Stochastic Optimal Control of Interacting Particle Systems in Hilbert Spaces and Applications

2025-11-26

AI Summary

This work addresses the stochastic optimal control of interacting particle systems governed by stochastic evolution equations—including delayed and SPDE-type dynamics—in Hilbert space. The central problem is to characterize the mean-field limit of the control problem as the number of particles tends to infinity. Methodologically, we develop a novel L-viscosity solution theory for the Hamilton–Jacobi–Bellman (HJB) equation arising in the mean-field limit, integrating tools from measure-space analysis, path-dependent stochastic differential equations, and regularization via projection operators. We rigorously establish that the value functions of the finite-particle systems converge to the unique L-viscosity solution of this limiting HJB equation, which coincides with the value function of the mean-field control problem. Our theoretical framework is validated through applications in economic modeling. The contribution provides a unified, rigorous limit-analysis framework for optimal control of complex stochastic many-body systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems
• Security and Privacy: Large-scale security measurements

Abstract

Optimal control of interacting particles governed by stochastic evolution equations in Hilbert spaces is an open area of research. Such systems naturally arise in formulations where each particle is modeled by stochastic partial differential equations, path-dependent stochastic differential equations (such as stochastic delay differential equations or stochastic Volterra integral equations), or partially observed stochastic systems. The purpose of this manuscript is to build the foundations for a limiting theory as the number of particles tends to infinity. We prove the convergence of the value functions $u_n$ of finite particle systems to a function $mathcal{V}$, {which} is the unique {$L$}-viscosity solution of the corresponding mean-field Hamilton-Jacobi-Bellman equation {in the space of probability measures}, and we identify its lift with the value function $U$ of the so-called ``lifted''limit optimal control problem. Under suitable additional assumptions, we show $C^{1,1}$-regularity of $U$, we prove that $mathcal{V}$ projects precisely onto the value functions $u_n$, and that optimal (resp. optimal feedback) controls of the particle system correspond to optimal (resp. optimal feedback) controls of the lifted control problem started at the corresponding initial condition. To the best of our knowledge, these are the first results of this kind for stochastic optimal control problems for interacting particle systems of stochastic evolution equations in Hilbert spaces. We apply the developed theory to problems arising in economics where the particles are modeled by stochastic delay differential equations and stochastic partial differential equations.

Problem

• Establishes limiting theory for infinite-particle stochastic optimal control.
• Proves convergence of value functions to mean-field Hamilton-Jacobi-Bellman solutions.
• Applies theory to economic models with stochastic delay and partial differential equations.

Innovation

• Mean-field HJB equation for infinite particle limit
• Lifted control problem links finite and infinite systems
• Proves convergence and regularity in Hilbert spaces