Published in IET Control Theory and Applications Received on 5th January 2009 Revised on 26th November 2009 doi: 10.1049/iet-cta.2009.0007 ISSN 1751-8644 Global exponential stability criteria for neural networks with probabilistic delays M.S. Mahmoud 1 S.Z. Selim 1 P. Shi 2,3 1 Systems Engineering Department, King Fahd University of Petroleum and Minerals, P. O. Box 985, Dhahran 31261, Saudi Arabia 2 Department of Computing and Mathematical Sciences, University of Glamorgan, Pontypridd CF 37 1DL, UK 3 ILSCM, School of Science and Engineering, Victoria University, Melbourne, Australia, and School of Mathematics and Statistics, University of South Australia, Adelaide, Australia E-mail: msmahmoud@kfupm.edu.sa Abstract: The problem of global exponential stability analysis for a class of neural networks (NNs) with probabilistic delays is discussed in this paper. The delay is assumed to follow a given probability density function. This function is discretised into arbitrary number of intervals. In this way, the NN with random time delays is transformed into one with deterministic delays and random parameters. New conditions for the exponential stability of such NNs are obtained by employing new Lyapunov–Krasovskii functionals and novel techniques for achieving delay dependence. It is established that these conditions reduce the conservatism by considering not only the range of the time delays, but also the probability distribution of their variation. Numerical examples are provided to show the advantages of the proposed techniques. 1 Introduction The last few decades have witnessed successful applications of neural networks (NNs) to a variety of information processing systems such as signal processing, pattern recognition, static image processing, associative memories, and combinatorial optimisation, where the rich dynamical behaviour has played a key role. These applications have revealed that signal transmission delays may cause oscillation and instability of the NNs [1]. Thus, the stability problem of delayed neural networks (DNNs) has become a topic of great theoretic and practical importance. A number of important results have been reported for the stability of general systems and NNs with time delays, including both delay-dependent and independent approaches [2–5]. In [2, 6–8], different criteria guaranteeing asymptotic stability are derived and in [4, 9–17], the problem of exponential stability is studied. It is well known that delay-dependent results are generally less conservative than delay-independent ones especially for systems with small delays, and thus increasing attention has been drawn to deriving delay-dependent stability conditions more recently. Focus on delay-dependent stability criteria has progressed as delay-independent stability criteria tend to be conservative particularly when the delay is relatively small or it varies in an interval. Further research activities of some of the recently stability criteria was carried out in [18–25] for different time-delay patterns. In the design of NNs, the issue of global exponential stability is of prime concern as it guarantees the NNs to converge fast enough in order to attain fast and satisfactory response. Accordingly, the problem of global exponential stability analysis for DNNs has been studied by many investigators in the past years. In the case with differentiable time-varying delays, sufficient conditions for global exponential stability were given in [26–28]. When time-varying delays may not be differentiable, global exponential stability results can be found in [5, 20–22]. It is noted that these exponential stability conditions are either with testing difficulty or with conservatism to some extent. This opens up the way for the development of improved stability methods, which is the subject of this paper. Consider a continuous time-delayed uncertain neural network which is described by the following non-linear IET Control Theory Appl., 2010, Vol. 4, Iss. 11, pp. 2405–2415 2405 doi: 10.1049/iet-cta.2009.0007 & The Institution of Engineering and Technology 2010 www.ietdl.org