An Inverse Problem With the Final Overdetermination for the Mean Field Games System

Abstract

The mean field games (MFG) theory has broad application in mathematical modeling of social phenomena. The Mean Field Games System (MFGS) is the key to the MFG theory. This is a system of two nonlinear parabolic partial differential equations with two opposite directions of time $t\in (0,T). $ The topic of Coefficient Inverse Problem (CIPs) for the MFGS is a newly emerging one. A CIP for the MFGS is studied. The input data are Dirichlet and Neumann boundary conditions either on a part of the lateral boundary (incomplete data) or on the whole lateral boundary (complete data). In addition to the initial conditions at $\left\{t=0\right\}, $ terminal conditions at $\left\{t=T\right\} $ are given. The terminal conditions mean the final overdetermination. The necessity of assigning all these input data is explained. H\"{o}lder and Lipschitz stability estimates are obtained for the cases of incomplete and complete data respectively. These estimates imply uniqueness of the CIP.