Exact and optimal controllability for scalar conservation laws with discontinuous flux

TL;DR

This paper develops a comprehensive framework for exact and optimal controllability of scalar conservation laws with discontinuous flux, providing criteria, algorithms, and proofs for reachability and control optimization.

Contribution

It introduces a new backward resolution method and explicit formulas to determine the reachable set and compute optimal controls for scalar conservation laws with discontinuous flux.

Findings

Established necessary and sufficient criteria for the reachable set

Proposed a backward algorithm to compute optimal controls

Proved existence of minimizers for the control problem

Abstract

This paper describes the reachable set and resolves an optimal control problem for the scalar conservation laws with discontinuous flux. We give a necessary and sufficient criteria for the reachable set. A new backward resolution has been described to obtain the reachable set. Regarding the optimal control problem we first prove the existence of a minimizer and then the backward algorithm allows us to compute it. The same method also applies to compute the initial data control for an exact control problem. Our methodology for the proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics curves.