LYAPUNOV FUNCTIONS AND STABILITY OF LINEAR TWO-DIMENSIONAL HYPERBOLIC SYSTEMS

This is a working document of a work in progress. We study the boundary stability and stabilization of two-dimensional linear hyperbolic systems. In contrast with the 1d case, in multi-dimensional hyperbolic systems, a quadratic Lyapunov function for the $L^2$ norm is usually a weighted $L^2$ norm, with weights that must be uniform across all components. Applying such Lyapunov functions to prove stability requires that the system itself already possess a good internal structure. We demonstrate that this limitation can be overcome by considering quadratic Lyapunov functions for higher norm, such as $H^1$ norm, thereby opening up the possibility of finding quadratic Lyapunov functions for systems that do not satisfy a dissipative structure. We successfully construct a quadratic Lyapunov function for the $H^1$ norm for a concrete example, confirming the feasibility of our approach. Furthermore, we can use the aforementioned quadratic Lyapunov function for the $H^1$ norm to construct a Lyapunov function for the $L^2$ norm, thereby establishing $L^2$ stability for such systems.