Exact distributed controllability for the semilinear wave equation (Q2706902)

The system is NEWLINE\[NEWLINE y_{tt} - \Delta y + f(y) = u NEWLINE\]NEWLINE in a bounded \(n\)-dimensional domain \(\Omega\) with Dirichlet boundary condition. Exact controllability (or null controllability) of this system depends on the growth of \(f\) and \(f'\) (and, of course, on the energy spaces and control spaces used). Under the superlinear growth condition NEWLINE\[NEWLINE |f'(y)|\leq k|y|^{p - 1} \qquad (p > 1) NEWLINE\]NEWLINE the system can be controlled to zero energy locally with boundary controls and globally with distributed controls supported by certain subsets \(\omega \subseteq \Omega\) if \(p < 1 + 2/n,\) with ``\(\leq\)'' allowed for \(n = 1.\) The proof uses compactness and doesn't extend to \(p = 1 + 2/n,\) \(n \geq 2.\) The author extends the result to \(p = 1 + 2/n\) using different techniques and shows that the controls can be supported by subsets \(\gamma \subseteq \Gamma\) and \(\omega \subseteq \Omega\) as long as controllability for the linear wave equation holds for these sets.