Energy Growth, Dissipation, and Control in Hamiltonian Systems

该项目将通过理论、模型和技术的发展，提高对动力系统的基本理解，这些理论、模型和技术可能解决一系列当代科学和技术问题，如可持续能源的生产和空间探索。本研究的主要目的是设计在受耗散和强迫影响的机械系统中获得能量的机制。该项目将研究能量收集装置的数学模型，该装置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，并将其应用于节能太空任务的设计。将向研究生和本科生提供部分支持，包括代表性不足的群体的成员。该项目将为一般扰动下的哈密顿系统的研究开辟新的研究方向。它将促进对阿诺德扩散现象的理解，描述受小的、一般的、哈密顿扰动影响的可积哈密顿系统，呈现出能量变化显著的轨道。该项目将从天体力学等具体系统中研究这种现象。它还将探讨由共形辛向量场给出的扰动的情况，它模拟了耗散的影响。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，以驱动系统从某个给定状态到任何期望状态。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.