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Global exact controllability for Schrödinger equations with Lipschitz and logarithmic nonlinearities

Lei Zhang, Xiuting Li · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.27658 · cited by 0 works (OpenAlex) · open-access PDF

Korean

이 논문에서, 우리는 지역화 된 내부 첨가 제어에 의해 반선 Schrödinger 방정식의 글로벌 정확한 제어 가능성을 연구합니다. 각 글로벌 Lipschitz 비선형 차원에서, 우리는 모든 두 $L^2$ 상태가 조정 된 긍정적 인 시간에 연결할 수 있음을 증명합니다. 한 차원에서, 우리는 형태의 logarithmic 비선형 비선형에 대한 제어 가능성을 추가로 설립합니다 \(z) = f(z) = f(z) + mathfrak g(z), $φ$가 실제 가치가있는, $Zmath g(z)는 글로벌 Lipschitzeffection이며, 경우의 글로벌 raffitzability $RR1의 경우,이 문제를 가져오는 D-logr1를 가져온다.

Machine translation (Meta M2M100); the English original follows.

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.

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Topics: Lipschitz continuity, Controllability, Mathematics, Bounded function, Logarithm, Interval (graph theory)

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