Boundary controllability for a degenerate and singular wave equation

TL;DR

This paper establishes boundary controllability for a one-dimensional degenerate and singular wave equation at the boundary, using duality, multiplier methods, and Hardy inequalities, for subcritical/critical potentials and large times.

Contribution

It introduces a novel approach combining Hardy inequalities and multiplier methods to prove boundary controllability for degenerate and singular wave equations.

Findings

Boundary controllability is achieved for subcritical/critical potentials.

Controllability holds for sufficiently large time.

A new Hardy-type inequality is developed for the analysis.

Abstract

In this paper, we deal with the boundary controllability of a one-dimensional degenerate and singular wave equation with degeneracy and singularity occurring at the boundary of the spatial domain. Exact boundary controllability is proved in the range of subcritical/critical potentials and for sufficiently large time, through a boundary controller acting away from the degenerate/singular point. By duality argument, we reduce the problem to an observability estimate for the corresponding adjoint system, which is proved by means of the multiplier method and a new special Hardy-type inequality.