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Exploiting a fluid dynamic formulation for which a probabilistic counterpartmight not be available, we extend the theory of Schroedinger bridges to thecase of inertial particles with losses and general, possibly singular diffusioncoefficient. We find that, as for the case of constant diffusion coefficientmatrix, the optimal control law is obtained by solving a system of two p.d.e.'sinvolving adjoint operators and coupled through their boundary values. In thelinear case with quadratic loss function, the system turns into two matrixRiccati equations with coupled split boundary conditions. An alternativeformulation of the control problem as a semidefinite programming problem allowscomputation of suboptimal solutions. This is illustrated in one example ofinertial particles subject to a constant rate killing.

Optimal steering of inertial particles diffusing anisotropically with losses