Energy Growth, Dissipation, and Control in Hamiltonian Systems

该项目将通过开发理论、模型和技术，潜在地解决一系列当代科学和技术问题，如生产可持续能源和空间探索，加强对动力系统的基础性理解。这项研究的主要目的是设计在受耗散和强迫的机械系统中获得能量的机制。该项目将研究能量收集设备的数学模型，该设备将外部振动转换为电能，并将优化其能量输出。它还将研究彗星、小行星和航天器的动力学，并将其应用于设计节油的空间任务。将向研究生和本科生提供部分支持，包括代表人数不足的群体的成员。该项目将为研究受一般扰动影响的哈密顿系统开辟新的研究方向。它将促进对阿诺德扩散现象的理解，描述受小的、一般的哈密顿扰动的可积哈密顿系统，呈现出能量沿轨道显著变化的轨道。该项目将在具体系统中研究这一现象，例如从天体力学角度。它还将探讨由共形辛矢量场给出的扰动的情况，它模拟了耗散效应。此外，该项目将开发新的方法来使用系统的自然扰动作为控制器，以将系统从某种给定状态驱动到任何所需状态。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project will enhance the foundational understanding of Dynamical Systems through the development of theories, models, and techniques that could potentially address a range of contemporary scientific and technological issues, such as production of sustainable energy and space exploration. The main objective of this research is to devise mechanisms to gain energy in mechanical systems subject to dissipation and forcing. The project will study mathematical models for energy harvesting devices, which convert external vibrations into electrical energy, and will optimize their energy output. It will also investigate the dynamics of comets, asteroids, and spacecraft, with applications to the design of fuel-efficient space missions. Partial support will be provided to graduate and undergraduate students, including members of underrepresented groups. The project will open new research directions in the study of Hamiltonian systems subject to general perturbations. It will advance the understanding of the Arnold diffusion phenomenon, describing that integrable Hamiltonian systems subject to small, generic, Hamiltonian perturbations, exhibit orbits along which the energy changes by a significant amount. The project will investigate this phenomenon in concrete systems, such as from celestial mechanics. It will also explore the case of perturbations given by conformally symplectic vector fields, which model the effect of dissipation. Additionally, the project will develop novel methods to use natural perturbations of a system as controllers, in order to drive the system from some given state to any desired state.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.