Contents I. Introduction
The so-called Hopfield neural networks were first introduced by Hopfield [8]. For a few decades, Hopfield neural networks have been extensively investigated. Many applications have been found in different fields such as combinatorial optimization, signal processing and pattern recognition, see for examples [8], [9], [12], [13], [19]. These applications are built upon the stability of the equilibrium of neural networks. Thus, the stability analysis is a necessary step for the design and applications of neural networks. Sometimes, neural networks have to be designed such that there is only one equilibrium and this equilibrium is globally stable. For example, when a neural network is applied to solve the optimization problem, it must have one unique equilibrium which is globally stable. On the other hand, both in biological and artificial neural networks, the interactions between neurons are generally asynchronous which inevitably result in time delays. In electronic implementation of analog neural networks, nevertheless, the delays are usually time-varying due to the finite switching speed of amplifiers. It is known that time delays are often a source of instability of neural networks [16]. Therefore, it is of great importance to study the global stability of neural networks with time-varying delays.
