  We consider optimal transport problems where the cost is optimized over
controlled dynamics and the end time is free. Unlike the classical setting, the
search for optimal transport plans also requires the identification of optimal
"stopping plans," and the corresponding Monge-Kantorovich duality involves the
resolution of a Hamilton-Jacobi-Bellman quasi-variational inequality. We
discuss both Lagrangian and Eulerian formulations of the problem, and its
natural connection to Pontryagin's maximum principle. We also exhibit a purely
dynamic situation, where the optimal stopping plan is a hitting time of a
barrier given by the free boundary problem associated to the dual variational
inequality. This problem was motivated by its stochastic counterpart, which
will be studied in a companion paper.
