An analysis of the input-to-state-stabilisation of linear hyperbolic systems of balance laws with boundary disturbances

TL;DR

This paper investigates the input-to-state stability of linear hyperbolic balance law systems with boundary disturbances, proposing an explicit Lyapunov function and validating stability through numerical experiments on test examples.

Contribution

It introduces an explicit ISS-Lyapunov function for hyperbolic systems and explores its discretisation to ensure stability in numerical schemes.

Findings

The Lyapunov function decays as expected in experiments.

Discretised systems maintain input-to-state stability.

Numerical results confirm theoretical predictions.

Abstract

In this paper, a linear hyperbolic system of balance laws with boundary disturbances in one dimension is considered. An explicit candidate Input-to-State Stability (ISS)-Lyapunov function in $ L^2- $norm is considered and discretised to investigate conditions for ISS of the discrete system as well. Finally, experimental results on test examples including the Saint-Venant equations with boundary disturbances are presented. The numerical results demonstrate the expected theoretical decay of the Lyapunov function.