Hölder stability and uniqueness for the mean field games system via Carleman estimates

Klibanov，Michael V.1; Li，Jingzhi2; Liu，Hongyu3

2023

10.1111/sapm.12633

Studies in Applied Mathematics   Impact Factor & Quartile

0022-2526

1467-9590

We are concerned with the mathematical study of the mean field games system (MFGS). In the conventional setup, the MFGS is a system of two coupled nonlinear parabolic partial differential equation (PDE)s of the second order in a backward–forward manner, namely, one terminal and one initial condition are prescribed, respectively, for the value function and the population density. In this paper, we show that uniqueness of solutions to the MFGS can be guaranteed if, among all four possible terminal and initial conditions, either only two terminals or only two initial conditions are given. In both cases, Hölder stability estimates are proven. This means that the accuracies of the solutions are estimated in terms of the given data. Moreover, these estimates readily imply uniqueness of corresponding problems for the MFGS. The main mathematical apparatus to establish those results is two new Carleman estimates, which may find application in other contexts associated with coupled parabolic PDEs.

Carleman estimates Hölder stability estimates ill-posed and inverse problems mean field games system uniqueness

SCI

English

Others

NSF of China[11971221] ; Guangdong NSF Major Fund[2021ZDZX1001] ; Shenzhen Sci-Tech Fund["RCJC20200714114556020","JCYJ20200109115422828","JCYJ20190809150413261"] ; Hong Kong RGC General Research Funds["12302919","12301420","11300821"] ; France-Hong Kong ANR/RGC Joint Research[A-CityU203/19]

Mathematics

Mathematics, Applied

WOS:001049418000001

WILEY

MATHEMATICS

2-s2.0-85168135524

Scopus

1.Department of Mathematics and Statistics,University of North Carolina at Charlotte,Charlotte,United States
2.Department of Mathematics & National Center for Applied Mathematics Shenzhen & SUSTech International Center for Mathematics,Southern University of Science and Technology,Shenzhen,China
3.Department of Mathematics,City University of Hong Kong,Kowloon,Hong Kong

Klibanov，Michael V.,Li，Jingzhi,Liu，Hongyu. Hölder stability and uniqueness for the mean field games system via Carleman estimates[J]. Studies in Applied Mathematics,2023,151(4).

Klibanov，Michael V.,Li，Jingzhi,&Liu，Hongyu.(2023).Hölder stability and uniqueness for the mean field games system via Carleman estimates.Studies in Applied Mathematics,151(4).

Klibanov，Michael V.,et al."Hölder stability and uniqueness for the mean field games system via Carleman estimates".Studies in Applied Mathematics 151.4(2023).