Controllability of linear deterministic and stochastic systems

Necessary and sufficient conditions, which are formulated in terms of convergence of a certain sequence of operators involving the resolvent of the negative of the controllability operator, are found for deterministic linear stationary control systems to be completely and approximately controllable, respectively. It is shown that the S-controllability of partially observable linear stationary control systems with additive Gaussian white noise disturbance, that is a property of attaining an arbitrarily small neighborhood of each point in the state space with a probability arbitrarily close to one, is equivalent to the approximate controllability of their deterministic part. Also, it is shown that the C-controllability of the same kind of systems, that is defined as S-controllability improved with some uniformity conditions, is equivalent to the complete controllability of their deterministic part