  We consider the optimal control problem of steering an agent population to a
desired distribution over an infinite horizon. This is an optimal transport
problem over dynamical systems, which is challenging due to its high
computational cost. In this paper, by using entropy regularization, we propose
Sinkhorn MPC, which is a dynamical transport algorithm integrating model
predictive control (MPC) and the so-called Sinkhorn algorithm. The notable
feature of the proposed method is that it achieves cost-effective transport in
real time by performing control and transport planning simultaneously, which is
illustrated in numerical examples. Moreover, under some assumption on
iterations of the Sinkhorn algorithm integrated in MPC, we reveal the global
convergence property for Sinkhorn MPC thanks to the entropy regularization.
Furthermore, focusing on a quadratic control cost, without the aforementioned
assumption we show the ultimate boundedness and the local asymptotic stability
for Sinkhorn MPC.
