Approximate controllability of a non-local hyperbolic coupled system

Abstract We analyse the interior approximate controllability of a non-local coupled system of damped wave equations, involving the fractional Laplacian. The single control is located in one of the equations. The other one depends on a non-negative parameter that makes the system converge to an integrodifferential parabolic–elliptic system, as it goes to zero. To obtain the main result, we first transformed the system into an abstract equation, revealing the main operator. Then, using the spectral properties of this operator and the semigroup theory, we established well-posedness results for the latter system and for its adjoint. Finally, we stated the approximate controllability of the initial system as a result of a strong unique continuation property for the fractional Laplacian.