A non-exchangeable mean field control problem with controlled interactions

Abstract: This paper introduces and analyzes a new class of mean-field control (\textsc{MFC}) problems in which agents interact through a \emph{fixed but controllable} network structure. In contrast with the classical \textsc{MFC} framework -- where agents are exchangeable and interact only through symmetric empirical distributions -- we consider systems with heterogeneous and possibly asymmetric interaction patterns encoded by a structural kernel, typically of graphon type. A key novelty of our approach is that this interaction structure is no longer static: it becomes a genuine \emph{control variable}. The planner therefore optimizes simultaneously two distinct components: a \emph{regular control}, which governs the local dynamics of individual agents, and an \emph{interaction control}, which shapes the way agents connect and influence each other through the fixed structural kernel. \medskip We develop a generalized notion of relaxed (randomized) control adapted to this setting, prove its equivalence with the strong formulation, and establish existence, compactness, and continuity results for the associated value function under minimal regularity assumptions. Moreover, we show that the finite $n$-agent control problems with general (possibly asymmetric) interaction matrices converge to the mean-field limit when the corresponding fixed step-kernels converge in cut-norm, with asymptotic consistency of the optimal values and control strategies. Our results provide a rigorous framework in which the \emph{interaction structure itself is viewed and optimized as a control object}, thereby extending mean-field control theory to non-exchangeable populations and controlled network interactions.