SIAM Journal on Applied Mathematics, Volume 80, Issue 3, Page 1153-1174, January 2020.
This paper introduces a system of SDEs of mean-field type that models pedestrian motion. The system lets the pedestrians spend time at and move along walls by means of sticky boundaries and boundary diffusion. As an alternative to Neumann-type boundary conditions, sticky boundaries and boundary diffusion have a “smoothing” effect on pedestrian motion. When these effects are active, the pedestrian paths are semimartingales with the first-variation part absolutely continuous with respect to the Lebesgue measure $dt$ rather than an increasing process (which in general induces a measure singular with respect to $dt$) as is the case under Neumann boundary conditions. We show that the proposed mean-field model for pedestrian motion admits a unique weak solution and that it is possible to control the system in the weak sense, using a Pontryagin-type maximum principle. We also relate the mean-field type control problem to the social cost minimization in an interacting particle system. We study the novel model features numerically, and we confirm empirical findings on pedestrian crowd motion in congested corridors.

中文翻译：

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均值场方法在人群动力学中的近壁行为

SIAM应用数学杂志，第80卷，第3期，第1153-1174页，2020年1月。
本文介绍了一种用于模拟行人运动的均场型SDE系统。该系统通过粘性边界和边界扩散，使行人在墙壁上花费时间并沿着墙壁移动。作为诺伊曼型边界条件的替代方法，粘性边界和边界扩散会对行人运动产生“平滑”效果。当这些效应起作用时，行人路径是半mart，勒贝格度量$ dt $的第一个变化部分绝对是连续的，而不是增加过程（通常会引起$ dt $奇异的度量）在诺伊曼边界条件下的情况。我们表明，提出的行人运动平均场模型接受了唯一的弱解，并且有可能在弱意义上控制系统，使用Pontryagin型最大原理。我们还将均场类型控制问题与相互作用粒子系统中的社会成本最小化联系起来。我们通过数值研究新颖的模型特征，并证实了拥挤走廊中行人运动的经验发现。