Global Exponential Robust Stability of Static Interval Neural Networks with Time Delay in the Leakage Term

The stability of a class of static interval neural networks with time delay in the leakage term is investigated. By using the method of -matrix and the technique of delay differential inequality, we obtain some sufficient conditions ensuring the global exponential robust stability of the networks. The results in this paper extend the corresponding conclusions without leakage delay. An example is given to illustrate the effectiveness of the obtained results. 1. Introduction Recently, neural networks have been widely studied because of their successful applications in different areas, such as pattern recognition, image processing, detection of moving objects, and optimization problems. The stability of the neural networks with time delay, upon which these applications largely depend, has been extensively studied (see [1–10]). However, to the best of our knowledge, there has been very little existing work on neural networks, especially, on static neural networks with time delay in the leakage term [11–18]. This is due to some theoretical and technical difficulties [13]. So, the main purpose of this paper is to study the stability of the static interval neural networks with time delay in the leakage term. By using the properties of -matrix and delay differential inequality, we obtain some sufficient conditions ensuring the global exponential robust stability. Our results extend the corresponding conclusions without leakage delay. 2. Model Description and Preliminaries In this section, we list all the notations which will be frequently used throughout the paper and give a few definitions, lemmas, and assumptions. Notations. Let be the set of real number, and let and be the space of -dimensional real vectors and real matrices, separately. denotes an unit matrix. . For or , the notation ( ) means that each pair of corresponding elements of and satisfies the inequality “ .” denotes the Euclidean norm. For any , is the sign function of . denotes the space of continuous mappings from the topological space to the topological space . Particularly, let denote the family of all continuous -valued function defined on with the norm . For , , we define , , , , and . denotes the upper-right-hand derivative of at time . Consider the following interval static neural network model with leakage delay: where , and denote the state and the external inputs of the neuron, separately. The integer corresponds to the number of units in a neural network, and denotes the signal propagation function of the unit. is a parameter, and represents the rate with which neuron will reset its