  We consider the problem to steer a linear dynamical system with full state
observation from an initial gaussian distribution in state-space to a final one
with minimum energy control. The system is stochastically driven through the
control channels; an example for such a system is that of an inertial particle
experiencing random "white noise" forcing. We show that a target probability
distribution can always be achieved in finite time. The optimal control is
given in state-feedback form and is computed explicitely by solving a pair of
differential Lyapunov equations that are coupled through their boundary values.
This result, given its attractive algorithmic nature, appears to have several
potential applications such as to active control of nanomechanical systems and
molecular cooling. The problem to steer a diffusion process between end-point
marginals has a long history (Schr\"odinger bridges) and therefore, the present
case of steering a linear stochastic system constitutes a Schr\"odinger bridge
for possibly degenerate diffusions. Our results, however, provide the first
implementable form of the optimal control for a general Gauss-Markov process.
Illustrative examples of the optimal evolution and control for inertial
particles and a stochastic oscillator are provided. A final result establishes
directly the property of Schr\"{o}dinger bridges as the most likely random
evolution between given marginals to the present context of linear stochastic
systems.
