Delay-dependent criterionfor asymptotic stability of a class of neutral equations

K e y w o r d s N e u t r a l equations, Asymptotic stability, Lyapunov method, Linear matrix inequality. 1. I N T R O D U C T I O N The main purpose of this article is to investigate the asymptotic behaviors of solutions of the neutral delay equation d d-t ix(t) + p x ( t T)] = a x ( t ) + b tanh x ( t a), t > 0, (1) where a, b, % and ~ are positive real numbers, a > ~and ]Pl < 1. Wi th each solution x(t) of equation (1), we assume the initial condition: x(s) = ¢(s), s e i a , 0], where ¢ E C([ -~ , 0], n ) . Delay differential equations of various types including equation (1) have been investigated by many authors for the s tudy of the dynamic characteristics of neural networks of Hopfield type (see [1] and references cited therein). Recently, the asymptot ic stability of equation (1) has been discussed in [2,3], and the delayindependent sufficient conditions for the stabili ty have been presented. In the work [2], only the case a = ~is considered. In general, abandonment of information on the delay causes conservativeness of the stabili ty criteria especially when delays are small. Delay-dependent criteria are often less conservative than delay-independent criteria. 0893-9659/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by ,4j~48-T~X doi: 10.1016/j.aml.2003.05.013