (𝑡) is bounded, the boundedness of 𝑥𝑖 (𝑡) is received. The proof is completed.

4. Simulation Example (44)

1 1 2 𝑁𝛾 𝑇 − 𝑙 − ) 𝑒𝑖 𝑒𝑖 2 2𝑖 4

To demonstrate the theoretical result obtained in Section 3, the following dynamical network with six non-identical nodes is considered: 𝑥𝑖̇ (𝑡) = 𝑓𝑖 (𝑥𝑖 (𝑡)) 2 𝑥𝑗1 (𝑡)

𝑁

𝛾 󵄨 󵄨󵄨 󵄨󵄨 󵄨 − ∑𝑓𝑖 tr [( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ 󵄨󵄨󵄨󵄨) 2 𝑖=1 [ 𝑗=1

𝑁

𝑗=1

Having the two sides of (44) integration, one can obtain

sin (2𝜋𝑡) 𝑁 + ∑ 𝑏𝑖𝑗 Γ (exp (sin (2𝜋𝑡)) 𝑗=1 sin (2𝜋𝑡) ̇ (𝑡)) sin (2𝜋𝑡) 0.1 cos (𝑥𝑗1 ̇ (𝑡)) sin (2𝜋𝑡))) − (0.1 cos (𝑥𝑗2 ̇ (𝑡)) sin (2𝜋𝑡) 0.1 cos (𝑥𝑗3

2

𝑁 𝛾𝐻 𝑁 𝑡 𝑉 (𝑡) ≤ 𝑉 (𝑇) − ∑ ∫ 𝑞𝑖 ( ∑𝑎𝑖𝑗2 𝑒𝑗𝑇 𝑒𝑗 ) 𝑑𝜏 2 𝑖=1 𝑇 𝑗=1 𝑡

𝑁

𝑖=1 𝑇

𝑗=1

2 𝑥𝑗3 (𝑡)

𝑇

󵄨 󵄨󵄨 󵄨󵄨 󵄨 × ( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ 󵄨󵄨󵄨󵄨) ] . 𝑗=1 ]

𝑁

2

+ ∑ 𝑎𝑖𝑗 Γ exp (−𝜑𝑖 (𝑡) (𝑥𝑗2 (𝑡)))

𝑁

− ∑ ∫ (𝐿∑ 𝑎𝑗𝑖2 −

(46)

It is worth mentioning that when 𝑘(⋅, ⋅) is bounded function, the boundedness of 𝑒𝑖̇ (𝑡) can be easy to get. Then, from Lemma 12 we can obtain the error globally asymptotical synchronization.

̇ (𝜃̃𝑖 (𝑡) + 𝐿) 𝜃̃𝑖 (𝑡)

𝑁 𝛾𝐻 𝑁 ≤− ∑𝑞𝑖 ( ∑𝑎𝑖𝑗2 𝑒𝑗𝑇 𝑒𝑗 ) 2 𝑖=1 𝑗=1

󵄨 󵄨󵄨 󵄨󵄨 󵄨 × ( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗𝑇̇ 󵄨󵄨󵄨󵄨) ] 𝑑𝜏. 𝑗=1 ] (45)

1 1 2 𝑁𝛾 − 𝑙 − ) (42) 2 2𝑖 4

Since 𝑉(𝑡0 ) is bounded in the interval [0, 𝑇), according to the convergence theorem of the sum of series and (42), the error 𝑒(𝑡) converges to zero asymptotically in 𝐿2𝑇 norm. That is to say, we have

−

𝑇

𝑁

1 1 2 𝑁𝛾 𝑇 − 𝑙 − ) 𝑒𝑖 𝑒𝑖 𝑑𝜏 2 2𝑖 4

+ 𝑢𝑖 (𝑡) ,

𝑖 = 1, . . . , 6, (47)


9 0.2

0.2

0

0

e2 (t)

e1 (t)

0 −0.2 −0.4

e3 (t)

0.2

−0.2

−0.4

−0.4 1

0

2

3

0

1

2

3

0

e11 e12 e13

e21 e22 e23

0

0.2 e6 (t)

0 e5 (t)

0.4

−0.2 −0.4

−0.4

0

1

2

3

−0.6

3

2

3

e31 e32 e33

0.2

−0.2

2 Time

0.2

−0.6

1

Time

Time

e4 (t)

−0.2

0 −0.2

0

1

2

3

−0.4

0

1

Time

Time e41 e42 e43

Time

e51 e52 e53

e61 e62 e63

Figure 1: The evaluations of synchronization errors 𝑒𝑖 (𝑡), 𝑖 = 1, . . . , 6.

where 2 −2.5𝑥11 + 0.3𝑥12 + 0.9𝑥13 + 3𝑥13 2 𝑓1 (𝑥1 ) = ( 0.6𝑥11 − 2.6𝑥12 + 3𝑥12 + 𝑥13 ) , −2.8𝑥11 + sin 𝑥11 cos 𝑥12 − 2.2𝑥13

