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Linear-Quadratic Mean Field Games in Hilbert spaces

Salvatore Federico, Fausto Gozzi, Daria Ghilli

Abstract

This paper represents the first attempt to develop a theory for linear-quadratic mean field games in possibly infinite dimensional Hilbert spaces. As a starting point, we study the case, considered in most finite dimensional contributions on the topic, where the dependence on the distribution enters just in the objective functional through the mean. This feature allows, similarly to the finite dimensional case, to reduce the usual mean field game system to a Riccati equation and a forward-backward coupled system of abstract evolution equations. Such system is completely new in infinite dimension and no results have been proved on it so far. We show existence and uniqueness of solutions for such system, applying a delicate approximation procedure. We apply the results to a production output planning problem with delay in the control variable.

Linear-Quadratic Mean Field Games in Hilbert spaces

Abstract

This paper represents the first attempt to develop a theory for linear-quadratic mean field games in possibly infinite dimensional Hilbert spaces. As a starting point, we study the case, considered in most finite dimensional contributions on the topic, where the dependence on the distribution enters just in the objective functional through the mean. This feature allows, similarly to the finite dimensional case, to reduce the usual mean field game system to a Riccati equation and a forward-backward coupled system of abstract evolution equations. Such system is completely new in infinite dimension and no results have been proved on it so far. We show existence and uniqueness of solutions for such system, applying a delicate approximation procedure. We apply the results to a production output planning problem with delay in the control variable.
Paper Structure (14 sections, 8 theorems, 91 equations)

This paper contains 14 sections, 8 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Some literature
  3. A Sketch of our setting
  4. Our results and methods
  5. On the applications
  6. Plan of the paper
  7. Notations and formal setting
  8. Existence of solutions to the MFG system
  9. Uniqueness of solutions
  10. Application: a Production Output Planning Problem with Delay
  11. The model
  12. Reformulation in Hilbert space
  13. A useful lemma
  14. On Assumption \ref{['ass:etan']}

Key Result

Proposition 3.1

The Riccati equation Riccati:P admits a unique mild solution $P \in C_s([0,T]; \Sigma^+(H))$. Moreover,

Theorems & Definitions (22)

  • Definition 2.2
  • Definition 2.3: LQM mild solution to MFG
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Remark 3.3
  • Theorem 3.4: Existence
  • proof
  • Proposition 4.1: Uniqueness for small time horizon
  • proof
  • ...and 12 more