  This paper studies an inverse hyperbolic problem for the wave equation with
dynamic boundary conditions. It consists of determining some forcing terms from
the final overdetermination of the displacement. First, the Fr\'echet
differentiability of the Tikhonov functional is studied, and a gradient formula
is obtained via the solution of an associated adjoint problem. Then, the
Lipschitz continuity of the gradient is proved. Furthermore, the existence and
the uniqueness for the minimization problem are discussed. Finally, some
numerical experiments for the reconstruction of an internal wave force are
implemented via a conjugate gradient algorithm.
