RUI: Combinatorial Control of Chaos, Symbolic Dynamics, Optimal Control and Inverse Frobenius-Perron Problem

RUI：混沌、符号动力学、最优控制和逆 Frobenius-Perron 问题的组合控制

• 批准号: 0071314
• 负责人: Erik Bollt
• 金额: $ 6.83万
• 依托单位: United States Naval Academy
• 依托单位国家: 美国
• 项目类别: Interagency Agreement
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2003-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071314&HistoricalAwards=false
• 关键词: RUI; Combinatorial; Control; Chaos; Symbolic

NSF Award Abstract - DMS-0071314Mathematical Sciences: RUI: Combinatorial Control of Chaos, Symbolic Dynamics, Optimal Control and Inverse Frobenius-Perron ProblemAbstract0071314 BolltThis project addresses global control strategies for chaotic dynamical systems, based on transfer operator representations of dynamics on an invariant subspace. Emphasis is on developing techniques useful to experimentalists. During the last decade, much research has been focused on the realization that sensitive dependence on initial conditions and parameters allows flexible and efficient control of a chaotic dynamical system. Major difficulties have been connected with representing the action of the dynamical system on its phase space. In contrast, a complete global representation of a dynamical system is available in terms of its symbolic dynamics, and a graph approximation of the symbolic grammar is a highly efficient way to completely encode coarse-grained control strategies. This project reduces the difficult problems of developing a global control strategy to much easier problems, concerning linear algebra if targeting invariant density, or combinatorics of path searching if targeting optimal trajectories. In this coarse-grained approximation, paths through graphs model trajectories of the dynamical system. Local feedback control and small parameter variations are used to realize paths through the graph as true trajectories. Questions under study are closely related to Ulam's method in the theory of Frobenius-Perron operators and also to the concept of partition in information theory and in symbolic dynamics.This project develops techniques for targeting, finding optimal trajectories, and highlighting desirable states. Potential engineering applications include Navy ship-to-ship transfer by cargo crane, space mission design, design of electronic circuits that stay within desirable operating regimes, and perhaps someday preemptive control of epileptic seizures. The work involves collaboration with experimentalists to pursue real world applications. In addition, the project investigates applications of controlling symbolic dynamics to encode information in chaotic oscillations.

NSF 奖项摘要 - DMS-0071314 数学科学：RUI：混沌、符号动力学、最优控制和逆 Frobenius-Perron 问题的组合控制摘要 0071314 Bollt 该项目基于不变子空间上的动力学传递算子表示，解决了混沌动力系统的全局控制策略。 重点是开发对实验者有用的技术。 在过去的十年中，许多研究都集中在认识到对初始条件和参数的敏感依赖性允许对混沌动力系统进行灵活有效的控制。 主要困难与表示动力系统在其相空间上的作用有关。 相比之下，动力系统的完整全局表示可以用其符号动力学来表示，并且符号语法的图近似是完全编码粗粒度控制策略的高效方法。 该项目将开发全局控制策略的困难问题简化为更简单的问题，如果目标是不变密度，则涉及线性代数，如果目标是最佳轨迹，则涉及路径搜索的组合学。 在这种粗粒度近似中，通过图形的路径模拟了动力系统的轨迹。局部反馈控制和小参数变化用于将通过图形的路径实现为真实轨迹。 正在研究的问题与 Frobenius-Perron 算子理论中的 Ulam 方法以及信息论和符号动力学中的划分概念密切相关。该项目开发了用于瞄准、寻找最佳轨迹和突出理想状态的技术。 潜在的工程应用包括通过货物起重机进行海军舰对舰转移、太空任务设计、保持在理想操作状态下的电子电路设计，以及也许有一天对癫痫发作进行先发制人的控制。 这项工作涉及与实验人员合作以追求现实世界的应用。 此外，该项目还研究了控制符号动力学在混沌振荡中编码信息的应用。