This paper is concerned with the stability analysis issue for coupled systems on networks with mixed delays and reaction-diffusion terms (CSNMRs). By employing Lyapunov method and Kirchhoff's Theorem in graph theory, a systematic method is proposed to guarantee exponential stability of CSNMRs. Two different kinds of sufficient criteria are derived in the form of Lyapunov function and coefficients of the system, respectively. Finally, a numerical example is given to show the effectiveness of the proposed criteria.
Coupled systems on networks (CSNs) have important practical significance; for example, neural networks can be seen as a coupled system, so that they are widely used in physics, biology, and engineering fields [
On the other hand, time delay is inevitably in practice because of finite transmission of interaction [
The main method that contributes to investigating stability of a system is Lyapunov function. In the literature, Zhu and Song [
However, it is complicated to straightly construct an appropriate Lyapunov function for a specific coupled system, for the reason that the stability of a system depends on not only the nature of the vertex system, but also the network topology. Considering that CSNMRs can be described in a digraph, in which each vertex represents an individual system called vertex system and the directed arcs stand for the interconnections and interactions among vertex systems, a novel method based on graph theory, which is used to construct the Lyapunov function, is proposed. The pioneering work based on graph theory to consider the global stability problem for coupled systems on networks has been done by Li et al. [
To the best of the authors' knowledge, the discussion about the exponential stability for CSNMRs is not deep enough, and much room remains to be explored. According to our survey, this method has not been applied to the systems with distributed time delay or reaction-diffusion terms. With few conclusions about the stability of CSNMRs obtained from the new method, we start the present research.
Compared with the previous results on the analysis of exponential stability, the main contribution of this paper is threefold. Firstly, distributed time delay and reaction diffusion are taken into account in the model. Secondly, a graph-theoretic approach is employed to get different kinds of sufficient stability criteria. Thirdly, some conditions that keep from finding a global Lyapunov function directly for CSNMRs are developed, through effective utilization of topological structure and coefficients of CSNMRs.
The remainder of this paper is organized as follows: in Section
In this section, we will give some useful notations, preliminaries about graph theory, and model descriptions.
For convenience, we use the following notations. Write
The following basic concepts on graph theory from [
Here, we show a result in Li et al. [
Then the following identity holds:
It is well known that CSNMRs have caught many researchers' attention. In this paper, exponential stability for delayed coupled networks is studied as stated in Section
Given a network represented by digraph
In order to complete the proof of this paper, we suppose that the system (
The boundary conditions of system (
Throughout this paper, the following definition, assumptions, and lemmas are needed to derive our main results. The functions
The trivial solution to system (
Let X be a cube
For any positive definite matrix
In the study of stability, Lyapunov method plays an important role. Combining Lyapunov method with graph theory, two kinds of sufficient criteria are investigated. One is given in the form of Lyapunov function, while the other is in terms of coefficients of system (
In order to propose a Lyapunov function for system (
Functions There are positive constants Along each directed cycle
Let
Let
Recently, attention was paid to stability analysis for coupled systems increasingly, and many methods were proposed. Gan and Xu [
However, the conditions in Theorem
Suppose that
Furthermore, in view that if for every
The conclusion of Theorem
Since the previous results are based on vertex-Lyapunov functions for system (
In the following, another sufficient exponential stability criterion is established in the form of coefficients of system (
Let
Let
From Lemma
In this section, we will give an example, showing the effectiveness and the correctness of our results. Consider the following system:
The dynamical behavior of the subsystem
The dynamical behavior of the subsystem
The dynamical behavior of the subsystem
In this paper, we have investigated the stability of CSNMRs. By applying some results in graph theory and Lyapunov method, we have derived two different types of novel exponential stability criteria. One is given in the form of Lyapunov functions and network topology, while the other is given in the form of coefficients of systems. Compared with the previous stability method, graph-theoretic approach in this paper is new and efficient. Furthermore, an illustrative example is given to validate the approach.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the NNSF of China (nos. 11301112, 11171081, and 11171056), China Postdoctoral Science Foundation funded project (no. 2013M541352), and the NNSF of Shandong Province (no. ZR2013AQ003).