Stochastic controllability of systems with variable delay in control

In the paper finite-dimensional time-variable dynamical, control systems described by linear stochastic ordinary differential state equations with single time-variable point delay in the control are considered. Using notations, theorems and methods taken directly from, deterministic controllability problems 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.