On the exact boundary controllability of semilinear wave equations

TL;DR

This paper proves the exact boundary controllability of semilinear wave equations under certain growth conditions on the nonlinearity, using fixed point theorems to establish controllability results for initial data in specific function spaces.

Contribution

It introduces new controllability results for semilinear wave equations with nonlinearities satisfying specific growth conditions, employing Schauder and Banach fixed point theorems.

Findings

Proves uniform controllability for initial data in L^2 x H^{-1}

Establishes strongly convergent sequences to state-control pairs

Extends controllability results to nonlinearities with subcritical growth

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 $\limsup_{\vert r\vert\to \infty} \vert f(r)\vert /(\vert r\vert \ln^p\vert r\vert)\leq \beta$ 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.