  This paper addresses the problem of steering the distribution of the state of
a discrete-time linear system to a given target distribution while minimizing
an entropy-regularized cost functional. This problem is called a maximum
entropy (MaxEnt) density control problem. Specifically, the running cost is
given by quadratic forms of the state and the control input, and the initial
and final distributions are Gaussian. We first reveal that our problem boils
down to solving two Riccati difference equations coupled through their boundary
values. Based on them, we give the closed-form expression of the unique optimal
policy. Next, we show that the optimal policy for the density control of the
time-reversed system can be obtained simultaneously with the forward-time
optimal policy. Finally, by considering the limit where the entropy
regularization vanishes, we derive the optimal policy for the unregularized
density control problem.
