Adaptive Fixed-Time Control for High-Order Stochastic Nonlinear Time-Delay Systems: An Improved Lyapunov–Krasovskii Function

In this article, the adaptive tracking control problem is considered for high-order stochastic nonlinear time-delay systems in fixed-time. Being different from existing results, an improved Lyapunov-Krasovskii function is designed, which can not only compensate for the time-delay term but also remove the obstacle from the high-order term. Due to the introduction of the Lyapunov-Krasovskii function into the total Lyapunov function, it makes it difficult to stabilize the controlled system within a fixed-time interval. L'Hopital's rule is used to determine the boundedness of the Lyapunov-Krasovskii function, and the fixed-time boundedness of the integral functions can be inferred. By utilizing the fixed-time Lyapunov stability theorem, it is proved that the controlled system is semi-globally practical fixed-time stable (SGPFS), all the closed-loop signals (CLSs) are bounded within the fixed-time interval, and the tracking error converges into a small region around zero. The validity of the designed scheme is substantiated via simulation results.