Distributed Adaptive Synchronization for Complex Dynamical Networks with Uncertain Nonlinear Neutral-Type Coupling

Abstract

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 open-loop 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 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, non-identical 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.

2. Problem Statement and Preliminaries

The complex dynamical network is described as 𝑥̇_𝑖(𝑡)=𝑓_𝑖𝑥_𝑖(𝑡)+^𝑁_𝐽=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑘𝑥̇_𝑗(𝑡),𝛽_𝑖(𝑡),𝑖=1,2,…,𝑁, (1) where 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the state vector of the 𝑖th node, 𝑓_𝑖(⋅)∶𝑅^𝑛→𝑅^𝑛 is a vector-valued continuous function, and 𝑔(⋅,⋅),𝑘(⋅,⋅)∶𝑅^𝑛×𝑅→𝑅^𝑛 are unknown continuous nonlinear vector-valued functions, 𝜑_𝑖(𝑡), 𝛽_𝑖(𝑡) represent the unknown time-varying functions, the inner-coupling matrix Γ=diag(𝛾₁,…,𝛾_𝑛) is positive definite. 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁 and 𝐵=(𝑏_𝑖𝑗)_𝑁×𝑁 are the coupling configuration matrices, which satisfying ∑^𝑁_𝑗=1𝑎_𝑖𝑗=∑^𝑁_𝑗=1𝑏_𝑖𝑗=0 and defined as follows: if there is a connection between node 𝑖 and node 𝑗(𝑖≠𝑗), then 𝑎_𝑖𝑗>0, 𝑏_𝑖𝑗>0, otherwise, 𝑎_𝑖𝑗=𝑏_𝑖𝑗=0.

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=1𝑏_𝑖𝑗Γ𝑘(𝑥̇_𝑗(𝑡),𝛽_𝑖(𝑡)). For the unknown parameter 𝛽_𝑖(𝑡) and the nonlinear function 𝑘(𝑥̇_𝑗(𝑡),𝛽_𝑖(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 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡),𝑖=1,2,…,𝑁, (2) where 𝑠(𝑡)∈𝑅^𝑛 is a solution of the dynamics of the isolated node to which all 𝑥_𝑖(𝑡) are expected to synchronize.
Then, (1) is said to be asymptotic synchronization in the 𝐿²_𝑇 norm sense if lim_𝑡→∞^𝑡_𝑡−𝑇𝑒_𝑖(𝜏)²𝑑𝜏=0,𝑖=1,2,…,𝑁 (3) and the system (1) is said to be globally asymptotically synchronized onto state 𝑠(𝑡) if lim_𝑡→∞𝑒_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 𝑥̇_𝑖(𝑡)=𝑓_𝑖𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑘𝑥̇_𝑗(𝑡),𝛽_𝑖(𝑡)+𝑢_𝑖(𝑡),𝑖=1,2,…,𝑁, (5) where 𝑢_𝑖(𝑡)=[𝑢_𝑖1(𝑡),𝑢_𝑖2(𝑡),…,𝑢_𝑖𝑛(𝑡)]^𝑇∈𝑅^𝑛, 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_𝑗=1𝑎_𝑖𝑗Γ𝑔(𝑠(𝑡),𝜑_𝑖(𝑡))=∑^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑘(𝑠(̇𝑡),𝛽_𝑖(𝑡))=0, which implies the dynamical error equations as 𝑒_𝑖̇(𝑡)=𝑥̇_𝑖(𝑡)−𝑠(̇𝑡)=𝑓_𝑖𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)−𝑔𝑠(𝑡),𝜑_𝑖(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑘𝑥̇_𝑗(𝑡),𝛽_𝑖(𝑡)−𝑘𝑠(̇𝑡),𝛽_𝑖(𝑡)−𝑠(̇𝑡)+𝑢_𝑖(𝑡). (6)
Our objective is to design a distributed adaptive control law 𝑢_𝑖(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.

Assumption 2 (Lipschits condition). For the system (1), there exist positive constants 𝑙_𝑖>0,𝑖=1,…,𝑁 such that 𝑓_𝑖(𝑥)−𝑓_𝑖𝑦≤𝑙_𝑖𝑥−𝑦,∀𝑥,𝑦∈𝑅^𝑛,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.

Assumption 4 (see [33]). For the unknown continuous functions 𝑔(⋅,⋅),𝑘(⋅,⋅), the following inequality holds: 𝑔(𝑥_𝑗(𝑡),𝜑_𝑖(𝑡))−𝑔𝑠(𝑡),𝜑_𝑖(𝑡)²≤𝑥_𝑗(𝑡)−𝑠(𝑡)²ℎ𝑥_𝑗(𝑡),𝑠(𝑡)𝜆𝜑_𝑖(𝑡),𝑘𝑥̇_𝑗(𝑡),𝛽_𝑖(𝑡)−𝑘𝑠(̇𝑡),𝛽_𝑖(𝑡𝛽)≤𝑀_𝑖(𝑡)𝑥̇_𝑗(𝑡)−𝑠(̇𝑡)≜𝑀_𝑖(𝑡)𝑒_𝑗̇(𝑡), (8) where ℎ(⋅,⋅) is an known nonnegative continuous function 𝜆(⋅) is a unknown nonnegative continuous function, 𝑀(⋅) or 𝑀_𝑖(⋅) is an 𝑛×𝑛 unknown nonnegative continuous function matrix.

