Robust mean field control: stochastic maximum principle and variational mean field games

We introduce a class of robust control problems formulated in min-max form, in which the principal agent is viewed as a central planner facing Nature. The agent's cost is a nonlinear function of all its possible realizations, encompassing in particular the mean field regime where the cost depends on the distribution of the states. In parallel, Nature favors the occurrence of outcomes that are least favorable to the agent, at an entropic cost. We establish existence and uniqueness of solutions under appropriate assumptions, including suitable convexity-concavity conditions, and derive a related stochastic maximum principle. We further address a corresponding class of robust variational mean field games in which the interaction term is subject to ambiguity, and prove existence and uniqueness of solutions.