An Occupation-Measure and Frank-Wolfe Framework for Heterogeneous Mean-Field Control

Heterogeneous mean-field control (MFC) problems involve multiple interacting populations with distinct dynamics, control constraints, and interaction patterns, making both analysis and computation substantially more difficult than in the homogeneous setting. In particular, existing formulations do not readily yield scalable solution methods that preserve population-level structure. To address this, we develop a heterogeneous occupation-measure mean-field control (OM-MFC) framework that lifts the problem to a population-level optimization over measures subject to dynamical constraints that are linear in the measures. We show that the resulting optimization problem is convex under a positive-semidefinite matrix-valued kernel condition, which captures coupled interactions across populations. Based on this formulation, we derive a Frank-Wolfe (FW) method whose linear minimization subproblem decomposes into independent population-wise optimal control problems, enabling parallel computation without requiring an a priori discretization of the measure space. Numerical examples on UAV coordination and search-and-rescue scenarios illustrate that the proposed framework captures symmetric coordination, asymmetric yielding, and directional interaction effects within a unified and computationally tractable trajectory-optimization framework.