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), 𝜑_𝑖(𝑡), 𝛽_𝑖(𝑡) are unknown time-varying periodic functions with a known period 𝑇.

From Assumption 6, it is easy to see that 𝜆(𝜑_𝑖(𝑡)) and each element in 𝑀_𝑖(𝑡) are periodic functions with the same period 𝑇. Suppose 𝜆𝜑_𝑖(𝑡)=𝜙_𝑖(𝑡)+𝜃_𝑖, (9) where 𝜙_𝑖(𝑡) is an unknown continuous periodic function with a known period 𝑇 and 𝜃_𝑖 is an unknown constant parameter.

Assumption 7. In network (1), the inner coupling matrix Γ and ℎ(⋅,⋅) satisfies max‖Γ‖,max𝛾_𝑖≤𝛾,ℎ𝑥_𝑖(𝑡),𝑠(𝑡)<𝐻,𝑖=1,2,…,𝑁, (10) 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.

Lemma 10 (Young’s inequality). For vectors 𝑥, 𝑦∈𝑅^𝑛, and any positive constant 𝑐, the following matrix inequality holds: 𝑥^𝑇𝑦≤𝑐𝑥^𝑇1𝑥+𝑦4𝑐^𝑇𝑦. (11)

Lemma 11. For any vectors 𝑥, 𝑦∈𝑅^𝑛, according to the properties of matrix trace, the following equality holds: 𝑥^𝑇𝑦=tr𝑦𝑥^𝑇=tr𝑥𝑦^𝑇. (12)

Lemma 12 (Barbalat’s Lemma [34]). If 𝑒(𝑡)∈𝐿₂ and 𝑒(̇𝑡)∈𝐿_∞, then lim_𝑡→∞𝑒(𝑡)=0, where 𝐿₂={𝑥∣𝑥∶[0,∞)→𝑅,(∫^∞₀|𝑥(𝑡)|²)^1/2<∞} and 𝐿_∞={𝑥∣𝑥∶[0,∞)→𝑅,sup_𝑡≥0|𝑥(𝑡)|<∞}.

Lemma 13. For a positive constant 𝑇, if ∀𝑡≥0, 𝑓(𝑡) is a continuous function and ∫^𝑡_𝑡−𝑇𝑓(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>0∀𝑘∈[𝜏−𝜀,𝜏+𝜀]⊂[𝑡−𝑇,𝑡],  𝑓(𝑘)>𝑁; then ∫^𝑡_𝑡−𝑇𝑓(𝜏)𝑑𝜏≥∫^𝜏+𝜀_𝜏−𝜀𝑓(𝜏)𝑑𝑡≥2𝜀𝑁. We can choose 𝑁=(𝑀/2𝜀)+1; then ∫^𝑡_𝑡−𝑇𝑓(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 𝑢_𝑖(𝑡)=𝑠(̇𝑡)−𝑓_𝑖(𝑠(𝑡))−𝛾𝐻𝑒_𝑖(𝑡)^𝑁_𝑗=1𝑎²_𝑗𝑖𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)−𝛾^𝑁_𝑗=1𝑏_𝑖𝑗diag_{1≤𝑗≤𝑁}sign𝑒_𝑖𝑗(𝑡)𝑀_𝑖(𝑡)𝑒_𝑗̇(𝑡), (13) where 𝛾, 𝐻 satisfy condition (10), 𝑁 is the node number, and 𝜙_𝑖(𝑡), and  𝜃_𝑖(𝑡), are 𝑀_𝑖(𝑡) used to estimate 𝜙_𝑖(𝑡), 𝜃_𝑖, 𝑀_𝑖(𝑡). |𝑒_𝑗̇(𝑡)|=(|𝑒̇_𝑗1(𝑡)|,…,|𝑒̇_𝑗𝑛(𝑡)|)^𝑇.

The constant parameter distributed update law is designed as follows: ̇𝜃_𝑖(𝑡)=𝑟_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗(𝑡)𝑒_𝑗(𝑡) (14) and the time-varying distributed periodic adaptive learning laws as 𝜙_𝑖(𝑡)=[)𝑞0,𝑡∈−𝑇,0_𝑖0(𝑡)∑^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗(𝑡)𝑒_𝑗(𝑡)[)𝜙,𝑡∈0,𝑇_𝑖(𝑡−𝑇)+𝑞∑_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗(𝑡)𝑒_𝑗(𝑡)[),𝑀,𝑡∈𝑇,+∞_𝑖(𝑡)=[)𝑓0,𝑡∈−𝑇,0_𝑖0(𝑡)∑^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖(𝑡)𝑒^𝑇_𝑗̇(𝑡[)𝑀),𝑡∈0,𝑇_𝑖(𝑡−𝑇)+𝑓∑_𝑖^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖(𝑡)𝑒^𝑇_𝑗̇(𝑡),𝑡∈[𝑇,+∞), (15)(16) where 𝑟_𝑖,𝑞_𝑖, and 𝑓_𝑖 are positive constants and 𝑞_𝑖0(𝑡), 𝑓_𝑖0(𝑡) are continuous and strictly increasing nonnegative functions satisfying 𝑞_𝑖0(0)=0, 𝑞_𝑖0(𝑇)=𝑞_𝑖, and 𝑓_𝑖0(0)=0, 𝑓_𝑖0(𝑇)=𝑓_𝑖 which ensure that 𝜙_𝑖(𝑡),𝑀_𝑖(𝑡) are continuous in 𝑡=𝑖𝑇, 𝑖=1,2,…,𝑁, |𝑒_𝑗(𝑡)|=(|𝑒_𝑗1(𝑡)|,…,|𝑒_𝑗𝑛(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 𝐿²_𝑇 norm sense. All closed-loop signals are bounded. Furthermore, if 𝑘(⋅,⋅) is bounded function, we can obtain globally asymptotical synchronization.

