  Fractional Pontryagin's systems emerge in the study of a class of fractional
optimal control problems but they are not resolvable in most cases. In this
paper, we suggest a numerical approach for these fractional systems. Precisely,
we construct a variational integrator allowing to preserve at the discrete
level their intrinsic variational structure. The variational integrator
obtained is then called shifted discrete fractional Pontryagin's system. We
provide a solved fractional example in a certain sense. It allows us to test in
this paper the convergence of the variational integrator constructed. Finally,
we also provide a discrete fractional Noether's theorem giving the existence of
an explicit computable discrete constant of motion for shifted discrete
fractional Pontryagin's systems admitting a discrete symmetry.
