Exponential Stability in the Sense of Lyapunov for a Linear Hyperbolic System of Balance Laws

Hyperbolic balance laws describe gas dynamics, open-channel flow, and transport processes. For such systems, the stability of the steady state is a fundamental issue of both theoretical and engineering interest [1]. The exponential stability of a linearized hyperbolic system is, in general, not equivalent to the local exponential stability of the underlying nonlinear system: even for smooth initial data close to a steady state, trajectories of nonlinear conservation laws may develop shock-type discontinuities in finite time. For this reason, the linear problem itself must be analyzed directly, in the sense of Lyapunov, without relying on indirect linearization arguments. A natural and robust tool for such an analysis is the method of weighted quadratic Lyapunov functionals, adapted to the hyperbolic setting [1,2].