An overdetermination is introduced in an initial condition for the second order mean field games system (MFGS). This makes the resulting problem close to the classical ill-posed Cauchy problems for PDEs. Indeed, in such a problem an overdetermination in boundary conditions usually takes place. A Lipschitz stability estimate is obtained. This estimate implies uniqueness. A new Carleman estimate is derived. This latter estimate is called “quasi-Carleman estimate”, since it contains two test functions rather than a single one in conventional Carleman estimates. These two estimates play the key role. Carleman estimates were not applied to the MFGS prior to the recent work of Klibanov and Averboukh in [M. V. Klibanov and Y. Averboukh, Lipschitz stability estimate and uniqueness in the retrospective analysis for the mean field games system via two Carleman estimates, preprint 2023, https://arxiv.org/abs/2302.10709].

中文翻译：

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平均场博弈系统：Carleman 估计、Lipschitz 稳定性和唯一性

在二阶平均场博弈系统 (MFGS) 的初始条件中引入了超定。这使得生成的问题接近 PDE 的经典病态柯西问题。事实上，在这样的问题中，通常会出现边界条件的过度确定。获得 Lipschitz 稳定性估计。这个估计意味着唯一性。推导出新的 Carleman 估计。后一种估计称为“准 Carleman 估计”，因为它包含两个测试函数，而不是传统 Carleman 估计中的一个函数。这两个估计起着关键作用。在 Klibanov 和 Averboukh 最近在 [MV Klibanov 和 Y. Averboukh，https://arxiv.org/abs/2302.10709].