Solution to LQ control for Itô-type stochastic system with time delay

In this paper, we consider the linear quadratic (LQ) optimal control problem for Itô-type stochastic system with time delay. The maximum principle is obtained by applying the variational method to the cost functional when the admissible control set is convex. Based on the derived maximum principle, we give the explicit adapting and optimal controller. The main procedure is to establish the non-homogeneous relationship between the state and the adjoint state.