Proof. Define a Lyapunov-Krasovskii-like function as follows: 𝑉(𝑡)=12^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖+𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)+𝑑𝜏𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿²+𝛾2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑓_𝑖⁻¹tr𝑀_𝑖(𝜏)𝑀^𝑇_𝑖(𝜏)[),𝑑𝜏,𝑡∈0,∞ (17) where 𝜙_𝑖(𝑡)=𝜙_𝑖(𝑡)−𝜙_𝑖(𝑡), 𝜃_𝑖(𝑡)=𝜃_𝑖−𝜃_𝑖(𝑡), 𝑀_𝑖(𝑡)=𝑀_𝑖(𝑡)−𝑀_𝑖(𝑡), and 𝐿 is a sufficiently large positive constant which will be determined later.
It should be noted that we define 𝜙_𝑖(𝑡)=𝜙_𝑖(0),  𝑀_𝑖(𝑡)=𝑀_𝑖(0)  𝑡∈[−𝑇,0); then, from the adaptive laws (15) and (16), we get 𝜙²_𝑖(𝑡−𝑇)=𝜙²_𝑖(𝑡−𝑇)=𝜙²_𝑖(0),𝑀_𝑖(𝑡−𝑇)𝑀^𝑇_𝑖(𝑡−𝑇)=𝑀_𝑖(𝑡−𝑇)𝑀^𝑇_𝑖(𝑡−𝑇)=𝑀_𝑖(0)𝑀^𝑇_𝑖(0),𝑡∈[0,𝑇]. (18)
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 𝑉̇(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)−𝑞_𝑖⁻¹𝜙²_𝑖(𝑡−𝑇)+𝛾𝐻^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)̇+𝐿𝜃_𝑖(𝑡)+𝛾2^𝑁_𝑖=1𝑓_𝑖⁻¹tr𝑀_𝑖(𝑡)𝑀^𝑇_𝑖(𝑡)−𝑓_𝑖⁻¹tr𝑀_𝑖(𝑡−𝑇)𝑀^𝑇_𝑖(𝑡−𝑇). (19)
Let us introduce some notations as 𝑓_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖(𝑠(𝑡𝑔𝑥))=Φ,_𝑗(𝑡),𝜑_𝑖(𝑡)−𝑔𝑠(𝑡),𝜑_𝑖(𝑡)𝑘𝑥̇=Λ,_𝑗(𝑡),𝛽_𝑖(𝑡)−𝑘𝑠_𝑗̇(𝑡),𝛽_𝑖(𝑡)=𝐾. (20)
From (6) and (13), the first term on the right hand side of (19) satisfies ^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑓_𝑖𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗+ΓΛ_𝑁𝑗=1𝑏_𝑖𝑗Γ𝐾−𝑠(̇𝑡)+𝑢_𝑖(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖Φ+^𝑁_𝑗=1𝑎_𝑖𝑗𝑒^𝑇_𝑖+ΓΛ_𝑁𝑗=1𝑏_𝑖𝑗𝑒^𝑇_𝑖Γ𝐾−𝛾𝐻^𝑁_𝑗=1𝑎²_𝑗𝑖𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)𝑒^𝑇_𝑖𝑒_𝑖−𝛾^𝑁_𝑗=1𝑏_𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑀_𝑖(𝑡)𝑒_𝑗̇(𝑡). (21)
According to Assumptions 2–7 and Lemma 10, from the above equation, we get ^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)≤^𝑁_𝑖=1𝑐𝑒^𝑇_𝑖𝑒_𝑖+1𝜙4𝑐^𝑇𝜙+^𝑁_𝑗=1𝑑𝑒^𝑇_𝑖𝑒_𝑖+1𝑎4𝑑^2𝑖𝑗Λ^𝑇Γ^𝑇ΓΛ+𝛾^𝑁_𝑗=1𝑏_𝑖𝑗𝑒^𝑇_𝑖|𝐾|−𝛾𝐻^𝑁_𝑗=1𝑎²_𝑗𝑖𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)𝑒^𝑇_𝑖𝑒_𝑖−𝛾^𝑁_𝑗=1𝑏_𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑀_𝑖(𝑡)𝑒_𝑗̇(𝑡≤_𝑁𝑖=1)𝑙𝑐+^2𝑖𝑒4𝑐^𝑇𝑖𝑒_𝑖+𝑁𝑑𝑒^𝑇_𝑖𝑒_𝑖+^𝑁_𝑗=1𝑎²_𝑖𝑗𝜙_𝑖(𝑡)+𝜃_𝑖𝐻𝛾²𝑒4𝑑^𝑇𝑗𝑒_𝑗+𝛾^𝑁_𝑗=1𝑏_𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑀_𝑖(𝑡)𝑒_𝑗̇(𝑡)−𝛾𝐻^𝑁_𝑗=1𝑎²_𝑗𝑖𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)𝑒^𝑇_𝑖𝑒_𝑖−𝛾^𝑁_𝑗=1𝑏_𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑀_𝑖(𝑡)𝑒_𝑗̇(𝑡), (22) where 