On the exact boundary controllability of semilinear wave equations

We address the exact boundary controllability of the semilinear wave equation ${\mathrm{\partial }}_{tt}y-\mathrm{\Delta }y+f(y)=0$ posed over a bounded domain $\mathrm{\Omega }$ of ${\mathbb{R}}^d$. Assuming that $f$ is continuous and satisfies the condition ${\mathrm{lim\hspace{0.17em}sup}}_{\mathrm{\mid }r\mathrm{\mid }\to \mathrm{\infty }}\mathrm{\mid }f(r)\mathrm{\mid }\mathrm{/}(\mathrm{\mid }r\mathrm{\mid }{\ln }^p\mathrm{\mid }r\mathrm{\mid })\le \beta$ for some $\beta$ small enough and some $p\in [0,3\mathrm{/}2)$, we apply the Schauder fixed point theorem to prove the uniform controllability for initial data in $L^2(\mathrm{\Omega })\times H^{-1}(\mathrm{\Omega })$. Then, assuming that $f$ is in ${\mathcal{C}}^1(\mathbb{R})$ and satisfies the condition ${\mathrm{lim\hspace{0.17em}sup}}_{\mathrm{\mid }r\mathrm{\mid }\to \mathrm{\infty }}\mathrm{\mid }{f}^{\mathrm{\prime }}(r)\mathrm{\mid }\mathrm{/}{\ln }^p\mathrm{\mid }r\mathrm{\mid }\le \beta$, we apply the Banach fixed point theorem and exhibit a strongly convergent sequence to a state-control pair for the semilinear equation.