  In this paper, we study the covariance steering (CS) problem for
discrete-time linear systems subject to multiplicative and additive noise.
Specifically, we consider two variants of the so-called CS problem. The goal of
the first problem, which is called the exact CS problem, is to steer the mean
and the covariance of the state process to their desired values in finite time.
In the second one, which is called the ``relaxed'' CS problem, the covariance
assignment constraint is relaxed into a positive semi-definite constraint. We
show that the relaxed CS problem can be cast as an equivalent convex
semi-definite program (SDP) after applying suitable variable transformations
and constraint relaxations. Furthermore, we propose a two-step solution
procedure for the exact CS problem based on the relaxed problem formulation
which returns a feasible solution, if there exists one. Finally, results from
numerical experiments are provided to show the efficacy of the proposed
solution methods.
