  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
Schr\"{o}dinger 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
$\{e^{-t(H-\chi_EK)}\}_{t\geq 0}$ 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, $E\subset \mathbb{R}^n$ 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 Schr\"{o}dinger operator with some
general potentials.
