Dynamics, Integrability, and Control of Mechanical and Nonholonomic Systems

该项目专注于力学、动力学和控制方面的问题。非完整动力学理论是研究受速度约束的机械系统，例如滚动而不打滑。我们在实际系统中使用这一理论的例子包括轮式车辆，如汽车(特别是自动驾驶汽车)和机器人。非完整系统控制背后的数学在一般机械系统的控制中起着关键作用，例如飞机的控制。同样重要的是，耗散或摩擦如何影响此类系统的行为和稳定性。这个研究项目探索如何显式求解动力学来预测这样的机械系统的行为。正在开发的方法预计也将用于研究量子控制问题，并应用于量子计算等。该项目涉及研究生和本科生参与研究。这项研究项目旨在拓宽和加深对机械系统的几何、动力学和控制的理解，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。研究人员将研究各种力学系统的动力学，包括有限和无限维可积的哈密顿系统，耦合的哈密顿和梯度系统，具有非完整约束的系统，流形上的最优控制方程，以及量子控制系统。这项研究将在几个新的背景下考虑可积系统的几何，包括Toda格流和刚体流的推广，以及在李群上某些系统的最优控制中的应用。期望类似的方法也可以用来研究包含耦合哈密顿和耗散动力学的开放量子系统的控制和动力学。

The project focuses on problems in mechanics, dynamics, and control. The theory of nonholonomic dynamics is the study of mechanical systems subject to constraints on velocities, such as rolling without slipping. Examples where we use this theory in practical systems include wheeled vehicles, such as cars (in particular self-steering cars) and robots. The mathematics behind the control of nonholonomic systems plays a key role in control of mechanical systems in general, such as the control of aircraft. Also important is how dissipation, or friction, affects the behavior and stability of such systems. This research project explores how to explicitly solve for the dynamics to predict the behavior of such mechanical systems. The methods under development are expected also to be useful in studying quantum control problems, with applications to quantum computing among others. The project involves graduate and undergraduate students in the research. This research project aims to broaden and deepen understanding 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, systems with nonholonomic constraints, optimal control equations on manifolds, and quantum control systems. The research will consider the geometry of integrable systems in several new contexts, including extensions of the Toda lattice flow and rigid body flows, as well as applications to optimal control of certain systems on Lie groups. It is expected that similar methods can be used to study the control and dynamics of open quantum systems that involve coupled Hamiltonian and dissipative dynamics.