  In this paper, we present new results on the covariance steering problem with
Wasserstein distance terminal cost. We show that the state history feedback
control policy parametrization, which has been used before to solve this class
of problems, requires an unnecessarily large number of variables and can be
replaced by a randomized state feedback policy which leads to more tractable
problem formulations without any performance loss. In particular, we show that
under the latter policy, the problem can be equivalently formulated as a
semi-definite program (SDP) which is in sharp contrast with our previous
results that could only guarantee that the stochastic optimal control problem
can be reduced to a difference of convex functions program. Then, we show that
the optimal policy that is found by solving the associated SDP corresponds to a
deterministic state feedback policy. Finally, we present non-trivial numerical
simulations which show the benefits of our proposed randomized state feedback
policy derived from the SDP formulation of the problem over existing approaches
in the field in terms of computational efficacy and controller performance.