−2.5𝑥21 + 𝑥22 + 𝑥21 𝑥22 + 𝑥23 2 𝑓2 (𝑥2 ) = (0.5𝑥21 − 0.8𝑥22 − sin 𝑥22 + 𝑥23 − 𝑥23 ), −2𝑥21 − 0.8𝑥23 − 0.5 sin (2𝑥23 ) −2.5𝑥31 + 𝑥31 sin 𝑥31 + 𝑥32 + 1.4𝑥33 0.5𝑥31 − 2.5𝑥32 + 𝑥33 ), 𝑓3 (𝑥3 ) = ( −2𝑥31 + 𝑥32 𝑥33 − 0.6𝑥33 − 𝑥33 cos 𝑥33 −2.5𝑥41 + 1.5𝑥42 + 1.5𝑥43 2 2 ), 𝑓4 (𝑥4 ) = (𝑥41 − 2.5𝑥42 + 𝑥42 + 1.5𝑥43 + 𝑥43 −2𝑥41 + 𝑥41 𝑥43 − 2.5𝑥43 2 0.5𝑥51

−2.1𝑥51 + + 1.4𝑥52 + 𝑥53 𝑓5 (𝑥5 ) = ( 0.9𝑥51 − 2.1𝑥52 + 𝑥52 𝑥53 + 𝑥53 ) , −2𝑥51 + 0.4𝑥52 − 2.1𝑥53 󵄨 󵄨 󵄨 󵄨 −3.2𝑥61 + 10𝑥62 + 2.95 (󵄨󵄨󵄨𝑥61 + 1󵄨󵄨󵄨 − 󵄨󵄨󵄨𝑥61 − 1󵄨󵄨󵄨) 𝑥61 − 𝑥62 + 𝑥63 𝑓6 (𝑥6 ) = ( ), −14.7𝑥62

1 0 0 Γ = [0 1 0] , [0 0 1] −3 0 2 1 0 [ 2 −5 0 0 2 [ [ 0 3 −4 0 0 𝐴=[ [ 1 1 0 −5 2 [ [ 0 0 1 3 −4 [0 1 1 1 0 −3 [0 [ [0 𝐵=[ [1 [ [0 [2

2 −3 1 0 0 0

0 0 −3 2 1 0

1 0 2 −4 1 0

0 1 0 0 −2 1

0 1] ] 1] ], 1] ] 0] −3]

0 2] ] 0] ]. 1] ] 0] −3]

(48) Nonlinearly parameterized function satisfies 2 󵄩󵄩2 󵄩󵄩 𝑥𝑗1 (𝑡) 󵄩󵄩 󵄩󵄩 2 𝑠 󵄩󵄩 󵄩󵄩 (𝑡) 2 󵄩󵄩 󵄩󵄩 𝑥 (𝑡) 2 𝑗2 󵄩󵄩 󵄩󵄩exp (−𝜑𝑖 (𝑡) ( ( )) )) − exp (−𝜑 (𝑡) 𝑠 (𝑡) 𝑖 󵄩󵄩 󵄩󵄩 2 2 󵄩󵄩 󵄩󵄩 𝑠 (𝑡) 𝑥 (𝑡) 󵄩󵄩 󵄩󵄩 𝑗3 󵄩 󵄩 󵄩󵄩2 󵄩󵄩 ≤ 󵄩󵄩󵄩𝑒𝑗 (𝑡)󵄩󵄩󵄩 2𝜑𝑖 (𝑡) exp (−1) ,

10

Journal of Applied Mathematics 40

150

80

20

100

60 40

u2

50

−20

0

1

2

3

−50

4

0 0

1

2

3

−20

4

60

60

15

40

40

10

20

0

0 1

2

3

4

−20

u6

20

u5

80

20

2

3

4

3

4

u31 u32 u33

80

0

1

Time

u21 u22 u23

u11 u12 u13

−20

0

Time

Time

u4

20

0

−40 −60

u3

u1

0

5 0

0

1

2

3

4

−5

0

1 u61 u62 u63

u51 u52 u53

u41 u42 u43

2 Time

Time

Time

Figure 2: The curve of control 𝑢𝑖 (𝑡) with 𝑖 = 1, . . . , 6.

󵄨󵄨 ̇ (𝑡)) sin (2𝜋𝑡) 0.1 cos (𝑥𝑗1 󵄨󵄨 sin (2𝜋𝑡) 󵄨󵄨 󵄨󵄨(exp (sin (2𝜋𝑡)) − (0.1 cos (𝑥𝑗2 ̇ (𝑡)) sin (2𝜋𝑡))) 󵄨󵄨 󵄨󵄨 sin (2𝜋𝑡) ̇ (𝑡)) sin (2𝜋𝑡) 0.1 cos (𝑥𝑗3 󵄨󵄨

