# On the optimization of conservation law models at a junction with inflow and flow distribution controls

## TL;DR

This paper develops a framework for optimizing conservation law models at network junctions with control over inflow and flow distribution, establishing solution existence and providing a variational formulation with numerical simulations.

## Contribution

It introduces a general control framework for conservation laws at junctions, proving solution compactness and existence, and formulates an equivalent variational problem with numerical insights.

## Key findings

- Established compactness of flux-traces in $L^1$
- Proved existence of solutions for optimization problems
- Provided a variational formulation and numerical simulations

## Abstract

The paper proposes a general framework to analyze control problems for conservation law models on a network. Namely we consider a general class of junction distribution controls and inflow controls and we establish the compactness in $L^1$ of a class of flux-traces of solutions. We then derive the existence of solutions for two optimization problems: (I) the maximization of an integral functional depending on the flux-traces of solutions evaluated at points of the incoming and outgoing edges; (II) the minimization of the total variation of the optimal solutions of problem (I). Finally we provide an equivalent variational formulation of the min-max problem (II) and we discuss some numerical simulations for a junction with two incoming and two outgoing edges.