Stability in determination of states for the mean field game equations

Abstract

We consider solutions satisfying the Neumann zero boundary condition and a linearized mean field game system in $ \Omega \times (0, T) $, where $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ and $ (0, T) $ is the time interval. We prove two kinds of stability results in determining the solutions. The first is Hölder stability in time interval $ ( \varepsilon, T) $ with arbitrarily fixed $ \varepsilon>0 $ by data of solutions in $ \Omega \times \{T\} $. The second is the Lipschitz stability in $ \Omega \times ( \varepsilon, T- \varepsilon) $ by data of solutions in arbitrarily given subdomain of $ \Omega $ over $ (0, T) $.