  In this paper, we study the optimal control problem for steering the state
covariance of a discrete-time linear stochastic system over a finite time
horizon. First, we establish the existence and uniqueness of the optimal
control law for a quadratic cost function. Then, we show the separation of the
optimal mean and the covariance steering problems. We also develop efficient
computational methods for solving for the optimal control law, which is
identified as the solution to a semi-definite program. The effectiveness of the
proposed approach is demonstrated through numerical examples. In the process,
we also obtain some novel theoretical results for a matrix Riccati difference
equation, which may be of independent interest.
