  In this paper, we introduce a novel approach to solve the (mean-covariance)
steering problem for a fairly general class of linear continuous-time
stochastic systems subject to input delays. Specifically, we aim at steering
delayed linear stochastic differential equations to a final desired random
variable with given mean and covariance. We first establish a controllability
result for these control systems, revealing the existence of a lower bound
under which the covariance of the control system can not be steered. This
structural threshold covariance stems from a unique combined effect due to
stochastic diffusions and delays. Next, we propose a numerically cheap approach
to reach any neighbor of this threshold covariance in finite time. Via an
optimal control-based strategy, we enhance the aforementioned approach to keep
the system covariance small at will in the whole control horizon. Under some
additional assumptions on the dynamics, we give theoretical guarantees on the
efficiency of our method. Finally, numerical simulations are provided to ground
our theoretical findings, showcasing the ability of our methods in optimally
approaching the covariance threshold.
