Lyapunov Functions for Linear Hyperbolic Systems

In this article, we develop new methods to construct a Lyapunov function for 1-D linear hyperbolic equations with variable coefficients. The main focus is on the nonstrictly hyperbolic case for which we give an example demonstrating that existing approaches cannot provide sufficient conditions for the asymptotic stability, but our approach does. Sufficient conditions for exponential
$L^{2}$
-stability for a connected
$2 \times 2$
system of linear 1-D hyperbolic systems are obtained. By means of examples, we compare the capabilities of our approach with the existing ones.