Subgradients of Optimal-Value Functions in Dynamic Programming: The Case of Convex Systems Without Optimal Paths

We study the first-order behaviour of the optimal-value function associated to a convex dynamic programming problem. The optimization process takes place in a certain environment characterized by some perturbation parameters affecting the transition costs and/or the evolution law of the dynamic system. An important aspect of this work is that we do not assume the existence of optimal paths to the unperturbed problem.