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A numerical method to solve matrix-valued differential inequalities with applications in dynamical systems

求解矩阵值微分不等式的数值方法及其在动力系统中的应用

基本信息

批准号：
2889464
负责人：
金额：
--
依托单位：
University of Sussex
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2023
资助国家：
英国
起止时间：
2023 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=studentship-2889464
关键词：
numerical method solve matrix valued

项目摘要

A contraction metric is a tool in dynamical systems to show that the distance of adjacent trajectories decreases and thus to determine the long-term behaviour of solutions. A contraction metric is a matrix-valued function, and the contraction condition is a linear matrix-valued differential inequality. The project seeks to develop a numerical method to discretise such problems using meshfree collocation and then solve the resulting problem using semidefinite optimisation methods. It is expected that the solutions of the discretised problems will converge to the solution of the original problem. This general numerical method will then be applied to problems in dynamical systems such as the computation of a contraction metric. A main area of applications will be stochastic differential equations, where we will explore the use and computation of contraction metrics.The project has aspects of numerical analysis, functional analysis, stochastics, coding and applications to dynamical systems: we will prove convergence of solutions, write a code to solve differential inequalities, and apply it to the computation of contraction metric in dynamical systems, in particular for stochastic differential equations.The student will be co-supervised by Peter Giesl and Nicos Georgiou, forming a new, strong link between the Analysis and PDEs and the Probability and Statistics group in the Department of Mathematics at Sussex. Research visits of the PhD student to Holger Wendland (Bayreuth, Germany) and Sigurdur Hafstein (Reykjavik, Iceland) will complement the training at Sussex.

收缩度量是动力系统中的一种工具，用来显示相邻轨迹的距离减小，从而确定解的长期行为。压缩度量是矩阵值函数，压缩条件是线性矩阵值微分不等式。该项目寻求开发一种数值方法来使用无网格配置来离散这类问题，然后使用半定优化方法来解决由此产生的问题。期望离散化问题的解收敛于原问题的解。然后，这种通用的数值方法将被应用于动力系统中的问题，例如压缩度量的计算。一个主要的应用领域将是随机微分方程，在那里我们将探索压缩度量的使用和计算。该项目包括数值分析、泛函分析、随机学、编码和动力系统的应用：我们将证明解的收敛，编写求解微分不等式的代码，并将其应用于动力系统中的压缩度量的计算，特别是随机微分方程。该学生将在Peter Giesl和Nicos Georgiou的共同指导下，在苏塞克斯数学系的分析和PDE与概率和统计小组之间形成新的、强有力的联系。博士生对德国拜罗伊特的Holger Wendland和冰岛的Reykjavik的Sigurdur Hafstein的研究访问将补充苏塞克斯的培训。