  This paper studies the set of terminal state covariances that are reachable
over a finite time horizon from a given initial state covariance for a linear
stochastic system with additive noise. For discrete-time systems, a complete
characterization of the set of reachable state covariances is given. For
continuous-time systems, we present an upper bound on the set of reachable
state covariances. Moreover, for both linear discrete-time and continuous-time
systems, necessary and sufficient conditions are provided for the
controllability of the state covariance over a finite horizon.
