CAREER: Multiscale Control of Mechanical Systems: Theory, Computation and Applications

这个项目寻求发展一个被称为多尺度控制的新领域。这种发展将提高我们控制实际系统的能力。由于物理和技术的限制，干扰机械系统的可用方法常常受到限制。例如，很难单独地强迫单个原子来控制一个分子系统。另一方面，局部干预往往更容易；例如，振荡电磁场可以与带电原子相互作用，并对系统产生扰动，但这种扰动的形式是有限的。当可用的扰动受到限制时，它们可能不足以引起系统立即发生有利的变化。然而，关键的思想是，如果在较小的尺度上引入，扰动可以积累并以较少的限制性方式对原始尺度做出贡献。基于这个想法，该项目开发了一个设计扰动的框架，以便其累积效应可以以期望的方式改变系统的宏观行为。数值计算和计算工具的发展也将与理论分析无缝集成。为了说明所提出的研究的创新，请注意传统的控制理论仍然很重要，尽管还不够，因为控制是在微观层面引入的，然而人们需要并且只需要控制系统的宏观行为。对于最优控制的蛮力计算搜索实际上也是不可行的，因为系统中呈现的尺度很宽。该研究是利用理论分析（基于动力系统和多尺度方法等领域的工具）直接设计或加速控制策略的计算搜索。该研究还将得到多个跨学科应用研究的补充，如DNA失活、运动遥控、时间超材料和材料缺陷修复。研究将与多个机构和不同学术水平的教育相结合。具体计划是扩大妇女和代表性不足的群体的参与，并传播PI直接研究所或学科以外的知识。更具体地说，该项目确定（主要是）振荡扰动来完成各种控制任务，例如将系统从宏观状态引导到另一个状态，或者在宏观水平上始终跟踪所需的动态行为。主要的方法是设计与系统共振相互作用的微观扰动，使不平凡的影响级联到宏观尺度。PI已经获得了关于这个想法的几个初步结果，并且提出的理论将是一个概括。研究策略是首先使用动力系统工具、渐近/多尺度方法和数值计算来量化小尺度扰动如何有效地在大尺度上积累，然后将局部理解拼凑在一起，以实现控制设计的全局目标。核心任务的一个例子是识别和跟踪非线性系统中缓慢变化的谐振频率。将考虑可积、近可积和随机系统，以便逐步接近控制复杂动力学的宏伟目标。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 seeks to develop a new area termed multiscale control. Such a development will increase our ability to control practical systems. Due to physical and technological limitations, available ways of perturbing a mechanical system are often constrained. For example, it is difficult to individually force single atoms in order to control a molecular system. On the other hand, partial intervention is often easier; for instance, an oscillatory electromagnetic field can interact with charged atoms and induce perturbations to the system, which are however in restricted forms. When the available perturbations are restricted, they may be inadequate to induce immediate favorable changes to the system. However, the key idea is that, if introduced at a smaller scale, perturbations can accumulate and contribute in less restrictive ways to the original scale. Based on this idea, the project develops a framework for designing the perturbation so that its accumulated effect can alter the system's macroscopic behavior in a desired fashion. Numerical computations and the development of computational tools will also be seamlessly integrated with the theoretical analysis. To illustrate the innovation of the proposed research, note that traditional control theory is still important, albeit insufficient, because the control is introduced at a microscopic level, yet one needs and only needs to control the system's macroscopic behavior. Brute-force computational search for the optimal control will also be practically infeasible due to the wide breadth of scales presented in the system. The research is to use theoretical analysis (based on tools from fields such as dynamical systems and multiscale methods) to either directly design or accelerate the computational search of control strategies. The study will also be complemented by investigations in multiple interdisciplinary applications, such as DNA deactivation, remote control of locomotion, temporal metamaterials, and recovery of material defects. Research will be integrated with education across multiple institutions and over the spectrum of academic levels. Specifics are planned for broadening women and underrepresented groups' participation and disseminating knowledge beyond PI's immediate institute or discipline.More specifically, this project identifies (mostly-)oscillatory perturbations for accomplishing various control tasks, such as to steer a system from a macroscopic state to another, or to track a desired dynamic behavior at the macroscopic level for all time. The main methodology is to design microscopic perturbations that resonantly interact with the system, so that nontrivial effects cascade to the macroscopic scale. The PI has already obtained several preliminary results on this idea, and the proposed theory will be a generalization. The strategy of investigation is to first quantify how small-scale perturbations effectively accumulate at large-scale using dynamical systems tools, asymptotic/multiscale methods, and numerical computations, followed by the patching together of the local understandings for the global objective of control design. An example of core tasks is to identify and then track slowly-varying resonant frequencies in nonlinear systems. Integrable, nearly-integrable, and stochastic systems will be considered, so that an ambitious goal of controlling complicated dynamics can be gradually approached.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.