Fixed‐Time Stability and Control for Stochastic Nonlinear Systems Based on Finite Time‐Varying Gain Functions

This article addresses the issue of infinite‐gain problems that exist in previous works concerning stochastic nonlinear systems, and provides more comprehensive fixed‐time stability results with an arbitrarily prescribed upper bound of the settling time (UBST). First, we propose a distinctive and uniformly bounded time‐varying gain function that can avoid the infinite‐gain problem. Building upon this, we present generalized fixed‐time Lyapunov theorems, along with several useful corollaries that incorporate prescribed UBST and offer additional flexibility for the Lyapunov differential operator constraint. These stability results generalize the concepts of finite/fixed‐time stability discussed in previous works, while providing a simpler method to prescribe the settling time. Based on the proposed stability theorem, we design a state‐feedback controller for stochastic nonlinear systems, which ensures the closed‐loop system exhibits fixed‐time stability in probability with prescribed UBST. The resulting controller relies on time‐varying gains, uniformly bounded over the infinite horizon, thereby yielding an attractive implementation opportunity which is opposed to the design with time‐varying gains, escaping to infinity as time goes to the prescribed‐time instant. Finally, we provide simulation results to illustrate the effectiveness of the proposed stability and controller designs.