论文标题
$ \ mathbb {r}^n $中某些抛物线方程的可稳定集的特征
Characterizations of stabilizable sets for some parabolic equations in $\mathbb{R}^n$
论文作者
论文摘要
我们考虑$ \ Mathbb {r}^n $：\ begin {align} \ label {equ-0}（\ partial_t+h）y（t，x）= 0，\，\，\，\，（t，x） \ quad y（0，x）\ in l^2（\ mathbb {r}^n），\ end {align}，其中$ h $可以是以下操作员之一：（i）转移的分数laplacian; （ii）转移的Hermite操作员； （iii）具有一些一般潜力的Schrödinger操作员。我们将子集$ e \ subset \ mathbb {r}^n $称为上述方程的可稳定设置，如果有$ l^2上有线性有界的操作员$ k $（\ mathbb {r}^n）$ 稳定的。 （在这里，$χ_e$表示$ e $的特征函数，该功能被视为$ l^2（\ Mathbb {r}^n）$的线性操作员。 本文介绍了上述方程式具有不同$ h $的可稳定套件的不同几何特征。 In particular, when $H$ is a shifted fractional Laplacian, $E\subset \mathbb{R}^n$ is a stabilizable set if and only if $E\subset \mathbb{R}^n$ is a thick set, while when $H$ is a shifted Hermite operator, $E\subset \mathbb{R}^n$ is a stabilizable set for if and只有$ e \ subset \ mathbb {r}^n $是一组积极的度量。我们的结果以及在\ cite {ab，ko，ko，li，m09}中获得的上述方程式的可观察到的结果，揭示了这种现象：对于一些$ h $，可稳定的套件严格包含一类可观察的套件，而对于其他$ h $，而其他$ h $，则可以可稳定的集合，可稳定的集合和可观察的集合和可观察的集合和可观察的套件。此外，本文为上述方程式提供了一些足够的条件，其中$ h $是具有一些一般潜力的Schrödinger运营商。
We consider the parabolic type equation in $\mathbb{R}^n$: \begin{align}\label{equ-0} (\partial_t+H)y(t,x)=0,\,\,\, (t,x)\in (0,\infty)\times\mathbb{R}^n;\;\; \quad y(0,x)\in L^2(\mathbb{R}^n), \end{align} where $H$ can be one of the following operators: (i) a shifted fractional Laplacian; (ii) a shifted Hermite operator; (iii) the Schrödinger operator with some general potentials. We call a subset $E\subset \mathbb{R}^n$ as a stabilizable set for the above equation, if there is a linear bounded operator $K$ on $L^2(\mathbb{R}^n)$ so that the semigroup $\{e^{-t(H-χ_EK)}\}_{t\geq 0}$ is exponentially stable. (Here, $χ_E$ denotes the characteristic function of $E$, which is treated as a linear operator on $L^2(\mathbb{R}^n)$.) This paper presents different geometric characterizations of the stabilizable sets for the above equation with different $H$. In particular, when $H$ is a shifted fractional Laplacian, $E\subset \mathbb{R}^n$ is a stabilizable set if and only if $E\subset \mathbb{R}^n$ is a thick set, while when $H$ is a shifted Hermite operator, $E\subset \mathbb{R}^n$ is a stabilizable set for if and only if $E\subset \mathbb{R}^n$ is a set of positive measure. Our results, together with the results on the observable sets for the above equation obtained in \cite{AB,Ko,Li,M09}, reveal such phenomena: for some $H$, the class of stabilizable sets contains strictly the class of observable sets, while for some other $H$, the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for the above equation where $H$ is the Schrödinger operator with some general potentials.
