Delayed Optimal Control of Stochastic LQ Problem

A stochastic linear-quadratic (LQ) problem with multiplicative noises and transmis-sion delay is studied in this paper; this LQ problem does not require any definiteness constraint on the cost weighting matrices. From some abstract representations of the system and cost functional, the solvability of this LQ problem is characterized by some conditions with operator form. Based on these, necessary and sufficient conditions are derived for the case with a fixed time-state initial pair and the general case with all the time-state initial pairs. For both cases, a set of coupled discrete-time Riccati-like equations can be derived to characterize the existence and the form of the delayed optimal control. In particular, for the general case with all the initial pairs, the existence of delayed optimal control is equivalent to the solvability of the Riccati-like equations with some algebraic con-straints, and both of them are also equivalent to the solvability of a set of coupled linear matrix equality-inequalities. Note that both the constrained Riccati-like equations and the linear matrix equality-inequalities are introduced for the first time in the literature for the proposed LQ problem. Furthermore, the convexity and the uniform convexity of the cost functional are fully characterized via certain properties of the solution of the Riccati-like equations.