Optimal Transport with Controlled Dynamics and Free End Times

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.