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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

Japanese

この論文では、私たちは、地元の内部添加コントロールによって半線形シュレディンガー方程式のグローバル正確なコントロール可能性を研究します。 任意の次元で各グローバルリプシッツ非線形性について、我々は、いかなる2つの$L^2$状態が、各州またはリプシッツ常数に小さい仮定なしに、任意の正確な正確な時間で接続することができることを証明します。 結果は、現地のコンヴェックス段階条件下で、および空白のオープンコントロール領域との間隔で、円滑に制限されたユークリッツ領域を保持します。 それはまた、適切な段階条件下でコンパクトリマルチ

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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