  In this paper, we present a new control policy parametrization for the
finite-horizon covariance steering problem for discrete-time Gaussian linear
systems (DTGLS) which can reduce the latter stochastic optimal control problem
to a tractable optimization problem. The covariance steering problem seeks for
a feedback control policy that will steer the state covariance of a DTGLS to a
desired positive definite matrix in finite time. We consider two different
formulations of the covariance steering problem, one with hard terminal LMI
constraints (hard-constrained covariance steering) and another one with soft
terminal constraints in the form of a terminal cost which corresponds to the
squared Wasserstein distance between the actual terminal state (Gaussian)
distribution and the desired one (soft-constrained covariance steering). We
propose a solution approach that relies on the affine disturbance feedback
parametrization for both problem formulations. We show that this particular
parametrization allows us to reduce the hard-constrained covariance steering
problem into a semi-definite program (SDP) and the soft-constrained covariance
steering problem into a difference of convex functions program(DCP). Finally,
we show the advantages of our approach over other covariance steering
algorithms in terms of computational complexity and computation time by means
of theoretical analysis and numerical simulations.
