多智能体网络系统的动力学与控制：一种finite difference方法
Dynamics and control for multi-agent networked systems: a finite difference approach
摘要：
我们分析了多智能体集体行为模型的动力学及其控制理论性质。我们首先导出抛物型扩散方程的一个大种群极限。我们还表明，非局部输运方程通常推导出的平均场极限，是从属于第一个。换句话说，非局部输运模型的解可以通过扩散模型的适当平均来获得。然后，我们解决控制问题的线性设置，连接多智能体模型与空间半离散化的抛物型方程。这使我们能够使用现有的技术，抛物控制问题，在目前的设置，并获得显式的估计控制这些系统的代理的数量趋于无穷大的成本。我们得到精确的估计的时间控制和控制的大小需要驱动系统的共识，这取决于考虑的人口规模。我们的方法，启发现有的结果抛物型方程，可能是分数型，在几个空间维度，表明共识的形成可以理解的基本扩散过程所描述的热半群。通过这种方式，我们能够精确地估计这些系统的可控性的成本，因为代理的数量增加，无论是在所需的控制时间范围和控制的大小。
英文摘要：
We analyze the dynamics of multi-agent collective behavior models and their control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the non-local transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the non-local transport model can be obtained by a suitable averaging of the diffusive one. We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered. Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time-horizon and the size of the controls.
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