Stability in determination of states for the mean field game equations

Hongyu Liu; Masahiro Yamamoto

doi:10.3934/cac.2023009

Article Contents

2023, Volume 1, Issue 2: 157-167. Doi: 10.3934/cac.2023009

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Stability in determination of states for the mean field game equations

Hongyu Liu^1, , and
Masahiro Yamamoto^2,

1.
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China
2.
Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan

^*Corresponding author: Hongyu Liu
^*Corresponding author: Hongyu Liu

Received: April 2023

Revised: April 2023

Early access: May 2023

Published: June 2023

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Abstract

We consider solutions satisfying the Neumann zero boundary condition and a linearized mean field game system in $ \Omega \times (0, T) $, where $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ and $ (0, T) $ is the time interval. We prove two kinds of stability results in determining the solutions. The first is Hölder stability in time interval $ ( \varepsilon, T) $ with arbitrarily fixed $ \varepsilon>0 $ by data of solutions in $ \Omega \times \{T\} $. The second is the Lipschitz stability in $ \Omega \times ( \varepsilon, T- \varepsilon) $ by data of solutions in arbitrarily given subdomain of $ \Omega $ over $ (0, T) $.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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