Characterizations of stabilizable sets for some parabolic equations in $\mathbb{R}^n$

TL;DR

This paper characterizes stabilizable sets for certain parabolic equations in ^n, revealing geometric conditions for stabilization depending on the operator, and compares these with observable sets from prior work.

Contribution

It provides new geometric characterizations of stabilizable sets for parabolic equations with different operators, extending understanding of stabilization and observability in infinite-dimensional systems.

Findings

For shifted fractional Laplacian, stabilizable sets are exactly thick sets.

For shifted Hermite operator, stabilizable sets are exactly sets of positive measure.

The class of stabilizable sets can be strictly larger than observable sets depending on the operator.

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)\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.