Stabilization in probability of nonlinear stochastic systems with guaranteed region of attraction and target set

We deal with nonlinear dynamical systems, consisting of a linear nominal part perturbed by model uncertainties, nonlinearities and both additive and multiplicative random noise, modeled as a Wiener process. In particular, we study the problem of finding suitable measurement feedback control laws such that the resulting closed-loop system is stable in some probabilistic sense. To this aim, we introduce a new notion of stabilization in probability, which is the natural counterpart of the classical concept of regional stabilization for deterministic nonlinear dynamical systems and stands as an intermediate notion between local and global stabilization in probability. This notion requires that, given a target set, a trajectory, starting from some compact region of the state space containing the target, remains forever inside some larger compact set, eventually enters any given neighborhood of the target in finite time and remains thereinafter, all these events being guaranteed with some probability. We give a Lyapunov-based sufficient condition for achieving stability in probability and a separation result which splits the control design into a state feedback problem and a filtering problem. Finally, we point out constructive procedures for solving the state feedback and filtering problem with arbitrarily large region of attraction and arbitrarily small target for a wide class of nonlinear systems, which at least include feedback linearizable systems. The generality of the result is promising for applications to other classes of stochastic nonlinear systems. In the deterministic case, our results recover classical stabilization results for nonlinear systems.