Stability of the multidimensional wave equation in port-Hamiltonian modelling

TL;DR

This paper analyzes the stability of a multidimensional wave equation with spatially varying coefficients, demonstrating semi-uniform stability via port-Hamiltonian formulation and spectral analysis.

Contribution

It introduces a port-Hamiltonian framework for the wave equation and proves semi-uniform stability, extending understanding of stability in multidimensional systems.

Findings

System is semi-uniformly stable under feedback control

Spectrum of the operator lies in the left half-plane

Operator has a compact resolvent and eigenvalues only

Abstract

We investigate the stability of the wave equation with spatial dependent coefficients on a bounded multidimensional domain. The system is stabilized via a scattering passive feedback law. We formulate the wave equation in a port-Hamiltonian fashion and show that the system is semi-uniform stable, which is a stability concept between exponential stability and strong stability. Hence, this also implies strong stability of the system. In particular, classical solutions are uniformly stable. This will be achieved by showing that the spectrum of the port-Hamiltonian operator is contained in the left half plane $\mathbb{C}_{-}$ and the port-Hamiltonian operator generates a contraction semigroup. Moreover, we show that the spectrum consists of eigenvalues only and the port-Hamiltonian operator has a compact resolvent.