# Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

## TL;DR

This paper establishes the asymptotic stability of the multidimensional wave equation with various physically relevant positive-real impedance boundary conditions, using spectral analysis and semigroup theory.

## Contribution

It introduces a unified framework for proving stability of wave equations with complex impedance boundary conditions via an abstract Cauchy problem approach.

## Key findings

- Proves asymptotic stability for wave equations with time-delayed boundary conditions.
- Extends stability results to diffusive and extended diffusive boundary conditions.
- Uses spectral conditions and semigroup theory for the proof.

## Abstract

This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and V\~u (Studia Math., 88 (1988)).