Stability of backward inverse problems for degenerate mean-field game systems
Résumé
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.
Mots clés
stability
2020 Mathematics Subject Classification. Primary: 35K65 35Q89; Secondary: 35R30 35R25 Degenerate system mean-field games backward problem Carleman estimate stability
2020 Mathematics Subject Classification. Primary: 35K65
35Q89; Secondary: 35R30
35R25 Degenerate system
Degenerate system
mean-field games
backward problem
Carleman estimate
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