𝑐, 𝑑 are positive constants. Choosing 𝑐=1/2, 𝑑=𝛾/4, and according to Lemma 11, we have ^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)≤^𝑁_𝑖=1𝑙_𝑖2+12+𝑁𝛾4+𝛾𝐻^𝑁_𝑗=1𝑎²_𝑗𝑖𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)𝑒^𝑇_𝑖𝑒_𝑖+𝛾^𝑁_𝑗=1𝑏_𝑖𝑗𝑒^𝑇_𝑖(𝑡)𝑀_𝑖(𝑡)𝑒_𝑗̇(𝑡=_𝑁𝑖=1)𝑙_𝑖2+12+𝑁𝛾4+𝛾𝐻^𝑁_𝑗=1𝑎²_𝑗𝑖𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)𝑒^𝑇_𝑖𝑒_𝑖+𝛾tr^𝑁_𝑗=1𝑏_𝑖𝑗𝑀_𝑖(𝑡)𝑒_𝑗̇(𝑡)𝑒^𝑇_𝑖(𝑡=_𝑁𝑖=1)𝑙_𝑖2+12+𝑁𝛾4+^𝑁_𝑗=1𝑎²_𝑗𝑖𝛾𝐻𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)𝑒^𝑇_𝑖𝑒_𝑖+𝛾tr^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖(𝑡)𝑒^𝑇_𝑗̇(𝑡)𝑀^𝑇_𝑖(𝑡). (23)
Applying (14), the third term on the right-hand side of (19) satisfies 𝛾𝐻^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)̇+𝐿𝜃_𝑖(𝑡)=−𝛾𝐻^𝑁_𝑖=1𝜃_𝑖(𝑡)+𝐿^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗. (24)
Let us focus on the second and fourth terms on the right-hand side of (19). In the interval [𝑇₁,𝑇), since 𝑞_𝑖0(𝑡),𝑓_𝑖0(𝑡) are continuous and strictly increasing functions, 𝑞_𝑖⁻¹≤𝑞⁻¹_𝑖0(𝑡)<∞, 𝑓_𝑖⁻¹≤𝑓⁻¹_𝑖0(𝑡)<∞ are ensured, we obtain 𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)−𝑞_𝑖⁻¹𝜙²_𝑖(=𝑡−𝑇)𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)−𝑞_𝑖⁻¹𝜙²_𝑖(0≤)𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)=𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)+2𝜙²_𝑖(𝑡)−2𝜙_𝑖(𝑡)𝜙_𝑖(𝑡)−𝜙²_𝑖(𝑡≤)𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)+2𝜙²_𝑖(𝑡)−2𝜙_𝑖(𝑡)𝜙_𝑖(𝑡=)𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)−𝛾𝐻^𝑁𝑁_𝑖=1𝑗=1𝑎²_𝑖𝑗𝜙_𝑖(𝑡)𝑒^𝑇_𝑗𝑒_𝑗. (25)
According to Lemma 11, we can get tr𝑀_𝑖(𝑡)𝑀^𝑇_𝑖(𝑡𝑀)=tr_𝑖−𝑀_𝑖(𝑡)𝑀_𝑖−𝑀_𝑖(𝑡)^𝑇=tr𝑀_𝑖𝑀^𝑇_𝑖−𝑀_𝑖𝑀^𝑇_𝑖−𝑀_𝑖𝑀^𝑇_𝑖+𝑀_𝑖𝑀^𝑇_𝑖=tr𝑀_𝑖𝑀^𝑇_𝑖+2𝑀_𝑖𝑀^𝑇_𝑖−𝑀_𝑖𝑀^𝑇_𝑖−𝑀_𝑖𝑀^𝑇_𝑖−𝑀_𝑖𝑀^𝑇_𝑖≤tr𝑀_𝑖𝑀^𝑇_𝑖+2𝑀_𝑖𝑀^𝑇_𝑖−𝑀_𝑖𝑀^𝑇_𝑖−𝑀_𝑖𝑀^𝑇_𝑖=tr𝑀_𝑖𝑀^𝑇_𝑖+2tr𝑀_𝑖𝑀^𝑇_𝑖−𝑀_𝑖𝑀^𝑇_𝑖=tr𝑀_𝑖𝑀^𝑇_𝑖−2tr𝑀_𝑖𝑀^𝑇_𝑖. (26)
