The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore–Gibson–Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. Assuming that \(b>0\) and \(\alpha -\frac{\tau c^2}{b}>0\), we show that if \(0<s<1\) and \(\varOmega \subset {\mathbb {R}}^N\) (\(N\ge 1\)) is a bounded domain with a Lipschitz continuous boundary \(\partial \varOmega \), then there is no control function g such that the following system

is exactly or null controllable in time \(T>0\). However, we prove that for \(0<s<1\), the system is approximately controllable for every \(g\in H^1((0,T);L^{2}({\mathcal {O}}))\), where \(\mathcal O\subset {\mathbb {R}}^N\setminus {\overline{\varOmega }}\) is an arbitrary non-empty open set.

从摩尔-吉布森-汤普森方程的外部分析了精确可控性、零可控性和近似可控性三个概念，该方程与非齐次狄利克雷型外部条件下的分数拉普拉斯算子相关。假设\(b>0\)和\(\alpha -\frac{\tau c^2}{b}>0\)，我们证明如果\(0<s<1\)和\(\varOmega \subset {\mathbb {R}}^N\) ( \(N\ge 1\) ) 是具有 Lipschitz 连续边界\(\partial \varOmega \)的有界域，则不存在控制函数g使得以下系统

在时间\(T>0\)上是完全可控的或为空可控的。然而，我们证明对于\(0<s<1\)，系统对于每个\(g\in H^1((0,T);L^{2}({\mathcal {O} }))\)，其中\(\mathcal O\subset {\mathbb {R}}^N\setminus {\overline{\varOmega }}\)是任意非空开集。