On the exact boundary controllability of semilinear wave equations

Abstract

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 ${\mathrm{limsup}}_{|r|\to \infty }|f(r)|/(|r|{\ln }^p|r|)\le \beta$ 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 ${\mathrm{limsup}}_{|r|\to \infty }|f^{\prime }(r)|/{\ln }^p|r|\le \beta$, we apply the Banach fixed point theorem and exhibit a strongly convergent sequence to a state-control pair for the semilinear equation.