Global boundary stabilization of 1d systems of scalar conservation laws

TL;DR

This paper analyzes the stabilization of coupled 1D scalar conservation laws using boundary feedback, establishing well-posedness and conditions for global exponential stability in $L^1$ and $L^inity$ norms.

Contribution

It introduces a framework for boundary feedback control that guarantees well-posedness and exponential stability for coupled scalar conservation laws.

Findings

Proved well-posedness in the space of $L^{ty}$ entropy solutions.

Derived sufficient dissipative conditions for exponential stability.

Achieved global stabilization results for coupled 1D scalar conservation laws.

Abstract

We study a system of several one-dimensional scalar conservation laws coupled through boundary feedback conditions that combine physical boundary constraints with static feedback control laws. Our first contribution establishes the well-posedness of the system in the space of $L^{\infty}$ entropy solutions. Our second contribution provides a set of sufficient dissipative conditions on the boundary coupling that ensure global exponential stability in the $L^1$ and $L^\infty$ norms.