Internal observability of the wave equation in a triangular domain

TL;DR

This paper extends observability results for the wave equation from rectangular to triangular domains using Fourier analysis and tessellation, providing new insights into observation times in such geometries.

Contribution

It introduces a method to relate observability in general domains to their tiles, specifically applying it to a triangular domain via tessellation theory.

Findings

Extended observability results to a triangular domain

Provided estimation of observation time for specific subdomains

Linked problems in general domains to their tiled counterparts

Abstract

We investigate the internal observability of the wave equation with Dirichlet boundary conditions in a triangular domain. More precisely, the domain taken into exam is the half of the equilateral triangle. Our approach is based on Fourier analysis and on tessellation theory: by means of a suitable tiling of the rectangle, we extend earlier observability results in the rectangle to the case of a triangular domain. The paper includes a general result relating problems in general domains to their tiles, and a discussion of the triangular case. As an application, we provide an estimation of the observation time when the observed domain is composed by three strips with a common side to the edges of the triangle.