We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation associated with the fractional Laplace operator subject to the non-homogeneous Dirichlet type exterior condition. In thefirst part, we show that if 0 < s < 1, Ω ⊂ RN(N ≥ 1) is a bounded Lipschitz domain and the parameter δ > 0, then there is no control function g such that the following system (formula present) is exact or null controllable at time T > 0. In the second part, we prove that for every δ ≥ 0 and 0 < s < 1, the system is indeed approximately controllable for any T > 0 and g 2 D(O ⊂(0; T)), where O Ω RN n is any non-empty open set.
