Lipschitz Stability Estimate and Uniqueness in the Retrospective Analysis for The Mean Field Games System via Two Carleman Estimates

TL;DR

This paper introduces novel Carleman estimates to analyze the mean field games system, establishing Lipschitz stability and uniqueness in retrospective problems, thus advancing the mathematical understanding of MFGS inverse problems.

Contribution

First application of Carleman estimates to MFGS, deriving stability and uniqueness results for retrospective analysis.

Findings

Derived two new Carleman estimates for MFGS

Established Lipschitz stability with respect to initial and terminal data

Proved uniqueness of the retrospective problem solution

Abstract

A retrospective analysis process for the mean field games system (MFGS) is considered. For the first time, Carleman estimates are applied to the analysis of the MFGS. Two new Carleman estimates are derived. They allow to obtain the Lipschitz stability estimate with respect to the possible error in the input initial and terminal data for a retrospective problem for MFGS. This stability estimate, in turn implies uniqueness theorem for the problem under the consideration. The idea of using Carleman estimates to obtain stability and uniqueness results came from the field of Ill-Posed and Inverse Problems.