Dissipative boundary conditions for $2\times 2$ hyperbolic systems of conservation laws for entropy solutions in BV

TL;DR

This paper establishes exponential stability for 2x2 hyperbolic conservation law systems with dissipative boundary conditions, using a front tracking method and a Lyapunov functional inspired by Glimm's approach.

Contribution

It provides new sufficient conditions for BV stability of hyperbolic systems with dissipative boundaries, extending previous stability results with a novel Lyapunov functional.

Findings

Proves exponential BV stability under dissipative boundary conditions.

Develops a Lyapunov functional inspired by Glimm's method.

Uses front tracking to control BV norms of solutions.

Abstract

In this article, we investigate the BV stability of $2\times 2$ hyperbolic systems of conservation laws with strictly positive velocities under dissipative boundary conditions. More precisely, we derive sufficient conditions guaranteeing the exponential stability of the system under consideration for entropy solutions in BV. Our proof is based on a front tracking algorithm used to construct approximate piecewise constants solutions whose BV norms are controlled through a Lyapunov functional. This Lyapunov functional is inspired by the one proposed in J. Glimm's seminal work [J. Glimm, Comm. Pure Appl. Math., 18:697--715, 1965], modified with some suitable weights in the spirit of the previous works [J.-M. Coron, G. Bastin, and B. d'Andr\'ea Novel, SIAM J. Control Optim., 47(3):1460--1498, 2008] and [J.-M. Coron, B. d'Andr\'ea Novel, and G. Bastin, IEEE Trans. Automat. Control, 52(1):2--11, 2007].