Local controllability around a regular solution and null-controllability of scattering solutions for semilinear wave equations

Abstract

On a Riemannian manifold with or without boundary, and whether bounded or unbounded, we consider a semilinear wave (or Klein-Gordon) equation with a subcritical nonlinearity (either defocusing or focusing). We establish local controllability around a partially analytic solution, under the Geometric Control Condition. Specifically, some blow-up solutions can be controlled. In the case of a Klein-Gordon equation on a non-trapping exterior domain of small dimension, we prove the null-controllability of scattering solutions. The proof is based on local energy decay and global-in-time Strichartz estimates. Several consequences are presented, including the null-controllability of a solution initiated near the ground state in some focusing cases, and exact controllability in some defocusing cases.