󵄨󵄨 ̇ (𝑡)) sin (2𝜋𝑡) 0.1 cos (𝑠𝑗1 󵄨󵄨 sin (2𝜋𝑡) 󵄨 ̇ (𝑡)) sin (2𝜋𝑡)))󵄨󵄨󵄨󵄨 − (exp (sin (2𝜋𝑡)) + (0.1 cos (𝑠𝑗2 󵄨󵄨 sin (2𝜋𝑡) 󵄨󵄨 ̇ (𝑡)) sin (2𝜋𝑡) 0.1 cos (𝑠𝑗3 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨𝑥𝑗̇ (𝑡) − 𝑠𝑗̇ (𝑡)󵄨󵄨 . (49)

̇ = 𝑓6 (𝑠(𝑡)), and the parameters are selected We choose 𝑠(𝑡) as follows: 𝑁 = 6, 𝑇 = 0.5, 𝛾 = 1, 𝐻 = 5, 𝜙2 (𝑡) = 2 cos 4𝜋𝑡 + 2, 𝜙1 (𝑡) = 0.2 sin 8𝜋𝑡 + 2, 𝜙3 (𝑡) = − sin 8𝜋𝑡 + 2, 𝜙4 (𝑡) = cos 8𝜋𝑡 + 2, (50) 𝜙6 (𝑡) = 2 sin 4𝜋𝑡 + 2, 𝜙5 (𝑡) = −2 sin 8𝜋𝑡 + 2, 𝜃 = (1, 2, 3, 1.1, 1.5, 1.3)𝑇 . In the following simulations, we choose 𝑞1 = 1, 𝑞2 = 3, 𝑞3 = 2, 𝑞4 = 5, 𝑞6 = 2, 𝑞10 (𝑡) = 2𝑡𝑞1 , 𝑞20 (𝑡) = 2𝑡𝑞2 , 𝑞5 = 1, 𝑞40 (𝑡) = 2𝑡𝑞4 , 𝑞30 (𝑡) = 2𝑡𝑞3 , 𝑞60 (𝑡) = 2𝑡𝑞6 . 𝑞50 (𝑡) = 2𝑡𝑞5 , (51)

The initially estimated values of the unknown parameters are 𝜙̂ (0) = (1 1 1 1 1 1)𝑇 ,

𝜃̂ (0) = (1, 2, 3, 1.1, 1.5, 0)𝑇 , (52)

and the initial states are chosen as 𝑥1 = [0.01 0.03 0.02]𝑇 ,

𝑥2 = [0.03 0.02 0.03]𝑇 ,

𝑥3 = [0.04 0.02 0]𝑇 ,

𝑥4 = [0 0.05 0.03]𝑇 ,

𝑥5 = [0.1 0 0.03]𝑇 ,

𝑥6 = [0.1 0.01 0]𝑇 ,

𝑠 = [0.1 0.5 0]𝑇 . (53) According to Theorem 14, the synchronization of the complex dynamical network can be guaranteed by the distributed adaptive controllers in (15) and the distributed adaptive learning laws when (16)–(18). Figure 1 shows the error evolutions under the designed controller. In this example, 𝑘(⋅, ⋅) is bounded function, we clearly see that the states of the network asymptotically synchronizes with the states of the desired orbit. Figure 2 depicts the time evolution of the controller, and Figure 3 shows the evolution of the estimated time-varying parameters. Figures 2 and 3 show that all signals in the network are bounded. Figures 4 and 5 show that the time-varying parameters are periodic and bounded.


11

1.4

×10−5 1

1.2

0.9 0.8

1 Estimated M1i

Estimated 𝜃i

0.7 0.8 0.6

0.6 0.5 0.4 0.3

0.4

0.2 0.2 0

0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.5

1

3

3.5

4

Figure 5: The evaluations of parameters 𝑀1𝑖 (𝑡).

×10−5 3

solved via distributed adaptive control method. The adaptive strategies are concerned with the networks topology. By combining inequality techniques and the parameter separation, introducing the composite energy function, the convergence of the tracking error and the boundedness of the system signals are derived. Moreover, the coupling matrix is not assumed to be symmetric or irreducible. Finally, a typical example was simulated to verify the proposed theoretical results.

2.5 Estimated 𝜙i

2.5

M11 M12 M13

𝜃4 𝜃5 𝜃6

Figure 3: The curve of control 𝜃𝑖 with 𝑖 = 1, . . . , 6.

2 1.5 1

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

0.5 0

2 Time

Time 𝜃1 𝜃2 𝜃3

1.5

0

0.5

𝜙1 𝜙2 𝜙3

1

1.5

2 Time

2.5

3

3.5

4

𝜙4 𝜙5 𝜙6

Figure 4: The evaluations of parameters 𝜙𝑖 with 𝑖 = 1, . . . , 6.

Remark 15. It is not difficult to draw the evolution of other elements in parameter 𝑀𝑖 (𝑡). Here, we only take the first row of 𝑀11 (𝑡) for example. Compared with existing results [26, 28], the biggest innovation of this paper is the asymptotical synchronization ability for the nonlinear neutral-type coupling complex networks under the designed controller.

5. Conclusion In this paper, the synchronization problem for a complex dynamical network with nonlinearly derivative couplings is

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