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 1d 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 partialtty - partialxxy + f(y) = 0 under an optimal growth assumption at infinity on f of the type sln2s. 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.