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In this paper, we study the covariance steering (CS) problem fordiscrete-time linear systems subject to multiplicative and additive noise.Specifically, we consider two variants of the so-called CS problem. The goal ofthe first problem, which is called the exact CS problem, is to steer the meanand 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 covarianceassignment constraint is relaxed into a positive semi-definite constraint. Weshow that the relaxed CS problem can be cast as an equivalent convexsemi-definite program (SDP) after applying suitable variable transformationsand constraint relaxations. Furthermore, we propose a two-step solutionprocedure for the exact CS problem based on the relaxed problem formulationwhich returns a feasible solution, if there exists one. Finally, results fromnumerical experiments are provided to show the efficacy of the proposedsolution methods.

Covariance Steering of Discrete-Time Linear Systems with Mixed Multiplicative and Additive Noise