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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555

Nonfragile Dissipative Synchronization of Reaction-diffusion Complex Dynamical Networks with Coupling Delays Xiaona Song*, Renzhi Zhang, Mi Wang, and Junwei Lu Abstract: This paper focuses on the synchronization of reaction-diffusion complex dynamical networks with coupling delay. In order to reflect the uncertainties of the controller, the nonfragile problem is considered. Furthermore, we also take into account the dissipativity analysis problem, which contains the H∞ performance and passivity performance in a unified framework. By utilizing the Lyapunov functional method, two sufficient delay-dependent conditions, which ensure the considered system is globally asymptotically synchronized onto the unforced node and strictly dissipative, are established in terms of linear matrix inequality. Finally, three numerical examples are employed to demonstrate the effectiveness of the design methods. Keywords: Complex dynamical networks, coupling delay, dissipative synchronization, nonfragile controller, reaction-diffusion.

1.

INTRODUCTION

In the past few decades, complex dynamical networks (CDNs) have received considerable attention due to their extensive applications, such as power grid regulation [1], parallel image processing [2] and robot formation [3]. Various complex networks have become an important part of our daily life, and they can be seen everywhere. The most famous examples include the World Wide Web, social networks, communication networks, food networks, neural networks, etc [4–6]. Therefore, the topological structure and dynamic behavior of CDNs have been extensively studied by researchers [7,8]. As one of the most important collective behaviors, the synchronization of CDNs attracts tremendous attention in recent years. So far, many important results have been obtained in the synchronization of various CDNs [9, 10]. Due to the limited switching speed of the amplifier and the inherent communication time of the nodes, the time delay inevitably exists. Its existence will affect the stability of the networks by generating oscillation and instability characteristics. Therefore, CDNs with coupling delay are more common, the synchronization for various CDNs with coupling delay has been widely studied by researchers. In [11, 12], the synchronization of both continuous and discrete CDNs with coupling delay is studied. However, in the real world, the time delay is always changing. In consequence, the stability of CDNs with

time-varying coupling delays has received much of the focus [13–16]. In general, the perturbation of the controller may not be considered in control design, that is to say, we assumed that the controller could be implemented precisely. Nevertheless, in practical applications, uncertainty is ubiquitous in controllers. This uncertainty may be due to the aging of system components, digital-to-analog conversion, and analog-to-digital conversion. It is worth noting that the uncertainty of the controller in the implementation process may a