LQ control for Itô-type stochastic systems with input delays

This paper considers the linear quadratic (LQ) control problem for the Itô-type stochastic system with input delays. Due to simultaneous appearances of diffusion terms (dependent on the state and the control) as well as delays in the dynamic system, the problem is very involved and remains to be solved. We not only provide the solvable condition of the problem but also the explicit expression of the causal and adapted controller for a kind of LQ problems. All of these are based on a stochastic Riccati equation. The key technique is to pursue the explicit cost value of the LQ problem by FBSDE and derive the analytical controller via the interplay between the original problem and its equivalent abstract description.