Global Stabilization of Stochastic Nonlinear Systems Via C¹ and C∞ Controllers

The problem of designing C1or C∞controllers for a class of stochastic nonlinear systems (SNSs) in lower-triangular form is studied in this note. By using the well-known backstepping method, the concept of homogeneity with monotone degrees (HWMD) and the sign function approach, we construct a C1state feedback controller recursively. Meanwhile, by employing a polynomial Lyapunov function with sign functions, we prove that the solution to SNSs is globally asymptotically stable in probability. Furthermore, based on the concept of HWMD, it shows that C1controllers for a class of three-dimensional SNSs can be modified to C∞ controllers under certain conditions. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed controllers.