Adaptive neural control for a class of non-affine stochastic non-linear systems with time-varying delay: A Razumikhin-Nussbaum method

This study focuses on the problem of adaptive neural control for a class of uncertain non-affine stochastic non-linear systems with time-varying delay. Major technical difficulties for this class of systems lie in: (i) the unknown control direction embedded in the unknown control gain functions and (ii) the unknown system function with unknown time-varying delay. A novel Razumikhin-Nussbaum lemma is proposed to overcome the mentioned difficulties. Moreover, based on the Razumikhin functional approach, an adaptive neural controller is developed for this class of systems by exploring the application of Nussbaum functions to stochastic non-linear systems. The proposed design guarantees that all the error variables in the closed-loop systems are four-moment semi-globally uniformly ultimately bounded in a compact set, while the tracking error remains in a neighbourhood of the origin. The effectiveness of the proposed design is verified by simulation results.