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- W3036651437 abstract "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)timesmathbb{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 Schrodinger operator with some general potentials. We call a subset $Esubset mathbb{R}^n$ 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 ${e^{-t(H-chi_EK)}}_{tgeq 0}$ is exponentially stable. (Here, $chi_E$ 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, $Esubset mathbb{R}^n$ is a stabilizable set if and only if $Esubset mathbb{R}^n$ is a thick set, while when $H$ is a shifted Hermite operator, $Esubset mathbb{R}^n$ is a stabilizable set for if and only if $Esubset mathbb{R}^n$ 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 Schrodinger operator with some general potentials." @default.
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- W3036651437 date "2020-06-18" @default.
- W3036651437 modified "2023-09-27" @default.
- W3036651437 title "Characterizations of stabilizable sets for some parabolic equations in $mathbb{R}^n$." @default.
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