Dynamics and Control of Nonholonomic and Quantum Systems

该项目是首席研究员对机械系统，特别是非完整系统和量子系统的几何，动力学和控制的研究的延续。研究者提出研究具有非完整约束、某些最优控制和导航问题的各种机械系统的动力学和控制问题；以及量子控制中的一些问题，特别是在离子阱中的应用。特别是研究者将研究具有对称性的非完整系统以及对称性与动力学的关系，以及非完整可积性及其与测量保存的关系。本文还将研究非完整Hamilton-Jacobi方程在可积性中的作用，以及拉格朗日系统逆理论与非完整系统的关系。研究者还将考虑具有无限多个自由度的非完整系统的动力学，并可能应用于弹性滚动固体和滑动柔性叶片的动力学。他还将研究量子系统的控制，特别是模拟离子阱的耦合振荡器/自旋系统的模型。这导致了无限维可控性的问题，以及在退相干和耗散存在下对这种系统的控制的研究。非完整动力学理论是研究受速度约束的机械系统。这样的约束是典型的系统组成的刚体滚动的表面上没有滑动。非完整系统经常出现在实际的机械问题中，包括轮式车辆，如汽车、自行车和机器人。一个特别有趣的问题是引导机器人绕过障碍物。非完整动力学在非线性控制和机械系统理论的发展中起着重要的作用。由于所涉及的数学类型，量子控制与非完整系统的控制密切相关。然而，许多有趣的新问题出现了，如耗散，退相干和测量的作用。由于量子计算机的应用，量子控制最近变得非常重要。我们在这里研究了一种已经被提出用于计算的特殊系统：离子阱。量子控制提供了非完整和非线性控制的一个非常有趣的扩展，并具有与之相关的重要数学问题。希望这项研究能在工程上取得进展。提议者继续与许多工程师和物理学家合作。拟议的计划对教育有很大的影响。与本研究相关的材料将用于高级动力学课程。这项研究也将涉及博士生和本科生的工作。

This project is a continuation of principal investigator's study of the geometry, dynamics and control of mechanical systems and in particular nonholonomic and quantum systems. The investigator proposes to study the dynamics and control of various mechanical systems with nonholonomic constraints, certain optimal control and navigation problems; and problems in quantum control with particular application to ion traps. In particular the investigator will study nonholonomic systems with symmetries and the relationship of symmetry to dynamics, and nonholonomic integrability and its relationship to measure preservation. The role of the nonholonomic Hamilton-Jacobi equation in integrability will also be studied, as well as the relationship between the inverse theory of Lagrangian systems and nonholonomic systems. The investigator will also consider he dynamics of nonholonomic systems with an infinite number of degrees of freedom with possible applications to the dynamics of elastic rolling solids and sliding flexible blades. He will also study the control of quantum systems, in particular models of coupled oscillator/spin systems which model ion traps. This leads to problems in infinite-dimensional controllability and the study of the control of such systems in the presence of decoherence and dissipation.The theory of nonholonomic dynamics is the study of mechanical systems subject to constraints imposed on velocities. Such constraints are typical for systems consisting of rigid bodies rolling on surfaces without slipping. Nonholonomic systems occur frequently in practical mechanical problems including wheeled vehicles such as cars, bicycles and robots. A particular problem of interest is navigating robots around obstacles. Nonholonomic dynamics is playing an important role in the development of nonlinear control and mechanical systems theory. Quantum control is closely related to the control of nonholonomic systems because of the type of mathematics involved. Many interesting new issues arise however such as the role of dissipation, decoherence and measurement. Quantum control has become very important recently because of applications to quantum computers. We study here a particular system which has been proposed for computing computing: the ion trap. Quantum control provides a very interesting extension of nonholonomic and nonlinear control and has important mathematical issues associated with it. It is hoped that this research will lead to advances in engineering. The proposer continues to collaborate with many engineers and physicists. The proposed program has a strong educational impact. Material related to this research will be used in an advanced dynamics class. The research will also involve the work of Ph.D students and undergraduates.