Analysis of the controllability from the exterior of strong damping nonlocal wave equations

We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. In the first part, we show that if $0<s<1$, $\Omega\subset\RR^N$ ($N\ge 1$) is a bounded Lipschitz domain and the parameter $\delta> 0$, then there is no control function $g$ such that the following system \begin{equation*} \begin{cases} u_{tt} + (-\Delta)^{s} u + \delta(-\Delta)^{s} u_{t}=0 & \mbox{ in }\; \Omega\times(0,T),\\ u=g\chi_{\mathcal O\times (0,T)} &\mbox{ in }\; (\Omc)\times (0,T) ,\\ u(\cdot,0) = u_0, u_t(\cdot,0) = u_1 &\mbox{ in }\; \Omega, \end{cases} \end{equation*} is exact or null controllable at time $T>0$. In the second part, we prove that for every $\delta\ge 0$ and $0<s<1$, the system is indeed approximately controllable for any $T>0$ and $g\in \mathcal D(\mathcal O\times(0,T))$, where $\mathcal O\subset\Omc$ is any non-empty open set.