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 ${\mathrm{\partial }}_{tt}y-{\mathrm{\partial }}_{xx}y+f(y)=0$ under an optimal growth assumption at infinity on $f$ of the type $s{\ln }^{2}s$. Moreover, assuming additional assumption on $f^{\mathrm{\prime }}$, 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 Münch and Trélat in $2022$.