  In this work, we analyze the properties of the solution to the covariance
steering problem for discrete time Gaussian linear systems with a squared
Wasserstein distance terminal cost. In our previous work, we have shown that by
utilizing the state feedback control policy parametrization, this stochastic
optimal control problem can be associated with a difference of convex functions
program. Here, we revisit the same covariance control problem but this time we
focus on the analysis of the problem. Specifically, we establish the existence
of solutions to the optimization problem and derive the first and second order
conditions for optimality. We provide analytic expressions for the gradient and
the Hessian of the performance index by utilizing specialized tools from matrix
calculus. Subsequently, we prove that the optimization problem always admits a
global minimizer, and finally, we provide a sufficient condition for the
performance index to be a strictly convex function (under the latter condition,
the problem admits a unique global minimizer). In particular, we show that when
the terminal state covariance is upper bounded, with respect to the L\"{o}wner
partial order, by the covariance matrix of the desired terminal normal
distribution, then our problem admits a unique global minimizing state feedback
gain. The results of this paper set the stage for the development of
specialized control design tools that exploit the structure of the solution to
the covariance steering problem with a squared Wasserstein distance terminal
cost.
