  We consider the problem of minimum energy steering of a linear stochastic
system to a final prescribed distribution over a finite horizon and to maintain
a stationary distribution over an infinite horizon. We present sufficient
conditions for optimality in terms of a system of dynamically coupled Riccati
equations in the finite horizon case and algebraic in the stationary case. We
then address the question of feasibility for both problems. For the
finite-horizon case, provided the system is controllable, we prove that without
any restriction on the directionality of the stochastic disturbance it is
always possible to steer the state to any arbitrary Gaussian distribution over
any specified finite time-interval. For the stationary infinite horizon case,
it is not always possible to maintain the state at an arbitrary Gaussian
distribution through constant state-feedback. It is shown that covariances of
admissible stationary Gaussian distributions are characterized by a certain
Lyapunov-like equation. We finally present an alternative to solving the system
of coupled Riccati equations, by expressing the optimal controls in the form of
solutions to (convex) semi-definite programs for both cases. We conclude with
an example to steer the state covariance of the distribution of inertial
particles to an admissible stationary Gaussian distribution over a finite
interval, to be maintained at that stationary distribution thereafter by
constant-gain state-feedback control.
