# Optimal Controlled Transports with Free End Times Subject to Import/Export Tariffs

## TL;DR

This paper develops a dual variational framework for optimal mass transportation with free end-times, incorporating import/export tariffs, and establishes existence and solution descriptions for the primal and Eulerian formulations.

## Contribution

It introduces a Kantorovich-like dual principle for transport problems with tariffs and free end-times, and links primal and Eulerian formulations via Hamilton-Jacobi-Bellman inequalities.

## Key findings

- Established a dual variational principle considering tariffs.
- Proved existence of solutions for primal and Eulerian problems.
- Derived Eulerian formulation involving Hamilton-Jacobi-Bellman inequalities.

## Abstract

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 \`a 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.