We address the exact boundary controllability of the semilinear wave equation
∂tty−Δy+f(y)=0 posed over a bounded domain Ω of
Rd. Assuming that f is continuous and satisfies the condition
limsup∣r∣→∞∣f(r)∣/(∣r∣lnp∣r∣)≤β for some β small enough and some p∈[0,3/2), we
apply the Schauder fixed point theorem to prove the uniform controllability for
initial data in L2(Ω)×H−1(Ω). Then, assuming that f is
in C1(R) and satisfies the condition limsup∣r∣→∞∣f′(r)∣/lnp∣r∣≤β, we apply
the Banach fixed point theorem and exhibit a strongly convergent sequence to a
state-control pair for the semilinear equation