Boundary feedback control for hyperbolic systems
Michael Herty, Ferdinand Thein
TL;DR
This work addresses boundary stabilization of general linear multi-dimensional first-order hyperbolic systems on bounded domains using a Lyapunov-based, weighted $L^2$ approach. It constructs a Lyapunov function with a weight $e^{\mu(\mathbf{x})}$ and derives a boundary LMI feasibility condition to design a stabilizing boundary feedback, linking decay to $C_A-C_B>0$. The main result proves exponential $L^2$ stability under these conditions and applies the framework to the barotropic Euler equations, illustrating broad applicability to symmetric hyperbolic systems. The findings provide a principled design tool for boundary control of multi-D hyperbolic PDEs and point to future work on nonlinear extensions and numerical implementations.
Abstract
We are interested in the feedback stabilization of general linear multi-dimensional first order hyperbolic systems in $\mathbb{R}^d$. Using a Lyapunov function with a suited weight function depending on the system under consideration we show stabilization in $L^2$ for the studied system using a suitable feedback control. Therefore the controllability of the studied system is related to the feasibility of an associated linear matrix inequality.We show the applicability discussing the barotropic Euler equations.