Linear-Quadratic Mean Field Games in Hilbert spaces

• Main Authors: Federico, Salvatore, Gozzi, Fausto, Ghilli, Daria
• Format: Journal Article
• Language: English
• Published: 22-02-2024
• Subjects: Mathematics - Analysis of PDEs; Mathematics - Dynamical Systems; Mathematics - Optimization and Control
• DOI: 10.48550/arxiv.2402.14935

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.