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Occupation-Measure Mean-Field Control: Optimization over Measures and Frank-Wolfe Methods

Di Yu, Sixiong You, Chaoying Pei

Abstract

Coordinating large populations of autonomous agents, such as UAV swarms or satellite constellations, poses significant computational challenges for traditional multi-agent control methods. This paper introduces a new optimization framework for large-population control, termed occupation-measure mean-field control (OM-MFC). The framework models the evolution of agent populations directly in the space of occupation measures and casts large-population control as an infinite-dimensional optimization problem over measures, which becomes convex under a positive-semidefiniteness condition on the interaction kernel. A Frank--Wolfe (FW) algorithm and its fully-corrective variant (FCFW) are developed to solve the resulting problem efficiently, where each iteration reduces to a classical optimal control subproblem. Theoretical results establish convexity, existence of optimal solutions, and convergence guarantees of the proposed algorithms. Owing to its measure-based formulation, the framework naturally accommodates systems with very large numbers of agents. Numerical experiments on UAV swarm coordination and satellite constellation control demonstrate the scalability and effectiveness of the proposed approach in high-dimensional and constrained environments.

Occupation-Measure Mean-Field Control: Optimization over Measures and Frank-Wolfe Methods

Abstract

Coordinating large populations of autonomous agents, such as UAV swarms or satellite constellations, poses significant computational challenges for traditional multi-agent control methods. This paper introduces a new optimization framework for large-population control, termed occupation-measure mean-field control (OM-MFC). The framework models the evolution of agent populations directly in the space of occupation measures and casts large-population control as an infinite-dimensional optimization problem over measures, which becomes convex under a positive-semidefiniteness condition on the interaction kernel. A Frank--Wolfe (FW) algorithm and its fully-corrective variant (FCFW) are developed to solve the resulting problem efficiently, where each iteration reduces to a classical optimal control subproblem. Theoretical results establish convexity, existence of optimal solutions, and convergence guarantees of the proposed algorithms. Owing to its measure-based formulation, the framework naturally accommodates systems with very large numbers of agents. Numerical experiments on UAV swarm coordination and satellite constellation control demonstrate the scalability and effectiveness of the proposed approach in high-dimensional and constrained environments.
Paper Structure (26 sections, 11 theorems, 71 equations, 8 figures, 2 algorithms)

This paper contains 26 sections, 11 theorems, 71 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation: The N-Agent Problem
  3. $N$-Agent Dynamics and Cost Functional
  4. Measure Representation of Single Trajectories
  5. Reformulation via Occupation Measures
  6. The OM-MFC Framework: A Measure-Theoretic Relaxation
  7. Problem Statement
  8. Relaxation and Relation to Classical MFC
  9. Properties of the Measure Optimization Problem
  10. Frank–Wolfe Methods for Optimization over Measures
  11. Complexity of Frank--Wolfe over Occupation Measures
  12. Fully-Corrective Frank--Wolfe
  13. Applications and numerical results
  14. UAV Swarm Coordination and Obstacle Avoidance
  15. Problem setup
  16. ...and 11 more sections

Key Result

Theorem 1

Consider agents $i=1,\dots,N$ with trajectories $\omega_i$ and associated occupation measures $(\mu[\omega_i],\nu[\omega_i])$. Define the empirical averages $\mu^N:=\tfrac{1}{N}\sum_{i=1}^N \mu[\omega_i]\in\mathcal{M}_+(\Sigma,T)$ and $\nu^N:=\tfrac{1}{N}\sum_{i=1}^N \nu[\omega_i]\in\mathcal{M}_+(\m Here the symbol $\delta(t-t')$ denotes the Dirac distribution and enforces $t=t'$ in the interactio

Figures (8)

  • Figure 1: 2D swarm with one obstacle (point-mass initial state): FCFW (purple points) vs. PDE (orange heatmap).
  • Figure 2: Objective and objective-gap convergence (point-mass initial state).
  • Figure 3: 2D swarm with one obstacle (distributed initial states): FCFW (purple points) vs. PDE (orange heatmap).
  • Figure 4: Objective and objective-gap convergence (distributed initial state).
  • Figure 5: 3D swarm with repulsion and 10 obstacle.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Theorem 1: Occupation–measure form of $J_N$
  • Example 1: Ghost Relaxation
  • Theorem 2: Convexity and compactness
  • Corollary 1: Solution Existence
  • Theorem 3: Gâteaux Differentiability
  • Theorem 4: Optimality Condition
  • Theorem 5: Solution to FW Subproblem
  • Definition 1: $L$-smoothness
  • Theorem 6: FW Complexity
  • Lemma 1: Disintegration
  • ...and 3 more