This paper is concerned with the problem of sampled-data synchronization for complex dynamical networks (CDNs) with time-varying coupling delay and random coupling strengths. The random coupling strengths are described by normal distribution. The sampling period considered here is assumed to be less than a given bound. By taking the characteristic of sampled-data system into account, a discontinuous Lyapunov functional is constructed, and a delay-dependent mean square synchronization criterion is derived. Based on the proposed condition, a set of desired sampled-data controllers are designed in terms of linear matrix inequalities (LMIs) that can be solved effectively by using MATLAB LMI Toolbox. Numerical examples are given to demonstrate the effectiveness of the proposed scheme.

In the real world, many practical and natural systems can be described by models of complex networks such as internet, food webs, electric power grids, scientific citation networks, and social networks. Therefore, a dynamical network can be regarded as a dynamical system with a special structure. In the last few years, complex dynamical networks (CDNs) have received extensive attention and increasing interest across many fields of science and engineering [

It is very common that many natural systems often exhibit collective cooperative behaviors among their constituents. Synchronization, as a kind of typical collective behavior, is one of key issues in the study of complex dynamical networks. The main reason is that network synchronization not only can explain many natural phenomena but also has wide applications in many fields including secure communications, synchronous information exchange in the internet, genetic regulatory process, the synchronous transfer of digital signals in communication networks, and so on. Over the past several decades, the synchronization in CDNs has been intensively investigated from various fields such as sociology, biology, and physics [

It is well known that the coupling strength of complex dynamical network plays an important role in the realizing synchronization. In general, the coupling strength of the considered CDNs is deterministic [

On the other hand, the sampled-data control system, whose control signals are allowed to change only at discrete sampling instants, can drastically reduce the amount of information transmitted and increase the efficiency of bandwidth usage. Therefore, sampled-data control has received notable attention [

Motivated by the aforementioned discussions, in this paper, the problem of sampled-data synchronization of CDNs with time-varying coupling delay and random coupling strengths is investigated. The sampling period is assumed to be time varying but less than a given bound. The random coupling strengths are described by normal distribution. By capturing the characteristic of sampled-data control system, a new discontinuous Lyapunov functional is constructed. By using the low bound lemma and convex combination approach, a mean square synchronization condition is formulated in terms of LMIs. The corresponding sampled-data controllers are designed that can achieve the synchronization of the considered CDNs. The proposed method can lead to a less conservative result than the existing ones. Finally, numerical examples are given to illustrate the effectiveness and less conservatism of the presented sampled-data control scheme.

Consider a class of linearly coupled complex dynamical networks consisting of

The coupling configuration matrix

In this paper, similar to [

We assume the coupling strengths satisfy the normal distribution, which can randomly vary around some given intervals. This is very different from the considered network models in [

There nonlinear function

The mathematical exception and variance of

On the basis of the property of variables

Let

The control signal is assumed to be generalized by using a zero-order-hold (ZOH) function with a sequence of hold times

By substituting (

Furthermore, by using the Kronecker product, system (

To proceed further, the following definition and useful lemmas are needed.

The coupled complex dynamical network (

Let

Let

The aim of this paper is to design a set of sampled-data controllers (

In this section, by considering the characteristic of sampled-data system, we first give a delay-dependent condition to ensure error system (

Under Assumptions

Consider the following Lyapunov functional:

It is clear that at any

Define the infinitesimal operator

Taking the derivative of (

If (

On the other hand, the following inequality is true for any matrix

Let

In addition, based on Assumption

Noting that

Inspired by [

Next, we will consider how to design the desired sampled-data controllers. Based on Theorem

Under Assumptions

Define matrices

If

Based on Theorems

Under Assumption

Under Assumption

Since the characteristic of sampled-data control system is fully considered, the conservatism of Corollary

It is worth pointing out that the main result here can be extended to some more general complex dynamical networks with probabilistic time-varying coupling delay [

In this section, two numerical examples are given to show the validity of the proposed results.

Consider complex network model (

Furthermore, choosing

In the numerical simulation, assume that the initial values are

Maximum sampling interval

| ||
---|---|---|

0.5 | 0.75 | |

[ |
0.5409 | 0.1653 |

[ |
0.5573 | 0.2277 |

[ |
0.9016 | 0.8957 |

Corollary |
0.9795 | 0.9121 |

Synchronization error states.

Sampled-data control inputs.

The isolated node of the dynamical networks and the coupling delay are the same as Example

Let

Random coupling strength

Random coupling strength

Synchronization error states.

Sampled-data control inputs.

In this paper, the sampled-data synchronization problem has been considered for a kind of complex dynamical networks with time-varying coupling delay and random coupling strengths. The sampling period and random coupling strengths considered here are assumed to be time varying but bounded and to obey normal distribution, respectively. By capturing the characteristic of sampled-data system, a novel discontinuous Lyapunov functional is defined. By using reciprocally convex approach and convex combination technique, a mean square synchronization criterion is proposed based on LMIs. The corresponding desired sampled-data controllers are designed. Numerical examples show the effectiveness of the proposed result.

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

The work is Supported by the National Natural Science Foundation of China (Grant nos. 61203049 and 61303020), the Doctoral Startup Foundation of Taiyuan University of Science and Technology (Grant no. 20112010), and Shanxi Education Department Foundation (Grant no. 20121068).