  Exploiting a fluid dynamic formulation for which a probabilistic counterpart
might not be available, we extend the theory of Schroedinger bridges to the
case of inertial particles with losses and general, possibly singular diffusion
coefficient. We find that, as for the case of constant diffusion coefficient
matrix, the optimal control law is obtained by solving a system of two p.d.e.'s
involving adjoint operators and coupled through their boundary values. In the
linear case with quadratic loss function, the system turns into two matrix
Riccati equations with coupled split boundary conditions. An alternative
formulation of the control problem as a semidefinite programming problem allows
computation of suboptimal solutions. This is illustrated in one example of
inertial particles subject to a constant rate killing.
