This paper considers the control problem of dynamically complex networks with saturation couplings. Two novel control schemes in terms of adaptive control are presented to deal with such saturation couplings. Based on the robust idea, the underlying complex network is firstly transformed into a strongly connected network having time-varying uncertainty. However, the upper bound of uncertainty is unknown. Because of such an unavailable bound, a kind of adaptive controller added to each node is proposed such that the closed-loop auxiliary network is uniformly bounded. In particular, the original system states are asymptotically stable. Moreover, in order to avoid adding the desired controller to every node, another different kind of adaptive controller based on the pinning control idea is proposed. Compared with the former method, it is only applied to a part of nodes and has a good operability. Finally, a numerical example is provided to show the effectiveness of our results.

Complex network is composed of a large number of interconnected dynamical units, which is ubiquitous in nature and human society, such as ecosystem network, biological network, food network, social network, and transportation network. It is regarded as a fundamental tool in understanding a variety of dynamic phenomena which are presented in real worlds [

On the other hand, it is known that the topological structure of complex network is important and could determine the characteristic of complex network in addition to controlling them effectively. By investigating the existing references about complex networks, it is found that most of them such as [

Inspired by the above discussions, the control problem of dynamically complex networks with saturation couplings is studied in this paper, where new adaptive controllers are proposed to handle the saturation couplings. The main contributions of this paper are as follows.

Consider a class of complex networks with saturation coupling and described as

It is worth mentioning that, based on the robust method, the originally complex network (

Before giving our main results, some necessary statements are needed here.

Kronecker product has the following properties:

Assuming that there is a matrix

For this assumption, it is said that it is without loss of generality and could be gotten from reference [

The complex network (

In this section, a kind of adaptive controller guaranteeing the states of the closed-loop complex network asymptotically stable is proposed as

Supposing that Assumption

For the adaptive closed-loop auxiliary system described by (

It means that

Based on this theorem, it is seen that the adaptive controller (

In order to remove the above constraint, another kind of adaptive controller based on the pinning control idea is proposed as

Supposing that Assumption

For the adaptive closed-loop auxiliary system described by (

It is worth mentioning that, different from Theorem

Consider a complex network with four nodes, each of which is the Sprott G chaotic system [

The chaotic attractor of the Sprott G system.

The state response of the closed-loop system by controller (

The curve of estimation

Next, we will design an adaptive controller (

The state response of the closed-loop system by controller (

The curve of estimation

In this paper, we have investigated the stabilization problem of complex networks with saturation coupling, whose saturation coupling is handled by the adaptive control method. By exploiting the robust viewpoint, an equivalent description of such a complex network is established, which is transformed to a strongly connected network having time-varying coupling uncertainty. However, the uncertainty is unknown in addition to its bound unavailable. Based on the proposed model, an adaptation law is proposed to estimate the unknown parameter. By applying the updated value, a kind of adaptive controller is proposed that the estimated parameter is bounded, whereas the system state is asymptotically stable. Because of the above adaptive controller added to each node and having some application restrictions, another different adaptive controller which could be added to a fraction of nodes is proposed based on the pinning control idea. Finally, the effectiveness of the theoretical results is illustrated by a numerical example.

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 under Grants 61104066, 61374043, and 61473140, the China Postdoctoral Science Foundation funded project under Grant 2012M521086, the Program for Liaoning Excellent Talents in University under Grant LJQ2013040, and the Natural Science Foundation of Liaoning Province under Grant 2014020106.