Exterior controllability properties for a fractional Moore–Gibson–Thompson equation

Abstract

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

$$\begin{aligned} {\left\{ \begin{array}{ll} \tau u_{ttt} + \alpha u_{tt}+c^2(-\varDelta )^{s} u + b(-\varDelta )^{s} u_{t}=0 &{} \text{ in } \; \varOmega \times (0,T),\\ u=g\chi _{{\mathcal {O}}} &{} \text{ in } \; ({\mathbb {R}}^N\setminus \varOmega )\times (0,T) ,\\ u(\cdot ,0) = u_0, u_t(\cdot ,0) = u_1, u_{tt}(\cdot ,0)=u_2 &{} \text{ in } \; \varOmega , \end{array}\right. } \end{aligned}$$

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.