  In this paper, we study the problem of how to optimally steer the state
covariance of a general continuous-time linear stochastic system over a finite
time interval subject to additive noise. Optimality here means reaching a
target state covariance with minimal control energy. The additive noise may
include a combination of white Gaussian noise and abrupt "jump noise" that is
discontinuous in time. We first establish the controllability of the state
covariance for linear time-varying stochastic systems. We then turn to the
derivation of the optimal control, which entails solving two dynamically
coupled matrix ordinary differential equations (ODEs) with split boundary
conditions. We show the existence and uniqueness of the solution to these
coupled matrix ODEs, and thus those of the optimal control.
