Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation
Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation
复制标题
使用无网格配置确定一维有限时间系统中的指数吸引力面积
DOI：
--
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
James McMichen
中科院分区：
文献类型：
--
作者：
P. Giesl;James McMichen
考虑有限时间区间上的一类非自治常微分方程解的初值问题。指数吸引区由解组成，使得到相邻解的距离从\Begin{Document}$T_1$\end{Document}到\Begin{Document}$T_2$\end{Document}指数收缩。人们可以使用收缩度量来确定指数吸引的面积，并提供吸引速率的界。在这篇文章中，我们将给出第一种方法来算法地构造一维有限时间系统的压缩度量。我们将证明一个压缩度量的存在性，该度量由满足一个带边界条件的二阶偏微分方程的函数给出。然后，我们使用无网格配置来近似求解该方程，并且证明了如果配置点足够稠密，所得到的近似本身定义了一个压缩度量。我们给出了误差估计，并将该方法应用到一个例子中。
We consider a non-autonomous ordinary differential equation over a finite time interval \begin{document}$[T_1,T_2]$\end{document} . The area of exponential attraction consists of solutions such that the distance to adjacent solutions exponentially contracts from \begin{document}$T_1$\end{document} to \begin{document}$T_2$\end{document} . One can use a contraction metric to determine an area of exponential attraction and to provide a bound on the rate of attraction. In this paper, we will give the first method to algorithmically construct a contraction metric for finite-time systems in one spatial dimension. We will show the existence of a contraction metric, given by a function which satisfies a second-order partial differential equation with boundary conditions. We then use meshless collocation to approximately solve this equation, and show that the resulting approximation itself defines a contraction metric, if the collocation points are sufficiently dense. We give error estimates and apply the method to an example.