Dynamics, Integrability, and Control of Mechanical and Physical Systems

该项目侧重于机械系统和控制，在工业上有许多应用，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（如相互作用细胞）的分析。研究者将与密歇根大学的工程师和物理学家合作。一些研究集中在有运动约束的系统上，包括汽车和有轮子的机器人，而另一些研究集中在有冲击的系统上，包括有腿的机器人和对腿运动的研究。机器学习在机器人领域也有应用，尤其是在视觉方面。在量子领域，研究者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。在生物学方面，研究者打算研究细胞类型的控制，这是关键，例如干细胞研究。该计划具有很强的教育影响，其思想将用于教学各种学科，包括动力系统，力学，机器人和控制。建议的部分活动适合博士项目和本科生项目。目前的项目是研究者对机械系统的几何，动力学和控制的研究的延续，包括哈密顿系统和拉格朗日系统，可积系统，非完整系统和梯度流。研究者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及其在人工智能中某些问题的应用，影响系统和非完整系统，光滑和离散设置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。他将分析各种新环境下可积系统的几何，包括由某些最优控制问题引起的系统，包括Stiefel流形上的流，Toda点阵流的扩展和刚体流。他还将研究相关的梯度流，这些梯度流在计算机视觉和人工智能的某些问题上有应用。此外，他将研究附加机械耗散的哈密顿系统。进一步的工作涉及具有Lindblad耗散的量子系统的控制和最优控制，这些系统是开放量子系统的模型，并应用于量子计算中的问题。他还将研究具有冲击的系统（混合动力系统），包括在具有冲击的非完整系统中的应用。非完整动力学理论是研究受速度约束的机械系统。这种约束经常出现在机器人系统中。此外，他将考虑各种系统中周期性行为的存在，包括具有冲击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。

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.