Stochastic LQ Optimal Control With Initial and Terminal Constraints

In this article, we consider the discrete-time stochastic linear quadratic (LQ) optimal control with both initial and terminal constraints. The main contribution includes two aspects: one is to provide a necessary and sufficient condition for the exact reachability of stochastic systems; the other is to characterize the solvability condition of the constrained stochastic LQ optimal control problem based on the exact reachability of the stochastic systems and obtain the explicitly optimal controller. The key technique is to innovatively transform the stochastic system governed by a forward stochastic difference equation into one governed by a backward stochastic difference equation. This way, we are able to solve the forward and backward stochastic difference equations derived from the stochastic maximum principle.