ON THE EXACT BOUNDARY CONTROLLABILITY OF SEMILINEAR WAVE EQUATIONS

We address the exact boundary controllability of the semilinear wave equation y tt- \Deltay y + f (y) y ) = 0 posed over a bounded domain \Omega of Rd. d . Assuming that f is continuous and satisfies the condition lim sup | r | \rightarrow \infty | f (r)|/(|r| r )| / (| r | lnp|r| ) p | r | ) G \beta for some \beta small enough and some p in [0, 3/2), / 2), we apply the Schauder fixed point theorem to prove the uniform controllability for initial data in L 2 (\Omega ) x H- 1 (\Omega ). Then, assuming that f is in C1(R) 1 (R) and satisfies the condition lim sup | r | \rightarrow \infty | f \prime ( r )| / lnp p | r | G \beta , we apply the Banach fixed point theorem and exhibit a strongly convergent sequence to a state- control pair for the semilinear equation.