Convergence results for control problems with unknown dynamic and applications to reinforcement learning

The study of optimal control problems under uncertainty plays an important role in many applications appearing, for instance, in engineering, biology, finance, and, in recent years, in machine learning and artificial intelligence. In this paper, we pursue the investigation in [13,14] and consider some affine optimal control problems where a term, $ f(x) $, is partially unknown. We assume that the admissible functions f vary in a space of functions X for which we know a measure π. We prove some convergence properties for the optimal policies and the multipliers of the problem as a family of measures πₙ weakly converging to π. The proofs strongly rely on Γ-convergence techniques. In the last section, we complete the discussion by proving the sufficiency of the Pontryagin-type optimality conditions under a suitable assumption.