Feedback stabilization for entropy solutions of a 2x2 hyperbolic system of conservation laws at a junction

TL;DR

This paper demonstrates exponential stabilization of entropy solutions for a 2x2 hyperbolic conservation law system on a network using a front-tracking method and Lyapunov functionals, extending previous strategies.

Contribution

It extends the stabilization strategy to a network setting with specific transmission and boundary conditions, using a novel Lyapunov functional approach.

Findings

Entropy solutions are exponentially stabilizable on star-shaped networks.

The method controls BV norms via a Glimm-type Lyapunov functional.

The approach generalizes previous stabilization techniques to networked systems.

Abstract

We consider the p-system in Eulerian coordinates on a star-shaped network. Under suitable transmission conditions at the junction and dissipative boundary conditions in the exterior vertices, we show that the entropy solutions of the system are exponentially stabilizable. Our proof extends the strategy by Coron et al. (2017) and is based on a front-tracking algorithm used to construct approximate piecewise constant solutions whose BV norms are controlled through a suitable exponentially-weighted Glimm-type Lyapunov functional.