A boundary feedback analysis for input-to-state-stabilisation of non-uniform linear hyperbolic systems of balance laws with additive disturbances

TL;DR

This paper develops boundary feedback control strategies for stabilizing non-uniform linear hyperbolic systems of balance laws with disturbances, using Lyapunov functions to prove decay rates both analytically and numerically.

Contribution

It introduces an input-to-state-stability framework with Lyapunov functions for hyperbolic systems, including discrete schemes with explicit decay rate derivations.

Findings

Lyapunov functions ensure $L^2$-stability of the system.

Explicit decay rates are derived for finite volume schemes.

Numerical simulations confirm theoretical stability results.

Abstract

A boundary feedback stabilisation problem of non-uniform linear hyperbolic systems of balance laws with additive disturbance is discussed. A continuous and a corresponding discrete Lyapunov function is defined. Using an input-to-state-stability (ISS) $ L^2- $Lyapunov function, the decay of solutions of linear systems of balance laws is proved. In the discrete framework, a first-order finite volume scheme is employed. In such cases, the decay rates can be explicitly derived. The main objective is to prove the Lyapunov stability for the $L^2$-norm for linear hyperbolic systems of balance laws with additive disturbance both analytically and numerically. Theoretical results are demonstrated by using numerical computations.