  In this paper we investigate how the subgradients of the value function of a
discrete-time convex Bolza problem evolve over time. In particular, we develop
a discrete-time version of the characteristic method introduced by Rockafellar
and Wolenski in the 2000s, by showing that the time-evolution of the
subgradients of the value functions can be associated with trajectories of a
discrete-time Hamiltonian system. To do so, we first prove that the value
function has a dual counterpart, which corresponds to the conjugate of the
value function of a suitable dual problem. We finally make a discussion about
the qualification conditions we require for our results, showing in particular
that classical problems, such as the Liner-Quadratic regulator, satisfy these
hypotheses.
