  Mean field limits are an important tool in the context of large-scale
dynamical systems, in particular, when studying multiagent and interacting
particle systems. While the continuous-time theory is well-developed, few works
have considered mean field limits for deterministic discrete-time systems,
which are relevant for the analysis and control of large-scale discrete-time
multiagent system. We prove existence results for the mean field limit of very
general discrete-time control systems, for which we utilize kernel mean
embeddings. These results are then applied in a typical optimal control setup,
where we establish the mean field limit of the relaxed dynamic programming
principle. Our results can serve as a rigorous foundation for many applications
of mean field approaches for discrete-time dynamical systems.
