[Paper Review] Control on Hilbert Spaces and Application to Mean Field Type Control Theory

Alain Bensoussan, P. Jameson Graber+1 more|arXiv (Cornell University)|May 21, 2020

Stochastic processes and financial applications7 citations

TL;DR

This paper introduces a novel Hilbert space-based approach to solving mean field type control problems by lifting the problem into an infinite-dimensional space resembling the tangent space of Wasserstein space. Using classical control theory on this structure, the authors establish a framework that simplifies the analysis of Bellman and Master equations, recovering classical solutions while preserving the advantages of Lions' lifting method with reduced complexity.

ABSTRACT

We propose a new approach to studying classical solutions of the Bellman equation and Master equation for mean field type control problems, using a novel form of the lifting idea introduced by P.-L. Lions. Rather than studying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytic techniques, we instead study a stochastic control problem on a specially constructed Hilbert space, which is reminiscent of a tangent space on the Wasserstein space in optimal transport. On this Hilbert space we can use classical control theory techniques, despite the fact that it is infinite dimensional. A consequence of our construction is that the mean field type control problem appears as a special case. Thus we preserve the advantages of the lifiting procedure, while removing some of the difficulties. Our approach extends previous work by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic.

Motivation & Objective

• To develop a new framework for analyzing mean field type control problems using Hilbert space techniques.
• To overcome limitations of traditional analytic methods for Hamilton-Jacobi/Fokker-Planck systems in mean field control.
• To extend prior lifting-based approaches to stochastic settings while maintaining the benefits of the lifting procedure.
• To demonstrate that mean field control problems emerge naturally as special cases within the proposed Hilbert space formulation.

Proposed method

• The authors construct a specially designed Hilbert space that mirrors the structure of the tangent space on Wasserstein space in optimal transport.
• They reformulate the mean field control problem as a stochastic control problem on this infinite-dimensional Hilbert space.
• Classical control theory techniques are applied directly on the Hilbert space, despite its infinite-dimensionality.
• The lifting procedure is adapted from P.-L. Lions' framework but extended to stochastic settings.
• The approach avoids direct analysis of coupled PDE systems by embedding the problem into a geometric structure amenable to standard control tools.
• The method preserves the advantages of lifting—such as simplifying nonlinear interactions—while reducing technical challenges.

Experimental results

Research questions

• RQ1 How can the Bellman and Master equations for mean field type control be analyzed without relying on traditional PDE techniques?
• RQ2 Can a Hilbert space framework be constructed such that mean field control problems become special cases of classical control on infinite-dimensional spaces?
• RQ3 What are the structural advantages of using a tangent-space-like Hilbert space in mean field control theory?
• RQ4 How does the proposed lifting method simplify the analysis compared to existing approaches?
• RQ5 In what way does the stochastic setting affect the applicability and structure of the lifting procedure?

Key findings

• The proposed Hilbert space formulation allows the application of classical control theory techniques to mean field type control problems, even in infinite dimensions.
• The mean field control problem is shown to naturally emerge as a special case within the new framework, preserving the geometric intuition of the lifting method.
• The approach avoids the analytical difficulties associated with solving coupled Hamilton-Jacobi/Fokker-Planck PDE systems directly.
• The lifting construction maintains the benefits of Lions' original method while extending it to stochastic settings.
• The framework provides a unified perspective that simplifies the study of classical solutions to the Bellman and Master equations.
• The method enables a more systematic and tractable analysis of mean field control problems through geometric and functional analytic tools.