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We present a novel distributionally robust framework for dynamic programmingthat uses kernel methods to design feedback control policies. Specifically, weleverage kernel mean embedding to map the transition probabilities governingthe state evolution into an associated repreducing kernel Hilbert space. Ourkey idea lies in combining conditional mean embedding with the maximum meandiscrepancy distance to construct an ambiguity set, and then design a robustcontrol policy using techniques from distributionally robust optimization. Themain theoretical contribution of this paper is to leverage functional analytictools to prove that optimal policies for this infinite-dimensional min-maxproblem are Markovian and deterministic. Additionally, we discuss approximationschemes based on state and input discretization to make the approachcomputationally tractable. To validate the theoretical findings, we conduct anexperiment on safe control for thermostatically controlled loads (TCL).

Distributionally Robust Optimal and Safe Control of Stochastic Systems via Kernel Conditional Mean Embedding