Then from (26), the last term on the right-hand side of (19) satisfies 𝛾2^𝑁_𝑖=1𝑓_𝑖⁻¹tr𝑀_𝑖(𝑡)𝑀^𝑇_𝑖(𝑡)−𝑓_𝑖⁻¹tr𝑀_𝑖(𝑡−𝑇)𝑀^𝑇_𝑖(≤𝛾𝑡−𝑇)2^𝑁_𝑖=1𝑓_𝑖⁻¹tr𝑀_𝑖(𝑡)𝑀^𝑇_𝑖(𝑡≤𝛾)2^𝑁_𝑖=1𝑓⁻¹_𝑖0tr𝑀_𝑖𝑀^𝑇_𝑖−2tr𝑀_𝑖𝑀^𝑇_𝑖=𝛾2^𝑁_𝑖=1𝑓⁻¹_𝑖0tr𝑀_𝑖𝑀^𝑇_𝑖−𝛾^𝑁_𝑖=1tr𝑓⁻¹_𝑖0𝑀_𝑖𝑀^𝑇_𝑖=𝛾2^𝑁_𝑖=1𝑓⁻¹_𝑖0tr𝑀_𝑖𝑀^𝑇_𝑖−𝛾^𝑁_𝑖=1tr^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖(𝑡)𝑒^𝑇_𝑗̇(𝑡)𝑀^𝑇_𝑖(𝑡). (27)
Substituting (23)–(27) into (19) we obtain 𝑉̇(𝑡)≤^𝑁_𝑖=1𝑙_𝑖2+12+𝑁𝛾4−𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖𝑒^𝑇_𝑖𝑒_𝑖+𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0(𝑡)𝜙²_𝑖(𝑡)+𝛾2^𝑁_𝑖=1𝑓⁻¹_𝑖0tr𝑀_𝑖𝑀^𝑇_𝑖 (28) It is obvious that there exist sufficiently large positive constants 𝐿 such that 12+12𝑙²_𝑖+𝑁𝛾4−𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖<0,𝑖=1,…,𝑁. (29) According to (28) we have 𝑉̇(𝑡)≤𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0(𝑡)𝜙²_𝑖(𝑡)+𝛾2^𝑁_𝑖=1𝑓⁻¹_𝑖0tr𝑀_𝑖𝑀^𝑇_𝑖,𝑡∈𝑇₁,𝑇. (30)
For ∀𝑡∈[𝑇₁,𝑇), since 𝜙_𝑖(𝑡) is continuous and periodic and every element in matrix 𝑀_𝑖(⋅) is continuous function, the boundedness of them can be obtained. The boundedness of 𝜙_𝑖(𝑡) and tr(𝑀_𝑖𝑀^𝑇_𝑖) leads to the boundedness of 𝑉̇(𝑡). That is, 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 (𝑡)(𝑡)()1Δ𝑉=𝑉−𝑉𝑡−𝑇=−2^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡−𝑇)𝑒_𝑖()+𝑡−𝑇𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿²−𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡−𝑇)+𝐿²+𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)−𝑞_𝑖⁻¹𝜙²_𝑖(+𝛾𝜏−𝑇)𝑑𝜏2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑓_𝑖⁻¹tr𝑀_𝑖(𝜏)𝑀^𝑇_𝑖(𝜏)−tr𝑀_𝑖(𝜏−𝑇)𝑀^𝑇_𝑖(𝜏−𝑇)𝑑𝜏. (31)
With Newton-Leibniz’s formula, the first two terms on the right-hand side of (31) satisfied 12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡−𝑇)𝑒_𝑖()=𝑡−𝑇_{𝑁𝑖=1𝑡𝑡−𝑇}𝑒^𝑇_𝑖(𝜏)𝑒_𝑖̇(𝜏)≤𝑑𝜏_{𝑁𝑖=1𝑡𝑡−𝑇}12+12𝑙²_𝑖+𝑁𝛾4𝑒^𝑇_𝑖(𝜏)𝑒_𝑖(𝜏)+𝑑𝜏_{𝑁𝑁𝑖=1𝑗=1𝑡𝑡−𝑇}𝑎²_𝑗𝑖𝛾𝐻𝑒^𝑇_𝑖(𝜏)𝑒_𝑖(𝜏)𝜙_𝑗(𝜏)+𝜃_𝑗(𝜏)+𝑑𝜏_{𝑁𝑡𝑖=1𝑡−𝑇}𝑏_𝑖𝑗𝛾tr^𝑁_𝑗=1𝑒_𝑖(𝜏)𝑒^𝑇_𝑗̇(𝜏)𝑀^𝑇_𝑖(𝜏)𝑑𝜏. (32)
Using the algebraic relation (𝑎−𝑏)^𝑇𝐺(𝑎−𝑏)−(𝑎−𝑐)^𝑇𝐺()=()𝑎−𝑐𝑐−𝑏^𝑇𝐺[2(𝑎−𝑏)+(𝑏−𝑐)], (33) where 𝑎, 𝑏, 𝑐∈𝑅^𝑝, 𝐺∈𝑅^𝑝×𝑝.
