Applied Mathematics and Computation 337 (2018) 315–328 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Quantized feedback control scheme on coupled systems with time delay and distributed delay: A finite-time inner synchronization analysis Yao Xu, Chenyin Chu, Wenxue Li ∗ Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, PR China a r t i c l e i n f o Keywords: Finite-time inner synchronization Coupled systems on a network Time delay and distributed delay Quantized feedback control Kirchhoff’s Matrix Tree Theorem a b s t r a c t In this paper, finite-time inner synchronization of coupled systems on a network with time delay and distributed delay (CSNTD) is investigated. And here, time delay and distributed delay are both taken into consideration when modelling a realistic network. Different from common feedback control, the controller we design is quantized, which is more realistic and reasonable. By using Lyapunov method and Kirchhoff’s Matrix Tree Theorem, some sufficient criteria are derived to guarantee finite-time inner synchronization of CSNTD. It should be underlined that the method is first applied to studying the issue of finite-time inner synchronization of CSNTD and the synchronization time we obtain has a close rela- tionship with the topological structure of the network. Moreover, to verify our theoretical results, we present an application to coupled oscillators with time delay and distributed delay, and a sufficient criterion is obtained. Ultimately, a numerical example is given to verify the validity and feasibility of theoretical results. © 2018 Elsevier Inc. All rights reserved. 1. Introduction In the past few decades, complex networks have gained great attention owing to their widespread applications in physics, biology, engineering [1–3], etc. Especially, coupled systems, as a vital structure of complex networks, have drawn rapidly interests. It is worth noting that the widespread applications principally depend on the dynamical behaviors of coupled systems which include stability [4], synchronization [5–7], stationary distribution [8], etc. In particular, synchronization, as a kind of typical collective behaviors and basic motions in nature, has become a hot research topic and different types of synchronization are investigated such as exponential synchronization [9], generalized synchronization [10], cluster synchro- nization [11], lag synchronization [12] and finite-time synchronization [13]. In a practical engineering process, since humans and machines are under a finite lifespan, people always expect systems to achieve synchronization in limited time. Thus, the concept of finite-time synchronization is offered. Compared with other kinds of synchronization, there are multitudinous ad- vantages of finite-time synchronization such as excellent disturbance rejection and strong robustness. For example, in secure communication, uncertainties can really be reduced, and the efficiency and confidentiality of the communication can also be greatly improved. Therefore, in recent years, many researchers have been devoted to studying finite-time synchronization of different sorts of systems [14–16]. ∗ Corresponding author. E-mail addresses: wenxuetg@hitwh.edu.cn, wenxue810823@163.com (W. Li). https://doi.org/10.1016/j.amc.2018.05.022 0096-3003/© 2018 Elsevier Inc. All rights reserved.