Subgradient Evolution of Value Functions of Discrete-Time Convex Bolza Problems

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 method of characteristics 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 discuss about the qualification conditions required for our results, showing in particular that classical problems, such as the Linear Quadratic Regulator, satisfy these hypotheses.