After choosing 𝐺=1,  𝑎=𝜙_𝑖(𝜏), 𝑏=𝜙(𝜏), 𝑎−𝑏=𝜙_𝑖(𝜏), 𝜙_𝑖(𝜏)=𝜙_𝑖(𝜏−𝑇), 𝑐=𝜙_𝑖(𝜏−𝑇), and 𝑎−𝑐=𝜙_𝑖(𝜏−𝑇), we can get 𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)−𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖()=𝜏−𝑇𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹2𝜙_𝑖(𝜏)+𝜙_𝑖(𝜏)−𝜙_𝑖(×𝜙𝜏−𝑇)_𝑖(𝜏−𝑇)−𝜙_𝑖(𝜏)=−𝛾𝐻^𝑁_𝑖=1𝜙_𝑖(𝜏)+12𝑞_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗^𝑁_𝑘=1𝑎²_𝑖𝑘𝑒^𝑇_𝑘𝑒_𝑘. (34)
From Lemma 11, we have tr𝑀_𝑖(𝜏)𝑀^𝑇_𝑖(𝜏)−tr𝑀_𝑖(𝜏−𝑇)𝑀^𝑇_𝑖(𝑀𝜏−𝑇)=tr_𝑖(𝜏)𝑀^𝑇_𝑖(𝜏)−𝑀_𝑖(𝜏−𝑇)𝑀^𝑇_𝑖(𝜏)+𝑀_𝑖(𝜏−𝑇)𝑀^𝑇_𝑖(𝜏)−𝑀_𝑖(𝜏−𝑇)𝑀^𝑇_𝑖()𝑀𝜏−𝑇=tr_𝑖(𝜏)−𝑀_𝑖(𝜏−𝑇)𝑀^𝑇_𝑖(𝜏)+𝑀_𝑖(𝜏−𝑇)𝑀^𝑇_𝑖(𝜏)−𝑀^𝑇_𝑖(𝑀𝜏−𝑇)=tr_𝑖(𝜏)−𝑀_𝑖(𝜏−𝑇)𝑀^𝑇_𝑖(𝜏)+𝑀^𝑇_𝑖(𝑀𝜏−𝑇)=tr_𝑖(𝜏)−𝑀_𝑖(×2𝑀𝜏−𝑇)_𝑇𝑖(𝜏)+𝑀^𝑇_𝑖(𝜏−𝑇)−𝑀^𝑇_𝑖(𝜏). (35)
According to (34) and (35), the last two terms on the right-hand side of (31) satisfy 𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)−𝑞_𝑖⁻¹𝜙²_𝑖(𝜏−𝑇)𝑑𝜏=−𝛾𝐻^𝑁_𝑖=1^𝑡_𝑡−𝑇𝜙_𝑖(𝜏)+12𝑞_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗×^𝑁_𝑘=1𝑎²_𝑖𝑘𝑒^𝑇_𝑘𝑒_𝑘𝛾𝑑𝜏,2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑓_𝑖⁻¹×tr𝑀_𝑖(𝜏)𝑀^𝑇_𝑖(𝜏)−tr𝑀_𝑖(𝜏−𝑇)𝑀^𝑇_𝑖()=𝛾𝜏−𝑇𝑑𝜏2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑓_𝑖⁻¹×tr𝑓_𝑖^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖𝑒^𝑇_𝑗̇×2𝑀^𝑇_𝑖(𝜏)−𝑓_𝑖^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖𝑒̇^𝑇_𝑗^𝑇𝑑𝜏=−𝛾^𝑁_𝑖=1^𝑡_𝑡−𝑇tr^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖𝑒^𝑇_𝑗̇𝑀^𝑇_𝑖(𝜏)−𝛾𝑑𝜏2^𝑁_𝑖=1^𝑡𝑁_{𝑡−𝑇𝑖=1}𝑓_𝑖tr^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖𝑒^𝑇_𝑗̇×^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖𝑒̇^𝑇_𝑗^𝑇𝑑𝜏. (36)(37)
The other terms of right-hand side of (31) can be simplified as follows: 𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿²−𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡−𝑇)+𝐿²=𝛾𝐻^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑟_𝑖⁻¹𝜃_𝑖(𝜏)̇+𝐿𝜃_𝑖(𝜏)𝑑𝜏=−𝛾𝐻^𝑁_𝑖=1^𝑡_𝑡−𝑇𝜃_𝑖(𝜏)+𝐿^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗(𝜏)𝑒_𝑗(𝜏)𝑑𝜏. (38)
Substituting (32) and (36)–(38) into (31), we can attain Δ𝑉(𝑡)≤^𝑁_𝑖=1^𝑡_𝑡−𝑇12+12𝑙²_𝑖+𝑁𝛾4−𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖𝑒^𝑇_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏. (39)
Choosing 1/2+(1/2)𝑙²_𝑖+𝑁𝛾/4−𝐿∑^𝑁_𝑗=1𝑎²_𝑗𝑖<0,𝑖=1,…,𝑁, we can obtain Δ𝑉(𝑡)<0. (40)
For any 𝑡∈[𝑙𝑇,(𝑙+1)𝑇],𝑙=1,2,…,  and denoting 𝑡₀=𝑡−𝑙𝑇,𝑡₀∈[0,𝑇), we have 𝑉(𝑡)=𝑉𝑡₀+^𝑙−1_𝑗=0Δ𝑉𝑡−𝑗𝑇. (41)
