Finite-time state feedback stabilization for a class of stochastic nonlinear systems

In this paper, the problem of global finite-time stabilization by state feedback for a class of continuous but non-smooth stochastic nonlinear systems is studied. By extending the adding-a-power-integrator technique and constructing quartic Lyapunov function, the state feedback finite time stabilization controller is designed systematically. Based on the criterion on finite time stability of stochastic systems, the closed loop system can be proved to be finite-time globally asymptotically stable in probability (FGSP). Further, the simulation result verifies the validity of the designed controller.

I. Introduction

From 1990s, the stabilization control of stochastic nonlinear systems has been extensively studied. This problem was initially studied by Florchinger [1]–[3]. In their work, the control Lyapunov function and Sontag stabilization formular was generalized to stochastic case. An essential improvement was that Pan and Basar ([5]) etc, provided the backstepping controller design method for strict feedback systems by using the quadratic Lyapunov functions and risk sensitive cost criteria([6]–[9]). Another essential improvement contributes to Krstić and Deng. By introducing quartic Lyapunov function, the asymptotical stabilization controller was designed by Deng, Krstić and Williams in [10]–[12] and [13]. From that, the stability and stabilization for many class of stochastic nonlinear systems ([14], [15], etc.) are studied.