Stability of backward inverse problems for degenerate mean-field game systems
S. E. Chorfi, A. Habbal, M. Jahid, L. Maniar, A. Ratnani
TL;DR
This work addresses the backward-in-time inverse problem for a one-dimensional degenerate Mean-Field Game system with vanishing diffusion at the boundary. The authors develop Carleman estimates for the linearized HJB and FP components using a simple time weight $\varphi(t)=e^{\lambda t}$ and weighted spaces, then combine them to obtain a Carleman inequality for the full linearized MFG system. They prove conditional Hölder and logarithmic stability for the backward problem, first for the linearized system and then for the nonlinear system under natural coefficient bounds such as $|p|\le C\sqrt{a}$, $|d|\le Ca$, and $|a_x|\le C\sqrt{a}$. The results provide a theoretical foundation for backward reconstruction in degenerate MFGs and open avenues for numerical methods and extensions to higher dimensions and different boundary conditions.
Abstract
We investigate inverse backward-in-time problems for a class of second-order degenerate Mean-Field Game (MFG) systems. More precisely, given the final datum $(u(\cdot, T),m(\cdot, T))$ of a solution to the one-dimensional mean-field game system with a degenerate diffusion coefficient, we aim to determine the intermediate states $(u(\cdot,t_{0}),m(\cdot,t_{0}))$ for any $t_{0} \in [0, T)$, i.e., the value function and the mean distribution at intermediate times, respectively. We prove conditional stability estimates under suitable assumptions on the diffusion coefficient and the initial state $(u(\cdot,0),m(\cdot,0))$. The proofs are based on Carleman's estimates with a simple weight function. We first prove a Carleman estimate for the Hamilton-Jacobi-Bellman (HJB) equation. A second Carleman estimate will be derived for the Fokker-Planck (FP) equation. Then, by combining the two estimates, we obtain a Carleman estimate for the mean-field game system, leading to the stability of the backward problems.