Lyapunov Functions for Linear Hyperbolic Systems

Abstract

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}$</tex-math></inline-formula> -stability for a connected <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2 \times 2$</tex-math></inline-formula> system of linear 1-D hyperbolic systems are obtained. By means of examples we compare the capabilities of our approach with the existing ones.