We investigate the problem of cluster anticonsensus of multiagent systems. For multiagent continuous systems, a new control protocol is designed based on the

Recently, multiagent systems have attracted much attention in various disciplines, such as mathematical, physical, biological, and social sciences [

On the other hand, synchronization of chaotic systems and complex networks has sparked the interest of many researchers. Many different types of synchronization phenomena have been observed such as complete synchronization, generalized synchronization, lag synchronization, antisynchronization, and cluster synchronization [

Very recently the signless Laplacian has attracted the attention of researchers. Several papers on the signless Laplacian spectrum have been reported since 2005 and a new spectral theory of graphs which is called the

In this paper, we investigate the problem of cluster anticonsensus of multiagent systems based on the

This paper is organized as follows. In Section

Throughout this paper, the superscripts “−1” and “T” stand for the inverse and transpose of a matrix, respectively;

In this section, we provide some results in the

An undirected graph

A weighted adjacency matrix

Let

From (

The least eigenvalue of the signless Laplacian of a connected graph is equal to 0 if and only if the graph is bipartite. In this case 0 is a simple eigenvalue.

Let

For a connected graph

In this section, we investigate the cluster anticonsensus problem for multiagent continuous systems.

Here, we consider a continuous system consisting of

Different from the traditional control protocol by the Lapalacian matrix [

Then, under the control protocol (

Let

For the system (

From Definition

Now we develop the cluster anticonsensus results of the system (

Consider the system (

Consider the Lyapunov function candidate

Taking the derivative of

Obviously,

Therefore, the cluster anticonsensus is achieved under the control protocol (

Observe that

In this section, we consider the cluster anticonsensus of multiagent discrete-time systems.

Here, we consider a discrete-time system consisting of

Similar to (

Then, under the control protocol (

Let

For the system (

Now sufficient conditions which guarantee the cluster anticonsensus of the system (

Consider the system (

Consider the Lyapunov function candidate

Then, we have

Let

Therefore, the cluster anticonsensus is achieved under the control protocol (

Observe that

Since

In this section, two numerical examples are provided to show the effectiveness of our theoretical results.

Here we consider a system consisting of 11 agents indexed by

The control input of agent

The signless Laplacian eigenvalues of the Herschel graph are

Herschel graph: original style.

Herschel graph: bipartite style.

The time histories of

Here we consider a system consisting of 30 agents indexed by

The control input of agent

Levi graph: original style.

Levi graph: bipartite style.

The histories of

In this paper, we investigate the problem of cluster anticonsensus of multiagent systems based on the

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

The authors are grateful to the anonymous reviewers for their valuable comments and suggestions that have helped to improve the presentation of the paper. This work is partially supported by the National Natural Science Foundation of China (Grants nos. 11202180, 61273106, and 11171290), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant no. 10KJB510026), and the Jiangsu Overseas Research and Training Program for University Prominent Young and Middle-aged Teachers and Presidents.