Considering 𝑡₀∈[0,𝑇) and the positive of 𝑉(𝑡), according to (41), we obtain 𝑉(𝑡)<max_{𝑡0∈[0,𝑇)}𝑉𝑡₀−^𝑙−1𝑁_{𝑗=0𝑖=1𝑇}^{𝑡−𝑗𝑇}_{𝑡−𝑗+1}𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖−12−12𝑙²_𝑖−𝑁𝛾4×𝑒^𝑇_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏. (42)
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 𝐿²_𝑇 norm. That is to say, we have lim_𝑡→∞^𝑡_𝑡−𝑇𝑒^𝑇_𝑖(𝜏)𝑒_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 𝑉̇(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)−𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡−𝑇)+𝛾𝐻^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)̇+𝐿𝜃_𝑖(𝑡)+𝛾2𝑓_𝑖⁻¹tr𝑀_𝑖(𝑡)𝑀^𝑇_𝑖(𝑡)−tr𝑀_𝑖(𝑡−𝑇)𝑀^𝑇_𝑖(𝑡−𝑇)≤−𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗²−^𝑁_𝑖=1𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖−12−12𝑙²_𝑖−𝑁𝛾4𝑒^𝑇_𝑖𝑒_𝑖−𝛾2^𝑁_𝑖=1𝑓_𝑖tr^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖𝑒^𝑇_𝑗̇×^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖𝑒̇^𝑇_𝑗^𝑇. (44)
Having the two sides of (44) integration, one can obtain 𝑉(𝑡)≤𝑉(𝑇)−𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑇𝑞_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗²−𝑑𝜏_{𝑁𝑖=1𝑡𝑇}𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖−12−12𝑙²_𝑖−𝑁𝛾4𝑒^𝑇_𝑖𝑒_𝑖−𝛾𝑑𝜏2^𝑁_𝑖=1𝑓_𝑖^𝑡_𝑡−𝑇tr^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖𝑒^𝑇_𝑗̇×^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑖𝑒̇^𝑇_𝑗^𝑇𝑑𝜏. (45)
Choosing (𝐿>1/2+((1/2)𝑙²_𝑖)+𝑁𝛾/4)/∑^𝑁_𝑗=1𝑎²_𝑗𝑖,𝑖=1,…,𝑁, we have 𝑉(𝑡)<𝑉(𝑇). (46)
From the boundedness of 𝑉(𝑡) and (17), we can conclude that 𝑒_𝑖, ∫^𝑡_𝑡−𝑇𝜙²_𝑖(𝜏)𝑑𝜏,𝜃_𝑖(𝑡) For 𝜃_𝑖 is constant and ∫^𝑡_𝑡−𝑇tr(𝑀_𝑖(𝜏)𝑀^𝑇_𝑖(𝜏))𝑑𝜏 are all bounded., it implies the boundedness of 𝜃_𝑖(𝑡). According to Lemma 13 and the continuity of 𝜙_𝑖(𝑡) and 𝑀_𝑖(𝑡), the boundedness of 𝜙_𝑖(𝑡) and 𝑀_𝑖(𝑡) are obviously obtained. As 𝜙_𝑖(𝑡) is a continuous periodic function and 𝑀_𝑖(𝑡) is continuous periodic matrix, we can get that 𝜙_𝑖(𝑡),𝑀_𝑖(𝑡) are bounded. According to (1), the boundedness of the control input 𝑢_𝑖(𝑡) is obtained. Since 𝑒_𝑖(𝑡) is bounded, the boundedness of 𝑥_𝑖(𝑡) is received. The proof is completed.

