Insights into the stability analysis of 2-by-2 linear weakly hyperbolic systems

In this paper, we address the exponential stability analysis of steady states for one-dimensional linear weakly hyperbolic systems of conservation laws and balance laws, characterized by non-diagonalizable coefficient matrices. We derive sufficient conditions to ensure exponential stability, conducting the analysis both in the Laplace domain for a system of conservation laws and in the time domain using Lyapunov-based techniques. Numerical examples are presented to demonstrate the effectiveness of the proposed methods.