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Stabilization of a Multi-Dimensional System of Hyperbolic Balance Laws

Michael Herty, Ferdinand Thein

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.

Stabilization of a Multi-Dimensional System of Hyperbolic Balance Laws

Abstract

We are interested in the feedback stabilization of systems described by Hamilton-Jacobi type equations in . A reformulation leads to a a stabilization problem for a multi-dimensional system of hyperbolic partial differential equations. Using a novel Lyapunov function taking into account the multi-dimensional geometry we show stabilization in for the arising system using a suitable feedback control. We further present examples of such systems partially based on a forming process.
Paper Structure (10 sections, 2 theorems, 69 equations)

This paper contains 10 sections, 2 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Stabilization of Multi-Dimensional Linear Hyperbolic Balance Laws
  3. Proof of Theorem \ref{['thm:main']}
  4. Estimating the Boundary Integral
  5. Estimating the Volume Integral
  6. Application to Stabilization of Hamilton--Jacobi Equations
  7. Potential Gradient Flow
  8. Vanishing Mixed Source Terms
  9. Constant Hamiltonian Gradient
  10. Summary

Key Result

Theorem 2.1

Assume ass1 and let $\mathbf{w}(t,\mathbf{x}) \in C^1\left((0,T),H^s(\Omega)\right)^n$, $s \geq 1+ d/2,$ be a solution to the IBVP eq:hyp_cons_sys. Define the Lyapunov function by and assume that there exists $\mu_i(\mathbf{x}) \in H^s(\Omega)$ such that holds true for some value $C_L \in \mathbb{R}_{>0}$. The boundary condition for eq:hyp_cons_sys is given by where we assume that for $i \in \{

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Corollary 3.5
  • proof
  • Remark 3.6