Global exact controllability for Schrödinger equations with Lipschitz and logarithmic nonlinearities

Abstract

In this paper, we study global exact controllability of semilinear Schrödinger equations by localized internal additive controls. For every globally Lipschitz nonlinearity in arbitrary dimensions, we prove that any two $L^2$ states can be connected in any prescribed positive time, without smallness assumptions on the states or the Lipschitz constant. The result holds on smooth bounded Euclidean domains under a local convex-phase condition, and on an interval with any nonempty open control region. It also extends to compact Riemannian manifolds under the corresponding phase condition. In one dimension, we further establish controllability for logarithmic nonlinearities of the form \[ f(z)=zϕ(|z|)+\mathfrak g(z), \] where $ϕ$ is real-valued, $\mathfrak g$ is globally Lipschitz, and $r|ϕ'(r)|\leq D(1+|\log r|)^γ$ with $0\leqγ<1/2$. For the endpoint case $γ=1/2$, we achieve the result by assuming the leading coefficient is sufficiently small. These results resolve the global Lipschitz controllability question raised by Zuazua \cite[Section~3]{Zuazua2003} in the stated geometric settings and establish its logarithmic extension within the specified one-dimensional classes.