• stochastic nonlinear system;
  • adaptive control;
  • neural network;
  • Razumikhin–Nussbaum lemma;
  • time-varying delay


This paper addresses the problem of adaptive neural control for a class of uncertain stochastic pure-feedback nonlinear systems with time-varying delays. Major technical difficulties for this class of systems lie in: (1) the unknown control direction embedded in the unknown control gain function; and (2) the unknown system functions with unknown time-varying delays. Based on a novel combination of the Razumikhin–Nussbaum lemma, the backstepping technique and the NN parameterization, an adaptive neural control scheme, which contains only one adaptive parameter is presented for this class of systems. All closed-loop signals are shown to be 4-Moment semi-globally uniformly ultimately bounded in a compact set, and the tracking error converges to a small neighborhood of the origin. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed control schemes. Copyright © 2012 John Wiley & Sons, Ltd.