论文标题
多维Ornstein-Uhlenbeck过程的Fokker-Planck方程的漏斗控制
Funnel control of the Fokker-Planck equation for a multi-dimensional Ornstein-Uhlenbeck process
论文作者
论文摘要
在本文中，探索了探索与多维的Ornstein-uhlenbeck工艺相对应的Fokker-Planck方程的漏斗控制技术的可行性。首先，使用加权的Lebesgue和Sobolev空间，通过合适的sesquilinear形式来定义辅助操作员。然后将该操作员转换为所需的Fokker-Planck操作员。我们表明，受控的Fokker-Planck方程的任何轻度解（这是概率密度）都有一个协方差矩阵，该矩阵将指数级收敛到恒定矩阵。讨论了简单的前馈控制方法之后，我们通过利用半群理论在存在干扰的情况下显示了漏斗控制的可行性。我们强调闭环系统是非线性和随时间变化的PDE。结果通过一些模拟说明了结果。
In this paper the feasibility of funnel control techniques for the Fokker-Planck equation corresponding to a multi-dimensional Ornstein-Uhlenbeck process on an unbounded spatial domain is explored. First, using weighted Lebesgue and Sobolev spaces, an auxiliary operator is defined via a suitable sesquilinear form. This operator is then transformed to the desired Fokker-Planck operator. We show that any mild solution of the controlled Fokker-Planck equation (which is a probability density) has a covariance matrix that exponentially converges to a constant matrix. After a simple feedforward control approach is discussed, we show feasibility of funnel control in the presence of disturbances by exploiting semigroup theory. We emphasize that the closed-loop system is a nonlinear and time-varying PDE. The results are illustrated by some simulations.
