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Mean Field Control by Stochastic Koopman Operator via a Spectral Method

Yuhan Zhao, Juntao Chen, Yingdong Lu, Quanyan Zhu

Abstract

Mean field control provides a robust framework for coordinating large-scale populations with complex interactions and has wide applications across diverse fields. However, the inherent nonlinearity and the presence of unknown system dynamics pose significant challenges for developing effective analytic or numerical solutions. There is a pressing need for data-driven methodologies to construct accurate models and facilitate efficient planning and control. To this end, we leverage Koopman operator theory to advance solution methods for mean field control problems. Our approach involves exploring stochastic Koopman operators using spectral analysis techniques. Through Koopman decomposition, we derive a linear model for mean field control problems in a data-driven fashion. Finally, we develop a model predictive control framework to achieve robust control and reduce the computational complexity for mean field control problems, thereby enhancing the efficacy and applicability of mean field control solutions in various domains.

Mean Field Control by Stochastic Koopman Operator via a Spectral Method

Abstract

Mean field control provides a robust framework for coordinating large-scale populations with complex interactions and has wide applications across diverse fields. However, the inherent nonlinearity and the presence of unknown system dynamics pose significant challenges for developing effective analytic or numerical solutions. There is a pressing need for data-driven methodologies to construct accurate models and facilitate efficient planning and control. To this end, we leverage Koopman operator theory to advance solution methods for mean field control problems. Our approach involves exploring stochastic Koopman operators using spectral analysis techniques. Through Koopman decomposition, we derive a linear model for mean field control problems in a data-driven fashion. Finally, we develop a model predictive control framework to achieve robust control and reduce the computational complexity for mean field control problems, thereby enhancing the efficacy and applicability of mean field control solutions in various domains.

Paper Structure

This paper contains 19 sections, 3 theorems, 24 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic Koopman Operator for Mean Field Control Problem
  4. Stochastic Koopman Operator without Control
  5. Stochastic Koopman Operator with Control
  6. Spectral Theorem and Spectral Measure
  7. Spectral Theorem
  8. Spectral Measure
  9. Lebesgue Decomposition of Measures
  10. Spectral Analysis of the stochastic Koopman Operator
  11. Measure-Preserving Dynamical Systems
  12. Data-Driven Spectral Measure Characterization of the Koopman Operator
  13. Characterizing A Real-valued Measure on An Interval
  14. Recovering the Spectral Measure
  15. Recovering the Eigenvalues and Eigenfunctions
  16. ...and 4 more sections

Key Result

Theorem II.1

For any bounded self-adjoint operator $A$ on a Hilbert space ${\mathcal{H}}$, there exists a resolution of identity, $E_t$, supported on $[-\|A\|, \|A\|]$ with $\|A\|$ being the operator norm of $A$. Furthermore, the operator $A$ has a spectral integration representation $A=\int t dE_t$.

Theorems & Definitions (9)

  • Definition 1
  • Remark II.1
  • Remark II.2
  • Definition 2: Resolution of Identity
  • Theorem II.1: Spectral Theorem simon2015operator
  • Remark II.3
  • Theorem III.1
  • Theorem III.2
  • proof