Abstract
In this thesis, we propose and study inverse problems for the Mean Field Games (MFGs) governed by the first-order master equation in different domains. Mean Field Games are differential games involving non-atomic players, where one aims to study the behaviours of a large population of symmetric agents as the number of agents goes to infinity. They provide quantitative modelling of the macroscopic behaviours of the agents who wish to minimise a certain cost.We first consider the MFGs system defined on torus, which means this system satisfies periodic boundary conditions. We thoroughly examine the essential requirements for recovering specific coefficients. Furthermore, if the Lagrangian corresponds to kinetic energy, we present unique identifiability results. These results demonstrate that it is possible to retrieve either the running cost or the terminal cost (both of which are unknown coefficients) by understanding the total cost. In order to tackle the non-linearity, a powerful strategy is based on the linearisation. Most of past studies are concerned with a single-type of nonlinear PDE. In this thesis, we develop this method for the MFG system which couples two nonlinear PDEs.
Next, we develop a novel approach to ensure the probability measure constraint while effectively tackle the MFG inverse problems. We term this approach as high-order variation in combination with the successive/high-order linearisation. This approach is particularly powerful in tackling the case that running cost depends on measure non-locally. The linearization method hinges on a specific solution of the MFG system. To satisfy the probability measure constraint, multiple scenarios are explored involving various forms of these solutions.
Finally, as supplementary content, we introduce some inverse problems for single parabolic type equation.
There are several salient features of our study. The MFG system couples two nonlinear parabolic PDEs with one forward in time and the other one backward in time, and moreover there is a probability measure constraint. These make the inverse problem study new to the literature and highly challenging. We develop an effective and efficient scheme in tackling the inverse problem. Our study opens up a new field of research on inverse problems for MFGs.
| Date of Award | 12 Jul 2024 |
|---|---|
| Original language | English |
| Awarding Institution |
|
| Supervisor | Hongyu LIU (Supervisor) |