Modified refinement algorithm to construct Lyapunov functions using meshless collocation

IF 1 Q3 Engineering

Journal of Computational Dynamics Pub Date : 2022-01-01 DOI:10.3934/jcd.2022022

N. Mohammed, P. Giesl

引用次数: 0

Abstract

Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sublevel sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshless collocation with radial basis functions.Recently, this method was combined with a grid refinement algorithm (GRA) to reduce the number of collocation points needed to construct Lyapunov functions. However, depending on the choice of the initial set of collocation point, the algorithm can terminate, failing to compute a Lyapunov function. In this paper, we propose a modified grid refinement algorithm (MGRA), which overcomes these shortcomings by adding appropriate collocation points using a clustering algorithm. The modified algorithm is applied to two- and three-dimensional examples.

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改进无网格配置构造Lyapunov函数的改进算法

李雅普诺夫函数是沿给定常微分方程的解具有负导数的函数。此外，Lyapunov函数的子水平集是平衡的吸引域的子集。李雅普诺夫函数的一种数值构造方法是径向基函数的无网格配点法。最近，该方法与网格细化算法(GRA)相结合，减少了构造Lyapunov函数所需的配点数。然而，由于初始配置点集的选择，该算法可能会因无法计算Lyapunov函数而终止。在本文中，我们提出了一种改进的网格细化算法(MGRA)，该算法通过使用聚类算法添加适当的并置点来克服这些缺点。将改进后的算法应用于二维和三维实例。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Computational Dynamics Engineering-Computational Mechanics

CiteScore

2.30

自引率

10.00%

发文量

期刊介绍： JCD is focused on the intersection of computation with deterministic and stochastic dynamics. The mission of the journal is to publish papers that explore new computational methods for analyzing dynamic problems or use novel dynamical methods to improve computation. The subject matter of JCD includes both fundamental mathematical contributions and applications to problems from science and engineering. A non-exhaustive list of topics includes * Computation of phase-space structures and bifurcations * Multi-time-scale methods * Structure-preserving integration * Nonlinear and stochastic model reduction * Set-valued numerical techniques * Network and distributed dynamics JCD includes both original research and survey papers that give a detailed and illuminating treatment of an important area of current interest. The editorial board of JCD consists of world-leading researchers from mathematics, engineering, and science, all of whom are experts in both computational methods and the theory of dynamical systems.