New results for global exponential stability of neural networks with varying delays Yajuan Liu a , Wanbiao Ma a , Magdi S. Mahmoud b,n a Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China b Systems Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 5067, Dhahran 31261, Saudi Arabia article info Article history: Received 20 October 2011 Received in revised form 28 April 2012 Accepted 5 May 2012 Communicated by S. Arik Available online 9 June 2012 Keywords: Neural networks Global exponential stability Interval time-varying delay Lyapunov–Krasovskii functional Linear matrix inequalities abstract In this paper, the problem of global exponential stability for a class of neural networks with interval time-varying delay is investigated. The time-delay pattern is quite general and including fast time- varyings. It is assumed that the time delay belongs to a given interval, but the derivative of a time- varying delay be less than 1 is removed, or the delay function is not necessary to be differentiable. By constructing a set of improved Lyapunov–Krasovskii functionals combined with a known integral inequality, new delay-dependent exponential stability criteria with explicitly exponential convergence rate are established in terms of LMIs (linear matrix inequalities). The stability criteria are less conservative than the existing results in the literatures. Numerical examples are given to illustrate the effectiveness of the results. & 2012 Elsevier B.V. All rights reserved. 1. Introduction During the last few decades, interests in studying neural networks have been steadily increasing due to their wide applica- tions such as associative memories, patten recognition, solving optimization problems, image processing, signal processing, and so on [1,2]. In some of these applications, it is required that the designed neural network has a unique equilibrium point which is globally stable. During the implementation of neural networks, the occurrence of time delays is unavoidable during the proces- sing and transmission of the signals because of the finite switch- ing speed of amplifiers in electronic networks or finite speed for signal propagation in biological networks, and the delays are often the source of instability, hidden oscillations, divergence, chaos or other poor performance behavior [3–5]. Therefore, global stability analysis of delayed neural networks have received considerable attention. In this regard, many sufficient conditions ensuring global asymptotic stability and global exponential stability for delayed neural networks have been derived [2–19]. But in most of the known results, the time-varying delay varies from 0 to an upper bound. In practice, the delay may vary in a range for which the lower bound is not restricted to be zero, i.e, interval time-varying. A typical real example of dynamic systems with interval time- varying delays is networked control systems [22]. To the best of our knowledge, there are few results about the exponential stability of neural networks with interval time-varying delay [19]. It is also worth mentioning that, in the existing results of exponential stability, the restriction that the derivative of a time- varying delay be less than 1 is imposed on stability criteria for neural networks with a time-varying delay [5,11–17]. In [23], the delay function is not necessary to be differentiable, and in [24], the time delay belongs to a given interval, and the derivative of a time-varying delay be less than 1 is removed. However, there are no results about considering the interval time-varying delay which is not necessary to be differentiable, and be less than 1 is removed at the same time. Motivated by the above statement, this paper considers global exponential stability of class of neural networks with interval time-varying delays. The time delay belongs to a given interval, which means that the lower and upper bounds for time the time- varying delay are variable, and the restriction that the derivative of a time-varying delay be less than 1 is removed in this paper. By constructing a set of new Lyapunov–Krasovskii functional com- bined with Newton–Leibniz formula, new criteria for exponential stability of the system are derived. The criteria is delay-depen- dent one which is less conservative than delay-independent one Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/neucom Neurocomputing 0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.05.003 n Corresponding author. Tel.: þ966 542019258; fax: þ966 38602965. E-mail addresses: yajuan_liu@126.com (Y. Liu), wanbiao_ma@ustb.edu.cn (W. Ma), msmahmoud@kfupm.edu.sa (M.S. Mahmoud). Neurocomputing 97 (2012) 357–363