Dynamics and Control of Nonholonomic and Quantum Systems

非完整和量子系统的动力学和控制

• 批准号: 0907949
• 负责人: Anthony Bloch
• 金额: $ 21.41万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907949&HistoricalAwards=false
• 关键词: Dynamics; Control; Nonholonomic; Quantum; Systems

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.

该项目是首席研究者对机械系统（尤其是非道理和量子系统的几何形状和控制）的研究的延续。研究人员建议研究具有非独立限制，某些最佳控制和导航问题的各种机械系统的动态和控制；以及特定应用于离子陷阱的量子控制中的问题。 特别是，研究者将研究具有对称性的非单面体系统，对称性与动力学的关系以及非实体性的整合性及其与衡量保存的关系。还将研究非自然学的汉密尔顿 - 雅各比方程在集成性中的作用，以及拉格朗日系统与非义学系统的反向理论之间的关系。 研究人员还将考虑具有无限数量自由度的非自我工艺系统的动态，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，特别是对离子陷阱建模的振荡器/自旋系统的模型。这导致了无限二维可控性的问题，并研究了在具有破坏性和耗散的存在下对此类系统的控制。非实体动力学理论是对受到速度施加约束的机械系统的研究。这些限制是典型的，对于由刚体在表面上滚动而不滑的系统组成的系统。 非独立系统经常发生在实际机械问题中，包括车辆，自行车和机器人等车辆。一个特定的兴趣问题是在障碍物周围导航机器人。 非单学动力学在非线性控制和机械系统理论的发展中起着重要作用。 由于所涉及的数学类型，量子控制与非单身系统的控制密切相关。 然而，出现了许多有趣的新问题，例如耗散，矫正和测量的作用。 由于对量子计算机的应用，量子控制已变得非常重要。我们在这里研究了一种用于计算计算的特定系统：离子陷阱。 量子控制提供了非常有趣的非虚构和非线性控制的扩展，并具有与之相关的重要数学问题。 希望这项研究将导致工程发展。提议者继续与许多工程师和物理学家合作。 拟议的计划具有强大的教育影响。与这项研究相关的材料将用于高级动力学类。 这项研究还将涉及博士学位学生和本科生的工作。