Sampled-Data Feedback Stabilization in Mean Square for Stochastic Homogeneous Systems

For stochastic nonlinear systems which are only continuous but not necessarily local Lipschitz nor linear growth, we study the problem of asymptotic stabilization in mean square (AS-in-MS) via sampled-data feedback. We begin by establishing the existence of solutions for a class of hybrid stochastic systems. With the aid of weighted homogeneity, we then prove that for stochastic homogeneous systems of degree zero, asymptotic stabilizability in mean square by homogeneous feedback implies asymptotic stabilizability in mean square by sampled-data feedback, under a fast sampling. The novelty and significance of the obtained sampled-data control results are illustrated by various applications to representative classes of stochastic nonlinear systems with uncontrollable unstable linearization including, but not limited to, stochastic systems with genuine nonlinearity in a lower triangular form, an upper triangular form and beyond.