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

4. Simulation Example

To demonstrate the theoretical result obtained in Section 3, the following dynamical network with six non-identical nodes is considered: 𝑥̇_𝑖(𝑡)=𝑓_𝑖𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γexp−𝜑_𝑖(𝑡)𝑥²_𝑗1(𝑡)𝑥²_𝑗2(𝑡)𝑥²_𝑗3(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ()()()−𝑥̇expsin2𝜋𝑡sin2𝜋𝑡sin2𝜋𝑡0.1cos_𝑗1(𝑡)()𝑥̇sin2𝜋𝑡0.1cos_𝑗2(𝑡)()𝑥̇sin2𝜋𝑡0.1cos_𝑗3(𝑡)sin(2𝜋𝑡)+𝑢_𝑖(𝑡),𝑖=1,…,6, (47) where 𝑓₁𝑥₁=−2.5𝑥₁₁+0.3𝑥₁₂+0.9𝑥₁₃+3𝑥²₁₃0.6𝑥₁₁−2.6𝑥₁₂+3𝑥²₁₂+𝑥₁₃−2.8𝑥₁₁+sin𝑥₁₁cos𝑥₁₂−2.2𝑥₁₃,𝑓₂𝑥₂=−2.5𝑥₂₁+𝑥₂₂+𝑥₂₁𝑥₂₂+𝑥₂₃0.5𝑥₂₁−0.8𝑥₂₂−sin𝑥₂₂+𝑥₂₃−𝑥²₂₃−2𝑥₂₁−0.8𝑥₂₃−0.5sin2𝑥₂₃,𝑓₃𝑥₃=−2.5𝑥₃₁+𝑥₃₁sin𝑥₃₁+𝑥₃₂+1.4𝑥₃₃0.5𝑥₃₁−2.5𝑥₃₂+𝑥₃₃−2𝑥₃₁+𝑥₃₂𝑥₃₃−0.6𝑥₃₃−𝑥₃₃cos𝑥₃₃,𝑓₄𝑥₄=−2.5𝑥₄₁+1.5𝑥₄₂+1.5𝑥₄₃𝑥₄₁−2.5𝑥₄₂+𝑥²₄₂+1.5𝑥₄₃+𝑥²₄₃−2𝑥₄₁+𝑥₄₁𝑥₄₃−2.5𝑥₄₃,𝑓₅𝑥₅=−2.1𝑥₅₁+0.5𝑥²₅₁+1.4𝑥₅₂+𝑥₅₃0.9𝑥₅₁−2.1𝑥₅₂+𝑥₅₂𝑥₅₃+𝑥₅₃−2𝑥₅₁+0.4𝑥₅₂−2.1𝑥₅₃,𝑓₆𝑥₆=−3.2𝑥₆₁+10𝑥₆₂+2.95𝑥₆₁+1−𝑥₆₁𝑥−1₆₁−𝑥₆₂+𝑥₆₃−14.7𝑥₆₂,,,.Γ=100010001𝐴=−3021002−5002103−4001110−5210013−4001110−3𝐵=−3201000−3001201−3200102−4010011−2020001−3 (48)

Nonlinearly parameterized function satisfies exp−𝜑_𝑖(𝑡)𝑥²_𝑗1(𝑡)𝑥²_𝑗2(𝑡)𝑥²_𝑗3(𝑡)−exp−𝜑_𝑖(𝑡)𝑠²(𝑡)𝑠²(𝑡)𝑠²(𝑡)²≤𝑒_𝑗(𝑡)²2𝜑_𝑖(𝑡)(),()()()−𝑥̇exp−1expsin2𝜋𝑡sin2𝜋𝑡sin2𝜋𝑡0.1cos_𝑗1(𝑡)()𝑥̇sin2𝜋𝑡0.1cos_𝑗2(𝑡)()𝑥̇sin2𝜋𝑡0.1cos_𝑗3(𝑡)()−()()()+𝑠̇sin2𝜋𝑡expsin2𝜋𝑡sin2𝜋𝑡sin2𝜋𝑡0.1cos_𝑗1(𝑡)()𝑠̇sin2𝜋𝑡0.1cos_𝑗2(𝑡)()𝑠̇sin2𝜋𝑡0.1cos_𝑗3(𝑡)()≤𝑥̇sin2𝜋𝑡_𝑗(𝑡)−𝑠_𝑗̇(𝑡). (49)

We choose 𝑠(̇𝑡)=𝑓₆(𝑠(𝑡)), and the parameters are selected as follows: 𝜙𝑁=6,𝑇=0.5,𝛾=1,𝐻=5,₁(𝑡)=0.2sin8𝜋𝑡+2,𝜙₂(𝑡)𝜙=2cos4𝜋𝑡+2,₃(𝑡)=−sin8𝜋𝑡+2,𝜙₄(𝑡)𝜙=cos8𝜋𝑡+2,₅(𝑡)=−2sin8𝜋𝑡+2,𝜙₆(𝑡)()=2sin4𝜋𝑡+2,𝜃=1,2,3,1.1,1.5,1.3^𝑇. (50)

In the following simulations, we choose 𝑞₁=1,𝑞₂=3,𝑞₃=2,𝑞₄𝑞=5,₅=1,𝑞₆=2,𝑞₁₀(𝑡)=2𝑡𝑞₁,𝑞₂₀(𝑡)=2𝑡𝑞₂,𝑞₃₀(𝑡)=2𝑡𝑞₃,𝑞₄₀(𝑡)=2𝑡𝑞₄,𝑞₅₀(𝑡)=2𝑡𝑞₅,𝑞₆₀(𝑡)=2𝑡𝑞₆. (51)

The initially estimated values of the unknown parameters are 𝜙(0)=(111111)^𝑇,𝜃(0)=(1,2,3,1.1,1.5,0)^𝑇, (52) and the initial states are chosen as 𝑥₁=[0.010.030.02]^𝑇,𝑥₂=[0.030.020.03]^𝑇,𝑥₃=[0.040.020]^𝑇,𝑥₄=[00.050.03]^𝑇,𝑥₅=[0.100.03]^𝑇,𝑥₆=[0.10.010]^𝑇,𝑠=[0.10.50]^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.

Figure 1 

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

Figure 2 

The curve of control 𝑢𝑖(𝑡) with 𝑖=1,…,6.

Figure 3 

The curve of control 𝜃𝑖 with 𝑖=1,…,6.

Figure 4 

The evaluations of parameters 𝜙𝑖 with 𝑖=1,…,6.

Figure 5 

The evaluations of parameters 𝑀1𝑖(𝑡).

Remark 15. It is not difficult to draw the evolution of other elements in parameter 𝑀_𝑖(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 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.

Conflict of Interests

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