The quadratic problem for stochastic linear control systems with delay

In this paper, we discuss the optimal control for stochastic linear system with delay in state and control variables and a quadratic criterion. This problem will lead to a kind of generalized forward-backward stochastic differential equations (FBSDEs) with Itô stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the system with time delay in control input via two different methods.