一类平均场对策系统收敛于聚集和群集模型
Convergence of some Mean Field Games systems to aggregation and flocking models
摘要：
对于两类平均场博弈系统，我们研究了当成本泛函中的利率变得很大，模型代理只关心很短的时间范围，并且控制成本变得很低时解的收敛性.在这两种情况下的限制是一个单一的一阶积分-偏微分方程的质量密度的演变。第一个模型是粘性为零的二阶MFG系统，极限是一个聚集方程。这一结果对集体动物行为和群体动力学模型有一个解释。第二类问题是加速度的一阶MFG，极限是与Cucker-Smale模型相关的动力学方程。第一个问题是用偏微分方程方法分析的，而第二个问题是用轨迹上的概率测度空间中的变分方法研究的。
英文摘要：
For two classes of Mean Field Game systems we study the convergence of solutions as the interest rate in the cost functional becomes very large, modeling agents caring only about a very short time-horizon, and the cost of the control becomes very cheap. The limit in both cases is a single first order integro-partial differential equation for the evolution of the mass density. The first model is a 2nd order MFG system with vanishing viscosity, and the limit is an aggregation equation. The result has an interpretation for models of collective animal behaviour and of crowd dynamics. The second class of problems are 1st order MFGs of acceleration and the limit is the kinetic equation associated to the Cucker-Smale model. The first problem is analyzed by PDE methods, whereas the second is studied by variational methods in the space of probability measures on trajectories.
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