论文标题
从内部到点控制的1D热方程和最小的控制时间
From internal to pointwise control for the 1D heat equation and minimal control time
论文作者
论文摘要
我们的目标是通过内部控制（或观察）设置$ω_ {\ varepsilon} =（x_ {0} - \ varepsilon，x_ {0}+\ varepsilon）$，以限制$ \ varepsilon \ rightarrow 0 $ {n $ （0,1）$。众所周知，取决于$ x_ {0} $的算术属性，可能存在最小的时间$ t_ {0} $ pointwise Control的$ x_ {0} $的热量方程。此外，对于任何$ \ varepsilon $固定，在任何时候$ t> 0 $中，加热方程可控制$ω_ {\ varepsilon} $。我们关联了这两个现象。我们表明，当$ t> t> t_ {0} $时，$ω__\ varepsilon $上的可观察性常数不会收敛到$ 0 $ as $ \ varepsilon \ rightarrow 0 $（在$ t> t> t_ {0} $时（在这种情况下，它与$ \ \ \ \ \ \ \ varepsilon^{1/2} $ t <t_ $ t <t_ $ cestrance concrance copitiance copitiance cobiots cobiotiage。我们还描述了$ω_ {\ varepsilon} $在限制$ \ varepsilon \ rightarrow 0 $上的最佳$ l^{2} $ null-controls的行为。
Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $ω_{\varepsilon}=(x_{0}-\varepsilon, x_{0}+\varepsilon )$, in the limit $\varepsilon\rightarrow 0$, where $x_{0}\in (0,1)$. It is known that depending on arithmetic properties of $x_{0}$, there may exist a minimal time $T_{0}$ of pointwise control at $x_{0}$ of the heat equation. Besides, for any $\varepsilon$ fixed, the heat equation is controllable with control set $ω_{\varepsilon}$ in any time $T>0$. We relate these two phenomena. We show that the observability constant on $ω_\varepsilon$ does not converge to $0$ as $\varepsilon\rightarrow 0$ at the same speed when $T>T_{0}$ (in which case it is comparable to $\varepsilon^{1/2}$) or $T<T_{0}$ (in which case it converges faster to $0$). We also describe the behavior of optimal $L^{2}$ null-controls on $ω_{\varepsilon}$ in the limit $\varepsilon \rightarrow 0$.
