On the controllability of a class of nonlinear stochastic systems

We study the controllability properties of the class of stochastic differential systems characterized by a linear controlled diffusion perturbed by a smooth, bounded, uniformly Lipschitz nonlinearity. We obtain conditions that guarantee the weak and strong controllability of the system. Also, given any open set in the state space we construct a control, depending only on the Lipschitz constant and the infinity-norm of the nonlinear perturbation, such that the hitting time of the set has a finite expectation with respect to all initial conditions.