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This paper studies the pinning synchronization in complex networks with node dynamics satisfying the one-sided Lipschitz condition which is less conservative than the well-known Lipschitz condition. Based on M-matrix theory and Lyapunov functional method, some simple pinning conditions are derived for one-sided Lipschitz complex networks with full-state and partial-state coupling, respectively. A selective pinning scheme is further provided to address the selection of pinned nodes and the design of pinning feedback gains for one-sided Lipschitz complex networks with general topologies. Numerical results are given to illustrate the effectiveness of the theoretical analysis.

A complex network is composed of a set of coupled dynamical systems where each system updates its state based on the local neighboring information such that the whole network might exhibit collective behaviors under some conditions. Nowadays, many natural and man-made systems in our daily life, such as biological networks, generic networks, electrical power grids, and the World Wide Web, can be described by complex networks.

The synchronization problem for complex networks has attracted much attention from various disciplines over the past decade [

The synchronization of complex networks is usually achieved by using full-state variables of network nodes which may not always be available in practice. To resolve this difficulty, some control algorithms have been proposed to synchronize complex networks by utilizing the observed states of network nodes. Based on the state observer approach, Jiang et al. [

If a complex network cannot achieve synchronization by itself, one can design some appropriate controllers to force the network to synchronize onto a homogenous trajectory. However, for a network consisting of a large number of nodes, the control cost will be very high when the control actions are applied to all network nodes. Fortunately, one can adopt the pinning control strategy [

It is well-known that each node in complex networks is usually described by a nonlinear dynamical system. In most existing results on pinning control of complex networks, the node dynamics is assumed to satisfy the QUAD condition [

The main contribution of this paper is threefold. First, the paper studies the pinning control problem for both full-state and partial-state coupled complex networks with one-sided Lipschitz-type node dynamics. To the best of our knowledge, the synchronization in one-sided Lipschitz complex networks has not been addressed up to date. Therefore, the pinning control results for one-sided Lipschitz complex networks in this paper fill in this gap in time. Second, by using the properties of M-matrices, some simple pinning conditions in terms of low-dimensional linear matrix inequalities (LMIs) are established for both full-state and partial-state coupled complex networks. With the derived stability criteria, the pinning control problem of a large-scale network can be reduced to the test of a linear matrix inequality whose dimension is the same as that of a single network node. Third, we discuss the selection of pinned nodes and the design of pinning feedback gains for one-sided Lipschitz complex networks with both directed and undirected topologies based on M-matrix and algebraic graph theories.

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

This section provides some mathematical preliminaries to derive the main results of this paper.

The notations in this paper are quite standard. Let

A nonlinear dynamical system which does not satisfy Lipschitz condition may satisfy the so-called one-sided Lipschitz condition [

A nonlinear function

The one-sided Lipschitz condition can be rewritten as

It is well-known that a nonlinear function

From Definition

Some properties of M-matrices are important to study the pinning control of complex networks.

For a nonsingular matrix

all entries of

all eigenvalues of

there exists a positive definite diagonal matrix

Consider a complex network composed of

The leader node (or isolated node) for complex network (

For complex network (

The node dynamics of complex network (

Suppose that a positive definite matrix

In this section, we consider the synchronization in complex network (

Letting

We see that complex network (

Let

Denote

Based on M-matrix and algebraic graph theories, Song et al. [

Suppose that there exists a positive definite matrix

Then, the pinning-controlled network (

Let

Construct the following Lyapunov function candidate:

By Assumption

It follows from (

From condition (

Note that the dimension of LMI (

In Theorem

Assume that

Obviously,

Take the Lyapunov function candidate as follows:

In view of the proof of Theorem

From (

When

In the previous section, complex network (

For complex network (

Let

Note that the output states are actually adopted to reach pinning synchronization in complex network (

If the matrix pair

Suppose that there exists a positive definite matrix

Then, the pinning-controlled network (

Following the similar line in the proof of Theorem

From (

Consider the following Lyapunov function candidate:

Calculate the time derivative of

Obviously,

Note that the dimension of LMI (

In Remark

Suppose that Assumption

Take the following Lyapunov function candidate as

Then, the time derivative of

By (

In this section, we discuss the selection of pinned nodes and the design of pinning feedback gains. A selective pinning scheme for complex networks with general topologies is proposed to satisfy the pinning conditions in Theorems

For complex network (

Now, we present the following selective pinning scheme for complex network (

Partition the digraph

Rearrange the remaining

Solve LMI (

In this section, some numerical results are provided to illustrate the effectiveness of our theoretical analysis. A digraph with ten nodes is shown in Figure

A digraph with ten nodes.

Note that the digraph in Figure

Consider a complex network with one-sided Lipschitz-type node dynamics in the form of (

Taking

Applying pinning control action to node

State evolutions of complex network with full-state coupling.

Choosing

Using algorithm (

State evolutions of complex network with partial-state coupling.

In this paper, the pinning synchronization problem for one-sided Lipschitz complex networks has been investigated by using M-matrix and algebraic graph theories. Some simple pinning criteria in terms of low-dimensional linear matrix inequalities have been established for full-state and partial-state coupled complex networks, respectively. In particular, the output states of network nodes are utilized to implement the distributed pinning control algorithm. A selective pinning scheme has been proposed to satisfy the derived pinning conditions for one-sided Lipschitz complex networks with general topologies. In the near future, it would be of interest to study the pinning control problem for one-sided Lipschitz-type complex networks with dynamically switching topologies and time delays.

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

This work was jointly supported by the National Science Foundation of China under Grants 61273218, 61304172, 61272530, and 61175119 and the Natural Science Foundation of Henan Province of China under Grants 122102210027, 122300410220 and 12B480005.