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.

The local controllability result has been strengthened compared to the first version of this article and now applies in higher dimensions