Stochastic linear quadratic optimal control problem with terminal state constraints

This paper addresses the stochastic linear quadratic (LQ) control problem with first- and second-order moment constraints on the terminal state. The problem is a modified version of the optimal covariance control problem, where the terminal state is steered to a given probability distribution. Studying a multiplicative-noise stochastic system rather than an additive-noise system is a salient feature. Unlike the existing ideas in the optimal steering, by using the Lagrange multipliers method and establishing the stochastic maximum principle, our problem is converted into solving forward-backward stochastic difference equations (FBSDEs), which is a special stochastic two-point boundary-value problem (TPBVP). We provide the optimal closed-loop controller and necessary and sufficient solvability conditions by developing a nonhomogeneous relationship between the optimal state and costate in FBSDEs. Finally, numerical examples are given to demonstrate our results.