Calculus of Variations and Geometric Measure Theory

S. Dweik - N. Ghoussoub - A. Z. Palmer

Optimal controlled transports with free end times subject to import/export tariffs

created by dweik on 04 Mar 2019
modified on 08 Aug 2019


Accepted Paper

Inserted: 4 mar 2019
Last Updated: 8 aug 2019

Journal: Journal of Dynamical and Control Systems
Year: 2019


We analyze controlled mass transportation plans with free end-time that minimize the transport cost induced by the generating function of a Lagrangian within a bounded domain, in addition to costs incurred as export and import tariffs at entry and exit points on the boundary. We exhibit a dual variational principle à la Kantorovich, that takes into consideration the additional tariffs. We then show that the primal optimal transport problem has an equivalent Eulerian formulation whose dual involves the resolution of a Hamilton-Jacobi-Bellman quasi-variational inequality with non-homogeneous boundary conditions. This allows us to prove existence and to describe the solutions for both the primal optimization problem and its Eulerian counterpart.