Dynamic programming for deterministic discrete-time systems with uncertain gain

We generalise the optimisation technique of dynamic programming for discrete-time systems with an uncertain gain function. We assume that uncertainty about the gain function is described by an imprecise probability model, which generalises the well-known Bayesian, or precise, models. We compare various optimality criteria that can be associated with such a model, and which coincide in the precise case: maximality, robust optimality and maximinity. We show that (only) for the first two an optimal feedback can be constructed by solving a Bellman-like equation.