We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two end-points of a given interval. Their motion in phase space can be modeled as an Ornstein-Uhlenbeck process conditioned at the two end-points-a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a time-varying drift is sufficient to modify the prior SDE and meet the end-point conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions.