Inverse boundary problem for a mean field game system with probability density constraint

TL;DR

This paper addresses an inverse boundary problem for a mean field game system with probability density constraints, introducing a novel scheme to handle the challenges of boundary conditions and population measure preservation.

Contribution

It develops a new method for inverse boundary problems in mean field games with probability density constraints, considering reflective boundary conditions.

Findings

Successfully constructs probing modes satisfying density constraints

Provides a new scheme for inverse problems with population measure preservation

Addresses challenges of reflective boundary conditions in mean field games

Abstract

By following the study in [24], we consider an inverse boundary problem for the mean field game system where a probability density constraint is enforced on the game agents. That is, we consider the case that reflective boundary conditions are enforced and hence the population distribution of the game agents should be treated as a probability measure which preserves both positivity and the total population. This poses significant challenges for the corresponding inverse problems in constructing suitable ``probing modes" which should fulfill such a probability density constraint. We develop an effective scheme in tackling such a case which is new to the literature.