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