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

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 $2022$.