Stochastic LQ problem with delayed control

A discrete-time stochastic LQ problem with multiplicative noises and state transmission delay is studied in this paper, which does not require any definiteness constraint on the cost weighting matrices. Necessary and sufficient conditions are derived for the case with a fixed time-state initial pair. A set of coupled discrete-time Riccati-like equations can be derived to characterize the existence and the form of the delayed optimal control. Furthermore, the convexity of the cost functional is fully characterized via certain properties of the solution of the Riccati-like equations.