  In this paper, we review the discrete Hamilton--Jacobi theory from a
geometric point of view. In the discrete realm, the usual geometric
interpretation of the Hamilton--Jacobi theory in terms of vector fields is not
straightforward.
  Here, we propose two alternative interpretations: one is the interpretation
in terms of projective flows, the second is the temptative of constructing a
discrete Hamiltonian vector field renacting the usual continuous
interpretation.
  Both interpretations are proven to be equivalent and applied in optimal
control theory. The solutions achieved through both approaches are sorted out
and compared by numerical computation.
