Automatica 64 (2016) 63–69 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Stability of nonlinear differential systems with state-dependent delayed impulses ✩ Xiaodi Li a,b , Jianhong Wu b a School of Mathematical Sciences, Shandong Normal University, Ji’nan, 250014, PR China b Laboratory for Industrial and Applied Mathematics, York University, Toronto, Ontario, Canada, M3J 1P3 article info Article history: Received 14 March 2015 Received in revised form 4 June 2015 Accepted 6 September 2015 Available online 18 November 2015 Keywords: State-dependent delay Impulsive control theory Lyapunov stability LMI abstract We consider nonlinear differential systems with state-dependent delayed impulses (impulses which involve the delayed state of the system for which the delay is state-dependent). Such systems arise naturally from a number of applications and the stability issue is complex due to the state-dependence of the delay. We establish general and applicable results for uniform stability, uniform asymptotic stability and exponential stability of the systems by using the impulsive control theory and some comparison arguments. We show how restrictions on the change rates of states and impulses should be imposed to achieve system’s stability, in comparison with general impulsive delay differential systems with state- dependent delay in the nonlinearity, or the differential systems with constant delays. In our approach, the boundedness of the state-dependent delay is not required but derives from the stability result obtained. Examples are given to demonstrate the sharpness and applicability of our general results and the proposed approach. © 2015 Elsevier Ltd. All rights reserved. 1. Introduction Impulsive delay differential systems have been used for mod- elling natural phenomena in many areas for many years, and there have been significant studies of such systems, as indi- cated by Churilov and Medvedev (2014), Dashkovskiy, Kosmykov, Mironchenko, and Naujok (2012), Lakshmikantham, Bainov, and Simeonov (1989), Li, Bohner, and Wang (2015), Sakthivel, Mah- mudov, and Kim (2009), Sakthivel, Ren, and Mahmudov (2010) and Samoilenko and Perestyuk (1995) and references therein. Of current interest is the delayed impulses of differential systems arising in such applications as automatic control, secure commu- nication and population dynamics (Akca, Alassar, Covachev, Cov- acheva, & Al-Zahrani, 2004; Akhmet & Yilmaz, 2014; Chen, Wei, & Lu, 2013; Chen & Zheng, 2011, 2009; Khadra, Liu, & Shen, 2005, 2009; Liu, Teo, & Xu, 2005), here and in what follows, a delayed impulse describes a phenomenon where impulsive transients de- pend on not only their current but also historical states of the sys- tem. For instance, in communication security systems based on ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Nathan Van De Wouw under the direction of Editor Andrew R. Teel. E-mail addresses: sodymath@163.com (X. Li), wujh@yorku.ca (J. Wu). impulsive synchronization, there exist transmission and sampling delays during the information transmission process, where the sampling delay created from sampling the impulses at some dis- crete instances causes the impulsive transients depend on their historical states (Chen et al., 2013; Khadra et al., 2005). The existing studies, however, such as those in Akca et al. (2004), Akhmet and Yilmaz (2014), Chen et al. (2013), Chen and Zheng (2011, 2009), Khadra et al. (2005, 2009) and Liu et al. (2005), assume the de- lays in impulsive perturbations are either fixed as constants or given by integrals with state-independent distributed kernels. For example, Khadra et al. (2005) considered the impulsive synchro- nization of chaotic systems with transmission delay and sampling delay, and then applied the results to the design of communica- tion security scheme. Chen and Zheng (2011) studied the nonlin- ear time-delay systems with two kinds of delayed impulses, that is, destabilizing delayed impulses and stabilizing delayed impulses, and derived some interesting results for exponential stability. But in both results, the delays in impulses are given constants. Akca et al. (2004) derived some results for global stability of Hopfield- type neural networks with delayed impulses, where the delays in impulses are in integral forms with state-independent distributed kernels. However, in many cases it is important to consider state- dependent delays in impulsive perturbations. For example, the sampling delay varies with the change of state variables since it is natural to consider sending control signals less frequently when http://dx.doi.org/10.1016/j.automatica.2015.10.002 0005-1098/© 2015 Elsevier Ltd. All rights reserved.