Fixed-time Lyapunov criteria of stochastic nonlinear systems revisited and its applications

In all the references on stochastic fixed-time stability, the customary treatment of the stochastic noise in the worst-case sense is that it is treated as the unfavorable factor for system stability. Consequently, stochastic fixed-time Lyapunov-type conditions are rather restrictive. Realizing this limitation, we revisit stochastic fixed-time stability and present a generalized fixed-time stability theorem for stochastic differential equations. This theorem completely removes the assumption in these references that the differential operator of the Lyapunov function must be strictly negative, and reveals a positive role of the stochastic noise in stochastic fixed-time stability. As the application of this theorem and its corollary, we solve the problem of fixed-time stabilization for stochastic nonlinear systems with high-order and low-order nonlinearities.