Lipschitz stability for determination of states and inverse source problem for the mean field game equations
Oleg Imanuvilov, Hongyu Liu, Masahiro Yamamoto
Abstract
In a bounded domain $ \Omega \subset \mathbb{R}^d $, $ d\geq 1 $, over a time interval $ (0,T) $, we consider mean field game equations whose principal coefficients depend on the time and the state variables with a general Hamiltonian. We attach a non-zero Robin boundary condition. We first prove the Lipschitz stability in $ \Omega \times ( \varepsilon, T- \varepsilon) $ with given $ \varepsilon>0 $ for the determination of the solutions by the associated Dirichlet data on an arbitrarily chosen subboundary of $ \partial \Omega $. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at an intermediate time.