Neural Networks 24 (2011) 370–377 Contents lists available at ScienceDirect Neural Networks journal homepage: www.elsevier.com/locate/neunet Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis Eva Kaslik a,b , Seenith Sivasundaram c,∗ a Institute eAustria Timisoara, Bd. V. Parvan nr. 4, room 045B, 300223, Timisoara, Romania b Department of Mathematics and Computer Science, West University of Timisoara, Bd. V. Parvan nr. 4, 300223, Romania c Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA article info Article history: Received 22 November 2010 Accepted 29 December 2010 Keywords: Hopfield-type neural networks Discrete time Distributed delays Impulses Lyapunov functionals Multistability Simulation Hybrid Delay kernels abstract In this paper we investigate multistability of discrete-time Hopfield-type neural networks with dis- tributed delays and impulses, by using Lyapunov functionals, stability theory and control by impulses. Example and simulation results are given to illustrate the effectiveness of the results. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction In recent years, there has been increasing interest in neural net- works such as (Chua & Yang, 1988; Cohen & Grossberg, 1983; Hop- field, 1982), and bidirectional associative memory (Kosko, 1988) neural networks, and their potential applications in many areas such as classification, optimization, signal and image processing, solving nonlinear algebraic equations, pattern recognition, associa- tive memories, cryptography and so on. The state of electronic networks is often subject to instan- taneous changes, and will experience abrupt changes at certain instants which can be caused by frequency change, switching phe- nomenon, or by some noise which do exhibit impulse effects. In the past decades, a number of research papers have dealt with dynamical systems with impulse effect as a class of general hybrid systems. Examples include the pulse frequency modulation, optimization of drug distribution in the human body and control systems with changing reference signal. Impulsive dynamical systems are characterized by the occurrence of abrupt change in the state of the system which occur at certain time instants over a period of negligible duration. The dynamical behavior of such ∗ Corresponding author. E-mail addresses: ekaslik@gmail.com (E. Kaslik), seenithi@gmail.com (S. Sivasundaram). systems is much more complex than the behavior of dynamical systems without impulse effects. The presence of impulse means that the state trajectory does not preserve the basic properties which are associated with non-impulsive dynamical systems. Thus, the theory of impulsive differential equations is quite interesting and has attracted the attention of many scientists. In general, most neural networks have been assumed to be in continuous time. Discrete-time counterparts of continuous-type neural networks have only been in the spotlight since 2000, even though they are essential when implementing continuous-time neural networks for practical problems such as image process- ing, pattern recognition and computer simulation. Discrete-time systems with delays have strong background in engineering appli- cations, among which network based control has been well recog- nized to be a typical example. Discrete-time neural networks are more applicable to problems that are inherently temporal in na- ture or related to biological realities. They perfectly can keep the dynamic characteristics, functional similarity, and even the bio- logical or physical resemblance of the continuous-time networks under certain mild conditions (restrictions) (Huo & Li, 2009; Mo- hamad, 2001, 2003, 2008; Mohamad & Gopalsamy, 2000). For this reason, the stability analysis of discrete-time neural networks have received more and more attention recently. In the following, we use the notations Z + ={1, 2, 3,...}, Z + 0 ={0, 1, 2,...}, Z − 0 ={..., −2, −1, 0}. 0893-6080/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.neunet.2010.12.008