  We study a generalization of the Brownian bridge as a stochastic process that
models the position and velocity of inertial particles between the two
end-points of a time interval. The particles experience random acceleration and
are assumed to have known states at the boundary. Thus, the movement of the
particles can be modeled as an Ornstein-Uhlenbeck process conditioned on
position and velocity measurements at the two end-points. It is shown that
optimal stochastic control provides a stochastic differential equation (SDE)
that generates such a bridge as a degenerate diffusion process. Generalizations
to higher order linear diffusions are considered.
