Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics

We study the problem of output feedback stabilization for stochastic nonlinear systems. We consider a class of stochastic nonlinear systems in observer canonical form with stable zero-dynamics. We introduce a sequence of state transformations that transform the system into a lower triangular structure that is amenable for integrator backstepping design. Then we design the output feedback controller and prove that the closed-loop system is bounded in probability. Under an infinite-horizon risk-neutral cost criterion, the controller designed can guarantee an arbitrarily small long term average cost. Furthermore, when the disturbance vector field vanishes at the origin, the closed-loop system is asymptotically stable in the large, and the risk-neutral cost is guaranteed to be zero. With special care, the controller preserves the equilibrium of the nonlinear system.