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Maximum principle for optimal control of interacting particle system: stochastic flow model

Andrey A. Dorogovtsev, Yuecai Han, Kateryna Hlyniana, Yuhang Li

TL;DR

The paper tackles stochastic control of interacting particle systems where the dynamics and costs depend on the random mass distribution $\mu_t=\mu_0\circ X(\cdot,t)^{-1}$. It develops a stochastic maximum principle by introducing a generalized backward SDE with interaction and proves its well-posedness using Lions differentiability for measure-valued arguments, enabling adjoint equations that account for the evolving mass. A Hamiltonian-based necessary condition is derived, leading to a forward–backward system with interaction that characterizes optimal controls, with explicit expressions in the linear-quadratic case. The results extend stochastic control beyond classical McKean–Vlasov dynamics by incorporating a random mass flow driven by a Brownian sheet, offering a principled route to optimal transport-like objectives for mass distributions and potential applications in controlled mass redistribution.

Abstract

In this paper, we consider the stochastic optimal control problem for the interacting particle system. We obtain the stochastic maximum principle of the optimal control system by introducing a generalized backward stochastic differential equation with interaction. The existence and uniqueness of the solution of this type of equation is proved. We derive the necessary condition that the optimal control should satisfy. As an application, the linear quadratic case is investigated to illustrate the main results.

Maximum principle for optimal control of interacting particle system: stochastic flow model

TL;DR

The paper tackles stochastic control of interacting particle systems where the dynamics and costs depend on the random mass distribution . It develops a stochastic maximum principle by introducing a generalized backward SDE with interaction and proves its well-posedness using Lions differentiability for measure-valued arguments, enabling adjoint equations that account for the evolving mass. A Hamiltonian-based necessary condition is derived, leading to a forward–backward system with interaction that characterizes optimal controls, with explicit expressions in the linear-quadratic case. The results extend stochastic control beyond classical McKean–Vlasov dynamics by incorporating a random mass flow driven by a Brownian sheet, offering a principled route to optimal transport-like objectives for mass distributions and potential applications in controlled mass redistribution.

Abstract

In this paper, we consider the stochastic optimal control problem for the interacting particle system. We obtain the stochastic maximum principle of the optimal control system by introducing a generalized backward stochastic differential equation with interaction. The existence and uniqueness of the solution of this type of equation is proved. We derive the necessary condition that the optimal control should satisfy. As an application, the linear quadratic case is investigated to illustrate the main results.
Paper Structure (6 sections, 8 theorems, 153 equations)

This paper contains 6 sections, 8 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Brownian sheet
  3. Differentiation of Function of Measure-Valued Argument
  4. Backward stochastic differential equation with interaction
  5. Stochastic Maximum Principle
  6. Linear Quadratic Case

Key Result

Lemma 2.1

Let $M_t$ be a square integrable $\mathcal{F}_t$-adapted martingale. Then there exists $z(\cdot,\cdot)\in L^2([0,T]\times \mathbb{R}^d)$, such that

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Definition 3.1
  • Example 1
  • Remark 4.1
  • Definition 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 13 more