Unique continuation and stabilization for nonlinear Schr\"odinger equations under the Geometric Control Condition

In this article we prove global propagation of analyticity in finite time for solutions of semilinear Schr\"odinger equations with analytic nonlinearity from a region $\omega$ where the Geometric Control Condition holds. Our approach refines a recent technique introduced by Laurent and the author, which combines control theory techniques and Galerkin approximation, to propagate analyticity in time from a zone where observability holds. As a main consequence, we obtain unique continuation for subcritical semilinear Schr\"odinger equations on compact manifolds of dimension $2$ and $3$ when the solution is assumed to vanish on $\omega$. Furthermore, semiglobal control and stabilization follow only under the Geometric Control Condition on the observation zone. In particular, this answers in the affirmative an open question of Dehman, G\'erard, and Lebeau from $2006$ for the nonlinear case.