Lipschitz stability for determination of states and inverse source problem for the mean field game equations

TL;DR

This paper establishes Lipschitz stability results for inverse problems related to mean field game equations, enabling the determination of solutions and source terms from boundary and spatial data, with implications for control and identification.

Contribution

It provides the first Lipschitz stability estimates for inverse problems in mean field game equations with general Hamiltonians and Robin boundary conditions.

Findings

Lipschitz stability for solution determination from boundary data.

Lipschitz stability for inverse source and coefficient identification.

Applicability to mean field game models with complex boundary conditions.

Abstract

In a bounded domain $\Omega \subset \mathbb{R}^d$ over time interval $(0,T)$, we consider mean field game equations whose principal coefficients depend on the time and state variables with a general Hamiltonian. We attach the 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 Dirichlet data on 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.