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This paper studies the pinning controllability of directed complex delayed dynamical networks by using periodic intermittent control scheme. The general and low-dimensional pinning synchronization criteria are derived to illustrate the design of periodic intermittent control scheme. According to our low-dimensional pinning criterion, especially, the constraint condition of coupling strength is obtained when the network structure and amounts of pinned nodes are fixed. An algorithm is presented to determine the amounts of periodically intermittent controllers and locate these intermittent controllers in a directed network, in which the significance of nodes out- (in-) degree in pinning control of complex network is also illustrated. Finally, a directed network consisting of 12 coupled delayed Chua oscillators is designed as numerical example to verify the effectiveness of the theoretical analysis.

Complex network has received increasing attention and has been intensively investigated in recent years due to its wide applications in many fields, such as the Internet, World Wide Web, social networks, citation network, and electrical power grids. The latest development of complex network focuses on the transition from a regular network to random network, in which the small-world and scale-free networks are mileposts in the field of complex network. Due to the pioneering work of Pecora and Carroll [

In the case where the network cannot synchronize by itself, some appropriate controllers are inevitably required to regulate the network to a desired state. Correspondingly, many novel control techniques [

Motivated by above novel results, we keep on studying pinning synchronization of a directed complex delayed network by utilizing the periodically intermittent control. In this paper, our attention is directed to delayed dynamical network and its synchronized criteria. The least number of pinned nodes can be determined under the given network configuration and coupling strength by the proposed synchronized criteria; on the other hand, the two constraints for coupling strength are provided by the network configuration and number of controllers. As a result, we propose a pinned-node selection algorithm, in which the amounts, type of pinned nodes, and feedback gain of controllers are determined in detail. Meanwhile, we point out especially that the node with zero in-degree must be selected as candidate.

The rest of this paper is organized as follows. In Section

Throughout this paper, the following standard notations are used.

To derive our main results, the following lemmas are essential.

Let

Let

Suppose that

This paper is devoted to a general delayed dynamical complex network consisting of

Let

The objective of control is to design the appropriate parameters

Denoting the error vectors by

To derive the synchronized criteria for pinning controlled network (

There exist two constant matrices

According to Assumption

Denote that

To realize the pinning synchronization of the controlled network (

Suppose that Assumptions

The proof is given in Appendix.

It should be noted that the outer parameters

For practical application, we are going to present some low-dimensional and convenient conditions to ensure global synchronization of the pinning process. Construct a symmetric matrix

According to the above analyses, we derive the following applicable corollary.

Suppose that Assumptions

Note that

In the case without time delayed term for network (

According to the processes of proving Theorem

It is well known that the undirected network can be regarded as a directed network with completely symmetric coupling matrix, that is,

In Corollary

Up to this point, we have presented the pinning synchronous criteria for complex delayed network (

For the fixed coupling strength and relevant configuration of network (

In order to satisfy

Select the first

If pinning condition

According to the number

According to the above discussion, we present an original frame for designing the intermittent pinning controllers (

For an undirected network, we have

In this section, we provide a numerical example to verify the effectiveness of the theoretical techniques. Construct a simple directed complex dynamical network with 12 nodes shown in Figure

Simple directed complex network with 12 nodes.

The dynamics of the Chua oscillator is given by

Chaotic behavior of delayed Chua attractor.

In the following, we can obtain

Examining the out-degree and in-degree of each node in network described by Figure

The decline curve of

When

Synchronous error

Synchronous error

Synchronous error

In this paper, the pinning synchronization problem has been investigated for a directed complex network with delayed dynamics. A pinning synchronous criterion based on periodically intermittent control strategy has been proposed, and its applicable versions are also given to illustrate two crucial techniques in pinning control scheme: (i) How many and what type of nodes can be chosen as pinning candidates? (ii) How do we allocate these pinning controllers to network’s nodes? Using the proposed algorithm and numerical simulation, it has been shown that pinning synchronization can be reached in a directed complex network by designing periodically intermittent control scheme. Further research direction would include the investigations on delay-inducing synchronization and application of intermittent control on consensus of multiagents system (including first- or second-order) such as those described in [

Construct the following Lyapunov function:

According to Assumption

On the other hand, when

According to Lemma

Generally, we have the following estimation of

Note that if

By (

According to definition (

According to condition (

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

This work was supported by the National Natural Science Foundation of China and the Natural Science Foundation of Honghe University (Grant no. XJ15SX03).