  This letter investigates dynamical optimal transport of underactuated linear
systems over an infinite time horizon. In our previous work, we proposed to
integrate model predictive control and the celebrated Sinkhorn algorithm to
perform efficient dynamical transport of agents. However, the proposed method
requires the invertibility of input matrices, which severely limits its
applicability. To resolve this issue, we extend the method to (possibly
underactuated) controllable linear systems. In addition, we ensure the
convergence properties of the method for general controllable linear systems.
The effectiveness of the proposed method is demonstrated by a numerical
example.
