  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.
