An observability estimate for the wave equation and applications to the Neumann boundary controllability for semi-linear wave equations

TL;DR

This paper establishes a boundary observability estimate for a 1D wave equation with potential and applies it to prove the existence of Neumann boundary controls for semi-linear wave equations, supported by numerical experiments.

Contribution

It extends previous work to the Neumann boundary control case for semi-linear wave equations with new observability estimates and control construction methods.

Findings

Boundary observability for 1D wave with potential proven.

Existence of Neumann boundary control established under growth conditions.

Numerical experiments validate theoretical results.

Abstract

We give a boundary observability result for a $1$d wave equation with a potential. We then deduce with a Schauder fixed-point argument the existence of a Neumann boundary control for a semi-linear wave equation $\partial_{tt}y - \partial_{xx}y + f(y) = 0$ under an optimal growth assumption at infinity on $f$ of the type $s\ln^2s$. Moreover, assuming additional assumption on $f'$, we construct a minimizing sequence which converges to a control. Numerical experiments illustrate the results. This work extends to the Neumann boundary control case the work of Zuazua in $1993$ and the work of M\"unch and Tr\'elat in $2022$.