Boundary stabilization and control of wave equations by means of a general multiplier method

TL;DR

This paper introduces a versatile multiplier method for boundary stabilization and control of wave equations, expanding applicability to new geometric configurations and handling mixed boundary conditions with singularities.

Contribution

It presents a novel general multiplier approach for boundary stabilization and control of wave equations, including new geometric cases and singularity management.

Findings

Achieved boundary stabilization using Neumann feedback under specific geometric conditions.

Extended the method to Dirichlet boundary control of wave equations.

Addressed singularities caused by mixed boundary conditions.

Abstract

We describe a general multiplier method to obtain boundary stabilization of the wave equation by means of a (linear or quasi-linear) Neumann feedback. This also enables us to get Dirichlet boundary control of the wave equation. This method leads to new geometrical cases concerning the "active" part of the boundary where the feedback (or control) is applied. Due to mixed boundary conditions, the Neumann feedback case generate singularities. Under a simple geometrical condition concerning the orientation of the boundary, we obtain a stabilization result in linear or quasi-linear cases.