Controllability of stochastic systems with delays

In the paper finite-dimensional stationary dynamical control systems described by linear stochastic ordinary differential state equations with multiple point delays in the control are considered. Using notations, theorems and methods taken directly from deterministic controllability problems for linear dynamical systems with delays in control, necessary and sufficient conditions for different kinds of stochastic relative controllability in a given time interval are formulated and proved. It will be proved that under suitable assumptions relative controllability of a deterministic linear associated dynamical system is equivalent to stochastic relative exact controllability and stochastic relative approximate controllability of the original linear stochastic dynamical system. Some remarks and comments on the existing results for stochastic controllability of linear dynamical systems are also presented.