Energy Growth, Dissipation, and Control in Hamiltonian Systems

该项目将通过理论，模型和技术的发展来加强对动力系统的基本理解，这些理论，模型和技术可能会解决一系列当代科学和技术问题，例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机制，以获得在机械系统的能量耗散和强迫。该项目将研究能量收集设备的数学模型，这些设备将外部振动转化为电能，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，并应用于设计燃料效率高的太空任务。部分支持将提供给研究生和本科生，包括代表性不足的群体的成员。该项目将在一般扰动下的哈密顿系统的研究中开辟新的研究方向。它将推进阿诺德扩散现象的理解，描述可积的哈密顿系统受到小的，通用的，哈密顿扰动，表现出轨道沿着的能量变化的一个显着的量。该项目将研究具体系统中的这种现象，例如天体力学。它还将探讨共形辛向量场的扰动情况，它模拟耗散的影响。 此外，该项目将开发新颖的方法，利用系统的自然扰动作为控制器，以便将系统从某个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，并且通过使用基金会的知识产权进行评估，被认为值得支持。优点和更广泛的影响审查标准。

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.