Solution to stochastic LQ control problem for Itô systems with state delay or input delay

This paper is concerned with the linear quadratic (LQ) control problem for Itô stochastic systems with state delay or input delay. The main contribution is to give the explicit optimal LQ controllers for Itô stochastic systems with input delay and with state delay respectively. For the case of state delay, the optimal LQ controller is a linear function of the state at the current time and the past time. For the case of input delay, the optimal LQ controller is a linear function of the state and the past inputs. The key technology is to define the value function and complete the square based on the coupled differential equations.