  We consider the problem of steering an initial probability density for the
state vector of a linear system to a final one, in finite time, using minimum
energy control. In the case where the dynamics correspond to an integrator
($\dot x(t) = u(t)$) this amounts to a Monge-Kantorovich Optimal Mass Transport
(OMT) problem. In general, we show that the problem can again be reduced to
solving an OMT problem and that it has a unique solution. In parallel, we study
the optimal steering of the state-density of a linear stochastic system with
white noise disturbance; this is known to correspond to a Schr\"odinger bridge.
As the white noise intensity tends to zero, the flow of densities converges to
that of the deterministic dynamics and can serve as a way to compute the
solution of its deterministic counterpart. The solution can be expressed in
closed-form for Gaussian initial and final state densities in both cases.
