Relative controllability of stochastic nonlinear systems with delay in control

The existence of solutions to systems is a natural premise to carry our study about controllability. Under the basic and readily verified conditions to guarantee the existence of the solutions to a system, in this paper, we prove the relative controllability (approximate controllability ) of the stochastic differential systems with delay in control. Sufficient conditions are given firstly for the relative controllability and relative approximate controllability in finite dimensional spaces, and these results are then generalized to infinite-dimensional Hilbert spaces. Finally, examples are given to illustrate the effectiveness of the proposed methods.