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A non-exchangeable mean field control problem with controlled interactions

Mao Fabrice Djete

TL;DR

This work extends mean-field control theory to non-exchangeable populations by treating the interaction structure as a controllable object encoded by a graphon-type kernel ${\rm G}$. It develops a generalized relaxed (randomized) control framework tailored to non-symmetric couplings, proves equivalence with the strong formulation, and establishes existence and continuity of the value function under minimal regularity. It also proves convergence of finite-$n$ cooperative control problems to a mean-field limit when the step-kernels converge in cut-norm, connecting discrete networks to graphon limits. The results enable optimizing both local dynamics and the interaction topology, enabling tractable analysis for networked, heterogeneous systems across social, economic, and engineered domains. Overall, the paper provides a rigorous, unifying framework for mean-field control with controlled interactions and non-exchangeable populations, with concrete implications for large-scale network design and optimization.

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.

A non-exchangeable mean field control problem with controlled interactions

TL;DR

This work extends mean-field control theory to non-exchangeable populations by treating the interaction structure as a controllable object encoded by a graphon-type kernel . It develops a generalized relaxed (randomized) control framework tailored to non-symmetric couplings, proves equivalence with the strong formulation, and establishes existence and continuity of the value function under minimal regularity. It also proves convergence of finite- cooperative control problems to a mean-field limit when the step-kernels converge in cut-norm, connecting discrete networks to graphon limits. The results enable optimizing both local dynamics and the interaction topology, enabling tractable analysis for networked, heterogeneous systems across social, economic, and engineered domains. Overall, the paper provides a rigorous, unifying framework for mean-field control with controlled interactions and non-exchangeable populations, with concrete implications for large-scale network design and optimization.

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 -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.

Paper Structure

This paper contains 22 sections, 21 theorems, 361 equations.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. The strong (closed--loop) formulation
  4. The relaxed/randomized formulation
  5. The strong (open--loop) formulation
  6. The n--particle formulation
  7. Examples and discussions
  8. Proofs of the main results
  9. A relaxed formulation
  10. Proofs of \ref{['thm:equivalence_relaxed_strong']}, \ref{['prop:existence_continuity']} and \ref{['prop:equiv_open']}
  11. Proof of \ref{['thm:equivalence_relaxed_strong']}
  12. Proof of \ref{['prop:existence_continuity']}
  13. Proof of \ref{['prop:equiv_open']}
  14. From strong formulation to n--particle: proof of \ref{['thm:strong_to_n']}
  15. From n--particle to relaxed formulation: proof of \ref{['thm:from_n_to_limit']}
  16. ...and 7 more sections

Key Result

Theorem 2.4

Let assum:main1_MF_CI hold, and let $\nu \in {\cal P}_p(\mathbb{R}^d)$ with $p \in \{0\} \cup [1,\infty)$. Then the set $\overline{\Pi}(\nu)$ is closed in the Wasserstein topology ${\cal W}_p$, and moreover, it coincides with the topological closure $($in ${\cal W}_p$$)$ of the set $\Pi(\nu)$. Conse

Theorems & Definitions (48)

  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.9
  • Remark 2.10
  • Remark 2.11
  • Theorem 2.12
  • ...and 38 more