  We consider an entropy-regularized version of optimal density control of
deterministic discrete-time linear systems. Entropy regularization, or a
maximum entropy (MaxEnt) method for optimal control has attracted much
attention especially in reinforcement learning due to its many advantages such
as a natural exploration strategy. Despite the merits, high-entropy control
policies induced by the regularization introduce probabilistic uncertainty into
systems, which severely limits the applicability of MaxEnt optimal control to
safety-critical systems. To remedy this situation, we impose a Gaussian density
constraint at a specified time on the MaxEnt optimal control to directly
control state uncertainty. Specifically, we derive the explicit form of the
MaxEnt optimal density control. In addition, we also consider the case where
density constraints are replaced by fixed point constraints. Then, we
characterize the associated state process as a pinned process, which is a
generalization of the Brownian bridge to linear systems. Finally, we reveal
that the MaxEnt optimal density control gives the so-called Schr\"odinger
bridge associated to a discrete-time linear system.
