Exact controllability for nonlocal wave equations with nonlocal boundary conditions

We are interested in the controllability of nonlocal wave equations subject to nonlocal dynamical boundary conditions, where the nonlocality stems from integral terms on the bulk and on the boundary of the domain considered. First, we establish, in two geometric settings satisfying the geometric control condition (GCC), the internal observability of the corresponding local system using multipliers together with compactness-uniqueness results. Then, we prove that under analyticity assumptions on the kernels, the nonlocal system is also observable. Moreover, assuming the kernels are symmetric, the spectral properties of our system and a result on simultaneous observability allow us to show that, in a rectangular domain, the kernel on the bulk being analytic is enough for the system to be observable.