Stabilization of a Multi-Dimensional System of Hyperbolic Balance Laws

TL;DR

This paper develops a new Lyapunov-based feedback control method to stabilize multi-dimensional hyperbolic systems described by Hamilton-Jacobi equations, demonstrating effectiveness through theoretical analysis and examples.

Contribution

It introduces a novel Lyapunov function tailored for multi-dimensional geometry to achieve stabilization of hyperbolic PDE systems.

Findings

Stabilization in L^2 norm achieved for the system

New Lyapunov function accounts for multi-dimensional geometry

Examples based on forming processes demonstrate practical applicability

Abstract

We are interested in the feedback stabilization of systems described by Hamilton-Jacobi type equations in $\mathbb{R}^n$. A reformulation leads to a a stabilization problem for a multi-dimensional system of $n$ hyperbolic partial differential equations. Using a novel Lyapunov function taking into account the multi-dimensional geometry we show stabilization in $L^2$ for the arising system using a suitable feedback control. We further present examples of such systems partially based on a forming process.