Almost Sure Asymptotic Stabilization of Stochastic Nonlinear Systems by Sampled-Data State and Output Feedback

The purpose of this article is to address the problem of global asymptotic stabilization almost surely (GAS a.s.) by sampled-data state and output feedback, respectively, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C^{0}$</tex-math></inline-formula> stochastic nonlinear systems that may be neither locally Lipschitz continuous nor linear growth, and are in general not smoothly stabilizable. Applying the homogeneous systems theory in a stochastic setting, combined with the existence of solutions of hybrid stochastic homogeneous systems established recently, we prove that for multi-input stochastic nonlinear systems with homogeneous degree zero, GAS a.s. by homogeneous state feedback implies GAS a.s. by sampled-data state feedback under a fast sampling. In the case of output feedback, we further show that for multi-input-multi-output stochastic homogeneous systems, GAS a.s. by homogeneous output feedback implies GAS a.s. by sampled-data output feedback. As illustrations, we present several applications of the new sampled-data control results to certain classes of stochastic nonlinear systems with special structures. Simulations are also given to validate the sampled-data control schemes developed in this work.