Terminal Conditions in Infinite-Horizon Dynamic Optimization

At the heart of many such problems is dynamic programming, a recursive method introduced by Richard Bellman, which offers powerful tools for tackling both finite and infinite horizon scenarios. In this thesis, we distinguish the possible cases of solving dynamic optimization problems, differentiated according to finite or infinite horizon, focusing in particular on infinite horizon discrete-time problems, with a specific emphasis on the role of terminal conditions. In classical approaches, terminal conditions are imposed to guarantee that the solution of the Bellman equation indeed corresponds to the value function. However, these traditional terminal conditions often prove to be too strong or inapplicable, especially when the problem admits unbounded payoffs or includes trajectories leading to pathological states, such as ecological collapse or economic ruin. The main objective of this work is to investigate a weaker form of terminal condition, which still suffices to ensure the optimality of the derived solution but allows for broader applicability. We rigorously demonstrate that this weaker condition, formulated in terms of the limit behavior of the value function along admissible trajectories, is the weakest sufficient condition guaranteeing that a function satisfying the Bellman equation is indeed the correct value function. Furthermore, we show that this condition is also necessary: neglecting it can lead to erroneous value functions and spurious optimal controls, even in seemingly simple models.