Adaptive neural control of pure-feedback stochastic nonlinear systems with multiple unknown time-varying delays

By the combination of the adaptive backstepping design with the dynamic surface control technique, an novel adaptive neural control approach is investigated for a class of pure-feedback stochastic nonlinear systems with multiple unknown time-varying delays. To overcome the design difficulty arising from the non-affine structure of pure-feedback stochastic systems, the mean value theorem is exploited. The design difficulties due to multiple unknown time-varying delay functions are overcome by using the function separation technique, the appropriate Lyapunov-Krasovskii functionals and the desirable property of hyperbolic tangent functions. The radial-basis-function (RBF) neural networks are utilized to approximate the unknown nonlinear functions. It is shown that the proposed control approach can guarantee that all signals of the closed-loop system are bounded in probability, and the tracking errors can be made arbitrarily small in probability by choosing suitable design parameters. Finally, simulation example is provided to demonstrate the effectiveness of the proposed control scheme.