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.
