Controllability of linear stochastic systems

We discuss several concepts of controllability for partially observable stochastic systems: complete controllability, approximate controllability, and stochastic controllability. We show that complete and approximate controllability notions are equivalent, and in turn they are equivalent to the stochastic controllability for linear stochastic systems controlled with Gaussian processes. We derive necessary and sufficient conditions for these concepts of controllability. These criteria reduce to the well-known rank condition.