复三次Ginzburg--Landau方程在可测集上的零能控性
Null controllability on measurable sets for the complex cubic Ginzburg--Landau equation
- School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过切片论证、谱不等式和分段反馈的时间迭代方案，证明了复三次Ginzburg--Landau方程在可测控制集上的局部零能控性。
AI中文摘要：
本文研究了控制支撑在一般时空可测集上的复三次Ginzburg--Landau方程的零能控性。从正测度控制集出发，我们使用切片论证获得空间测度一致为正的时间切片。这在这些切片上产生一致谱不等式，这是构造定量稳定反馈的关键要素。密度点论证提供了一列收缩区间，这些区间在密度点处累积，且在这些区间上活跃时间具有一致正密度。然后我们应用具有分段反馈律和递增衰减率的时间迭代方案。迭代所得稳定估计表明闭环解在选定的密度点处达到零状态。这为非线性抛物方程在可测控制集上的局部零能控性提供了一种构造性方法。
英文摘要：
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.
