Elsevier

Applied Mathematics and Computation

Volume 331, 15 August 2018, Pages 341-357
Applied Mathematics and Computation

Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control

https://doi.org/10.1016/j.amc.2018.03.020Get rights and content

Abstract

This article concerns the inner exponential synchronization (IES) problem of a class of stochastic coupled systems with time-varying delays on networks (SCSTDNs). Periodically intermittent control is imposed on SCSTDNs to realize IES. By combining graph theory with Lyapunov method, a Lyapunov function is constructed and some sufficient conditions guaranteeing IES of SCSTDNs under periodically intermittent control are derived. Additionally, the analysis of IES for a stochastic coupled oscillators model with time-varying delays is performed to show the applicability of the analytical results. Finally, an example with numerical simulation is presented to illustrate the validity of our theoretical results.

Section snippets

introduction

Over the past decades, coupled system on networks (CSNs), as a special kind of complex networks, have been studied extensively because they have many important applications in various areas such as moving image processing, optimization, speed detection of moving subjects, secure communication and so on, see [1], [2], [3], [4] and references therein. Especially, the synchronization phenomena in CSNs have secured growing attention among the research community since the fact that synchronization

Preliminaries and model formulation

In this section, we first introduce some basic notations and a vital lemma in graph theory. Then, we give the model formulation and finally present two important definitions.

Main results

In this section, we consider the IES problem of system (1) under periodically intermittent control (3). Some IES criteria for system (1) will be derived by combing Lyapunov method with graph theory.

An application to stochastic coupled oscillators with time-varying delays

Synchronization of oscillators model is extensively studied in physical and biological systems for underlying interests ranging from novel communications strategies to understand how large and small neural assemblies efficiently and sensitively achieve desired functional goals. In this section, we will apply the main results to discuss IES of stochastic coupled oscillators with time-varying delays. Firstly, a common oscillator reads x¨(t)+φx˙(t)+x(t)=0,t0,where φ ≥ 0 is the dimpling

Numerical Test

In this section, we provide an example with numerical simulation to demonstrate the effectiveness of the proposed theoretical results.

Here we proceed to consider the IES problem of stochastic coupled oscillators (30) under periodically intermittent control. Take n=5, system (30) can be expressed by a digraph G, and the coupling strengths dik, hik and mik are shown as follows: (dik)(5×5)=(00.02150.03150.04000.01850.015500.01150.00650.00250.00500.0100000.00120.01050.03000.003000.01250.02500.1000

Conclusion

In this paper, we have studied IES of SCSTDNs via periodically intermittent control. By combining graph theory with Lyapunov method, several simple yet generic sufficient conditions for IES of SCSTDNs under periodically intermittent control have been derived. As illustrations, the IES analysis of stochastic coupled oscillators with time-varying delays has been carried out by applying the derived theoretic results. A numerical simulation has been given to show the effectiveness of the

Acknowledgment

The authors are very grateful to the reviewers for carefully reading the paper and for their valuable comments and suggestions which have improved the paper.

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