Dynamics, Integrability, and Control of Mechanical and Physical Systems

该项目侧重于机械系统和控制，并在工业中有许多应用，例如机器人系统的控制和车辆和其他复杂机械系统的设计，量子计算中使用的量子系统的设计和控制，以及耦合生物系统的分析，如相互作用的细胞。研究人员将与密歇根大学的工程师和物理学家合作。一些研究集中在具有运动约束的系统上，包括汽车和带轮子的机器人，而其他部分则集中在具有影响的系统上，包括带腿的机器人和腿运动的研究。机器学习在机器人技术中也有应用，特别是在视觉方面。在量子领域，研究人员将研究将一个量子态转向另一个量子态的问题，这是量子计算的关键。在生物学领域，研究人员打算研究细胞类型的控制，这是干细胞研究等领域的关键。该计划具有很强的教育影响力，它的想法将用于教学各种科目，包括动力系统，力学，机器人和控制。本项目是研究者对力学系统的几何学、动力学和控制的研究的延续，包括哈密顿系统、拉格朗日系统、可积系统、非完整系统和梯度流。研究人员将研究各种机械系统的动力学，包括有限维和无限维的可积哈密顿系统，耦合哈密顿和梯度系统，应用于人工智能中的某些问题，具有影响和非完整系统的系统，光滑和离散设置中流形上的最优控制方程，以及量子系统和某些生物系统的控制和动力学。他将分析几何学的可积系统在各种新的情况下，包括系统所产生的某些最佳控制问题，包括流Stiefel流形，扩展的户田格流和刚体流量。他还将研究相关的梯度流，这些梯度流应用于计算机视觉和人工智能中的某些问题。此外，他还将研究具有附加机械耗散的哈密顿系统。进一步的工作涉及的控制和最优控制的量子系统与Lindblad耗散模型开放的量子系统和量子计算中的问题有应用。他还将研究系统的影响（混合动力系统），包括应用到非完整系统的影响。非完整动力学理论是研究受速度约束的力学系统的理论。这样的约束经常发生在机器人系统中。此外，他还将考虑各种系统中周期性行为的存在，包括具有影响的系统和合成生物学中出现的某些生物系统。该奖项反映了NSF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而被认为值得支持。

The project focuses on mechanical systems and control and has many applications in industry, such as in the control of robotic systems and design of vehicles and other complex mechanical systems, the design and control of quantum systems used in quantum computing, and the analysis of coupled biological systems such as interacting cells. The investigator will collaborate with engineers and physicists at University of Michigan. Some of the research focuses on systems with motion constraints which includes cars and robots with wheels, while other parts focus on systems with impacts which includes robots with legs and the study of legged locomotion. There are also applications to machine learning in robotics particularly with respect to vision. In the quantum realm the investigator will study the problem of steering one quantum state to another which is key in quantum computing. In the biological regime the investigator intends to study the control of cell type, which is key, for example, to stem cell research. The program has a strong educational impact and ideas from it will be used in teaching various subjects, including dynamical systems, mechanics, robotics and control. Parts of the proposed activity are appropriate for PhD projects and undergraduate projects.The present project is a continuation of the investigator's study of the geometry, dynamics and control of mechanical systems including Hamiltonian and Lagrangian systems, integrable systems, nonholonomic systems, and gradient flows. The investigator will study the dynamics of various mechanical systems including integrable Hamiltonian systems in finite- and infinite-dimensions, coupled Hamiltonian and gradient systems with applications to certain problems in artificial intelligence, systems with impacts and nonholonomic systems, optimal control equations on manifolds in both the smooth and discrete setting, and the control and dynamics of quantum systems and certain biological systems. He will analyze the geometry of integrable systems in various new contexts including systems arising from certain optimal control problems including flows on Stiefel manifolds, extensions of the Toda lattice flow and rigid body flows. He will also study related gradient flows which have applications to certain problems in computer vision and artificial intelligence. In addition, he will study Hamiltonian systems with added mechanical dissipation. Further work concerns the control and optimal control of quantum systems with Lindblad dissipation which model open quantum systems and which have application to problems in quantum computing. He will also study systems with impacts (hybrid dynamical systems) including application to nonholonomic systems with impacts. The theory of nonholonomic dynamics is the study of mechanical systems subject to constraints imposed on velocities. Such constraints often occur in robotic systems. In addition, he will consider the existence of periodic behavior in various systems including systems with impacts and certain biological systems that arise in synthetic biology.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.