We consider an infinite strip ${\mathrm{\Omega }}_L={(0,2\pi L)}^{d-1}\times \mathbb{R}$, $d\ge 2$, $L>0$, and study the control problem of the heat equation on ${\mathrm{\Omega }}_L$ with Dirichlet or Neumann boundary conditions, and control set $\omega \subset {\mathrm{\Omega }}_L$. We provide a sufficient and necessary condition for null‐controllability in any positive time $T>0$, which is a geometric condition on the control set ω. This is referred to as “thickness with respect to ${\mathrm{\Omega }}_L$” and implies that the set ω cannot be concentrated in a particular region of ${\mathrm{\Omega }}_L$. We compare the thickness condition with a previously known necessity condition for null‐controllability and give a control cost estimate which only shows dependence on the geometric parameters of ω and the time T.

中文翻译：

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无限带上热方程的零可控性与控制成本估计

我们考虑无限条带 ${\mathrm{\Omega }}_{\mathrm{大号}}={（0，\mathrm{2个}\pi \mathrm{大号}）}^{d-\mathrm{1个}}\times \mathbb{[R}$， $d\ge \mathrm{2个}$， $\mathrm{大号}>0$，并研究了热方程的控制问题 ${\mathrm{\Omega }}_{\mathrm{大号}}$ 带有Dirichlet或Neumann边界条件和控制集 $\omega \subset {\mathrm{\Omega }}_{\mathrm{大号}}$。我们提供了在任何积极时间进行零可控性的充分必要条件$Ť>0$，这是控制集ω上的几何条件。这被称为“相对于厚度${\mathrm{\Omega }}_{\mathrm{大号}}$”，这意味着集合ω不能集中在的特定区域 ${\mathrm{\Omega }}_{\mathrm{大号}}$。我们将厚度条件与先前已知的对零可控性的必要条件进行了比较，并给出了控制成本估算，该估算仅显示了对ω的几何参数和时间T的依赖性。