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A Variational Approach to Mean Field Type Control

Alain Bensoussan, Ziyu Huang, Sheung Chi Phillip Yam

TL;DR

This work develops a variational, PDE-based approach to mean field type control by focusing on the Hamilton-Jacobi-Bellman equation for the linear functional derivative of the value function. It constructs a non-local, measure-dependent HJB and uses a Girsanov transformation to define the probability-measure flow, enabling analysis in the finite-dimensional state space ${\mathbb R^n}$ with modest regularity assumptions. The authors prove local-in-time and global-in-time solvability under convexity, derive comprehensive a priori estimates for $V$ and $DV$, establish time and space regularity, and show how master, Bellman, and PMP can be recovered from the HJB framework. The results provide a robust variational perspective on mean field control, yielding higher-order regularity and a flexible, less stringent analytical route compared to traditional HJB-FP or master equation approaches.

Abstract

Variational methods have been used to study stochastic control for long, see Bensoussan (1982) and Bensoussan-Lions (1978) for the early works. More precisely, variational approaches apply to the study of Bellman equation as a parabolic quasi-linear equation, when the nonlinearity affects only the gradient of the solution, and the second order derivative term is linear and not degenerate. This corresponds to a stochastic control problem, where the state equation is a diffusion process. The primary objective of this article is to extend this approach to mean field control theory, as an alternative to the current approach, which considers a coupled system of Hamilton-Jacobi (HJ) and Fokker-Planck (FP) equations, since the introduction of the theory by Lasry-Lions (2007). The main novelty lies in that the equation studied here is the HJB equation, neither the HJ-FP system nor the master equation; and our results also provide another perspective for probabilistic approaches; see Chassagneux-Crisan-Delarue (2022), Bensoussan-Wong-Yam-Yuan (2024), Bensoussan-Tai-Yam (2025) and Bensoussan-Huang-Tang-Yam (2025) for instance. Within the scope of the PDE methods, the advantage of this article is to solve a larger class of mean field control problems, with moderate regularity; and this kind of variational methods fairly require few conditions on the regularity of the coefficients.

A Variational Approach to Mean Field Type Control

TL;DR

This work develops a variational, PDE-based approach to mean field type control by focusing on the Hamilton-Jacobi-Bellman equation for the linear functional derivative of the value function. It constructs a non-local, measure-dependent HJB and uses a Girsanov transformation to define the probability-measure flow, enabling analysis in the finite-dimensional state space with modest regularity assumptions. The authors prove local-in-time and global-in-time solvability under convexity, derive comprehensive a priori estimates for and , establish time and space regularity, and show how master, Bellman, and PMP can be recovered from the HJB framework. The results provide a robust variational perspective on mean field control, yielding higher-order regularity and a flexible, less stringent analytical route compared to traditional HJB-FP or master equation approaches.

Abstract

Variational methods have been used to study stochastic control for long, see Bensoussan (1982) and Bensoussan-Lions (1978) for the early works. More precisely, variational approaches apply to the study of Bellman equation as a parabolic quasi-linear equation, when the nonlinearity affects only the gradient of the solution, and the second order derivative term is linear and not degenerate. This corresponds to a stochastic control problem, where the state equation is a diffusion process. The primary objective of this article is to extend this approach to mean field control theory, as an alternative to the current approach, which considers a coupled system of Hamilton-Jacobi (HJ) and Fokker-Planck (FP) equations, since the introduction of the theory by Lasry-Lions (2007). The main novelty lies in that the equation studied here is the HJB equation, neither the HJ-FP system nor the master equation; and our results also provide another perspective for probabilistic approaches; see Chassagneux-Crisan-Delarue (2022), Bensoussan-Wong-Yam-Yuan (2024), Bensoussan-Tai-Yam (2025) and Bensoussan-Huang-Tang-Yam (2025) for instance. Within the scope of the PDE methods, the advantage of this article is to solve a larger class of mean field control problems, with moderate regularity; and this kind of variational methods fairly require few conditions on the regularity of the coefficients.

Paper Structure

This paper contains 26 sections, 20 theorems, 294 equations, 1 table.

Table of Contents

  1. Introduction
  2. Basics of Mean Field Type Control
  3. Functionals on the Wasserstein Space
  4. Differential Calculus
  5. Mean Field Control Problem
  6. Setting of the Problem
  7. Pontryagin Maximum Principle and Bellman equation
  8. HJB Equation for Linear Functional Derivative of Value Function
  9. Master Equation
  10. The Hamilton Jacobi Bellman Equation
  11. The Measure $m^{m,t}_s$
  12. Recovering All Equations from the HJB Equation
  13. Study of the HJB Equation
  14. A Priori Estimate on the boundedness of $|V(s,x)|$
  15. A Priori Estimate on the Boundedness of $|DV(s,x)|$
  16. ...and 11 more sections

Key Result

Theorem 3.1

Suppose that FBSDEs eq:2-18 has an adapted solution $\left(Y^{m,t}_s(x),P^{m,t}_s(x),Q^{m,t}_s(x)\right)$. Then, the value function $\Phi$ is functionally differentiable in $m$ and satisfies Moreover, if $\Phi$ is continuous in $t$ and the derivative $D^2\frac{d\Phi}{d\nu}(t,m)(x)$ is continuous, then $\Phi$ is a solution of the following Bellman equation

Theorems & Definitions (40)

  • Remark 3.1
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.1: Important Observation
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • ...and 30 more