Null controllability on measurable sets for the complex cubic Ginzburg--Landau equation
Abstract
This paper investigates the null controllability of the complex cubic Ginzburg--Landau equation with controls supported on general space-time measurable sets. Starting from a positive-measure control set, we use a slicing argument to obtain time slices of uniformly positive spatial measure. This yields uniform spectral inequalities on the slices, which are the key ingredient in constructing quantitative stabilizing feedbacks. A density-point argument provides a sequence of shrinking intervals accumulating at the density point, on which the active times have a uniform positive density. We then apply a time-iteration scheme with piecewise feedback laws and increasing decay rates. Iterating the resulting stabilization estimates shows that the closed-loop solution reaches the zero state at the selected density point. This provides a constructive approach to the local null controllability on measurable control sets for nonlinear parabolic equations.
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BibTeX
@misc{xiao2026null,
title = {{Null controllability on measurable sets for the complex cubic Ginzburg--Landau equation}},
author = {Yu Xiao and Can Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29008},
eprint = {2609.29008},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29008},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Null controllability on measurable sets for the complex cubic Ginzburg--Landau equation
AU - Yu Xiao
AU - Can Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29008
UR - https://arxiv.org/abs/2609.29008
AB - This paper investigates the null controllability of the complex cubic Ginzburg--Landau equation with controls supported on general space-time measurable sets. Starting from a positive-measure control set, we use a slicing argument to obtain time slices of uniformly positive spatial measure. This yields uniform spectral inequalities on the slices, which are the key ingredient in constructing quantitative stabilizing feedbacks. A density-point argument provides a sequence of shrinking intervals accumulating at the density point, on which the active times have a uniform positive density. We then apply a time-iteration scheme with piecewise feedback laws and increasing decay rates. Iterating the resulting stabilization estimates shows that the closed-loop solution reaches the zero state at the selected density point. This provides a constructive approach to the local null controllability on measurable control sets for nonlinear parabolic equations.
ER -