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

数学科学：混沌的组合控制，符号动力学，最优控制和逆Frobenius-Perron问题[Abstract] 0071314 bollt本项目研究混沌动力系统的全局控制策略，基于不变子空间上动力学的传递算子表示。重点是发展对实验人员有用的技术。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可以实现混沌动力系统的灵活有效控制。主要的困难与表示动力系统对其相空间的作用有关。相比之下，动态系统的完整全局表示就其符号动态而言是可用的，符号语法的图逼近是一种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降低为更容易的问题，涉及线性代数，如果目标是不变密度，或路径搜索的组合，如果目标是最优轨迹。在这种粗粒度近似中，通过图的路径模拟动力系统的轨迹。使用局部反馈控制和小参数变化来实现通过图的路径为真轨迹。所研究的问题与乌拉姆在Frobenius-Perron算子理论中的方法密切相关，也与信息论和符号动力学中的划分概念密切相关。该项目开发了定位、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海军船对船的货物起重机转移，太空任务设计，保持理想运行状态的电子电路设计，也许有一天会先发制人地控制癫痫发作。这项工作涉及与实验家合作，以追求现实世界的应用。此外，本课题还研究了控制符号动力学在混沌振荡信息编码中的应用。

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.