Controlling conservation laws I: entropy-entropy flux

TL;DR

This paper develops a variational framework for regularized conservation laws using entropy-entropy flux pairs, introducing a modified optimal transport space and demonstrating that these laws are flux-gradient flows, with applications to traffic flow and Burgers' equation.

Contribution

It introduces a novel variational approach and a modified optimal transport space for conservation laws, establishing their flux-gradient flow structure and enabling computational control methods.

Findings

Conservation laws with diffusion are flux-gradient flows.

Dual PDE systems are derived for regularized conservation laws.

Primal-dual algorithms effectively compute control strategies.

Abstract

We study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we demonstrate that conservation laws with diffusion are "flux--gradient flows". We next construct variational problems for these flows, for which we derive dual PDE systems for regularized conservation laws. Several examples, including traffic flow and Burgers' equation, are presented. Incorporating both primal-dual algorithms and monotone schemes, we successfully compute the control of conservation laws.