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In this paper, we introduce a novel approach to solve the (mean-covariance)steering problem for a fairly general class of linear continuous-timestochastic systems subject to input delays. Specifically, we aim at steeringdelayed linear stochastic differential equations to a final desired randomvariable with given mean and covariance. We first establish a controllabilityresult for these control systems, revealing the existence of a lower boundunder which the covariance of the control system can not be steered. Thisstructural threshold covariance stems from a unique combined effect due tostochastic diffusions and delays. Next, we propose a numerically cheap approachto reach any neighbor of this threshold covariance in finite time. Via anoptimal control-based strategy, we enhance the aforementioned approach to keepthe system covariance small at will in the whole control horizon. Under someadditional assumptions on the dynamics, we give theoretical guarantees on theefficiency of our method. Finally, numerical simulations are provided to groundour theoretical findings, showcasing the ability of our methods in optimallyapproaching the covariance threshold.

Mean-Covariance Steering of a Linear Stochastic System with Input Delay and Additive Noise