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A pinning stabilization problem of complex networks with time-delay coupling is studied under stochastic noisy circumstances in this paper. Only one controller is used to stabilize the network to the equilibrium point when the network is connected and the minimal number of controllers is used when the network is unconnected, where the structure of complex network is fully used. Some criteria are achieved to control the complex network under stochastic noise in the form of linear matrix inequalities. Several examples are given to show the validity of the proposed control criteria.

Complex networks have been a major research topic and attracted increasing attention from various fields including physics, biology, sociology, and engineering. Many real phenomena can be described as complex networks, such as the World Web, telephone call graphs, and social organization. Recently, the stabilization and synchronization problem and stabilization problem of complex network have become more and more important.

In fact, the synchronization problem is a special stabilization one since it can be converted into the stabilization problem of the error system between the complex network and the synchronization manifold [

However, it is true that the inner coupling strength sometimes cannot be adjustable for a complex network. Consequently, the whole network cannot be synchronized or stabilized by itself [

All of the above-mentioned contributions focus on the complex networks coupled with no time delaying. However, there often exists time delaying when the signals are transmitted over networks [

Motivated by the above discussion, we study the pinning stabilization problem on complex networks coupled with time delaying and disturbed by the stochastic noise from three kinds of topology matrices: symmetrical and irreducible, asymmetrical and irreducible, and

The rest of this paper will be organized as follows. In Section

Consider a general complex dynamical network consisting of

For the function

Network (

The function

The noise intensity function matrix

Condition (

The inner coupling matrix

Assumption

The following definitions and lemmas are required for the derivation of our main results in this paper.

Matrix

The pinning controlled network (

For an

Consider a reducible matrix

Let

For an

let

suppose that

For an

For any vectors

If matrix

In this section, the pinning controllers are designed for complex network according to the structure of the network topologies (resp., symmetric irreducible, asymmetric irreducible, and

If the coupling matrix

From (

Suppose that Assumptions

Consider using the following Lyapunov functional:

From Lemma

Noticing that

From Assumption

Equality (

From (

Taking the mathematical expectation of both sides of (

Since

By Definition

If the coupling matrix

From Definition

Suppose that Assumptions

Consider using the following Lyapunov functional:

From Lemma

Noticing that

From Assumption

From (

Since

Taking the mathematical expectation of both sides of (

By following the proof of Theorem

If the coupling matrix

From Definition

By referring to [

From [

Then, we can obtain the error networks by using the controllers (

Let matrix

Then, the error subnetworks (

Let matrices

Every subnetwork can be seen as a network with symmetric irreducible coupling matrix

Suppose that Assumptions

For

Therefore complex network (

In this section, two examples are given to show the effectiveness of the proposed synchronization scheme. The networks are composed of ten coupled nodes, and each node satisfies the chaotic Lorenz system as its dynamics. A chaotic Lorenz system can be described as follows:

The attractor of the three states of the first nodes.

We consider a complex network consisting of ten nodes which are described by Lorenz system. Assume that the network is coupled with symmetric and irreducible matrix

In simulation, we assume that the coupling delay of the network is

The states of complex network with a symmetric irreducible coupling matrix

We assume that the network is composed of ten Lorenz nodes coupling with asymmetric and irreducible matrix

The states of complex network coupled with an asymmetic and irreducible matrix.

A complex network is coupled with matrix

In simulation, the other conditions are consistent with Example

The states of complex networks with an

It can be inferred that the stabilization criteria obtained in Theorems

The stabilization problem of time-delayed coupling complex networks with a stochastic perturbation by utilizing pinning controllers is studied in this paper. A minimal number of controllers are designed to force the states of complex networks to the origin point in mean square by fully using the structure of the network topology matrix. Some stabilization criteria are achieved to pinning control the complex network, which are described in the form of LMIs. Some examples are given to show the effectiveness of the proposed pinning controller in this paper.

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

This work was supported in part by the National Natural Science Foundation of China under 61104103, 61203028, and 61374180, the Natural Science Foundation for Colleges and Universities in Jiangsu Province, China, under 10KJB120001, and Climbing Program of Nanjing University of Telecommunications & Posts, China, under NY210013 and NY210014.