Perturbations of superstable linear hyperbolic systems

TL;DR

This paper studies the stability of linear hyperbolic systems under perturbations, showing they remain exponentially stable and solutions become smooth over time, which is important for understanding their robustness.

Contribution

It proves that superstable linear hyperbolic systems maintain exponential stability under small bounded perturbations and solutions become eventually smooth.

Findings

Systems remain exponentially stable in $L^2$ and $C^1$ under perturbations.

Solutions to perturbed problems become eventually $C^1$-smooth.

Stability results apply to initial-boundary value problems for hyperbolic systems.

Abstract

The paper deals with initial-boundary value problems for linear non-autonomous first order hyperbolic systems whose solutions stabilize to zero in a finite time. We prove that problems in this class remain exponentially stable in $L^2$ as well as in $C^1$ under small bounded perturbations. To show this for $C^1$, we prove a general smoothing result implying that the solutions to the perturbed problems become eventually $C^1$-smooth for any $L^2$-initial data.