Dynamics and Control of Mechanical Systems

PI: Anthony M. bloch机械系统动力学与控制摘要：这是对提出者在有限维和无限维非线性机械系统动力学与控制研究的延续和扩展。特别是在以下领域提出了研究建议：有限和无限维的可积动力系统，包括哈密顿系统和非完整系统，非线性机械系统的稳定和控制通过能量方法和这些方法的扩展到非完整（不可积）约束系统，刚体的光滑和离散动力学的几何，以及量子力学系统的控制。包含本提案几个部分的一般兴趣领域是动力系统中能量守恒与渐近行为之间以及可逆行为与不可逆行为之间的关系。在可积系统领域，作者主要研究了非贝尔的Toda格、Toda的无限维推广以及广义光滑和离散刚体系统。最后，他打算分析各种耦合力学系统在经典和量子状态下的动力学和控制。本提案旨在研究各种机械系统的行为，这些系统在科学和工程应用中很重要。这些系统包括轮式或铰接式机器人、航空航天系统、潜艇和量子（微观）设备。原子和分子的运动规律在量子器件中起着关键作用，在通信和编码理论等领域变得越来越重要。除了研究这些系统的行为外，作者还打算分析它们的控制和最优控制——也就是说，提供在工程应用中以实用、稳定和有效的方式使用这些系统的方法。建议者打算用数学结构分析这种机械系统的各种关键例子，这种数学结构特别适合于详细的定性和定量分析。此外，作者打算分析系统行为在量子（微观）水平和大尺度（宏观）水平之间的转换。这种转变对于物理设备在现实世界中的应用非常重要，从科学的角度来看也很有趣。

DMS-0103895 PI: Anthony M. BlochDynamics and Control of Mechanical SystemsAbstract:This is a proposal for the continuation and extension of the proposer's research into the dynamics and control of nonlinear mechanical systems in finite- and infinite-dimensions. In particular, research is proposed in the following areas: integrable dynamical systems in finite- and infinite- dimensions including both Hamiltonian and nonholonomic systems, the stabilization and control of nonlinear mechanical systems via energy methods and the extension of these methods to systems with nonholonomic (nonintegrable) constraints, the geometry of the smooth and discrete dynamics of rigid bodies, and the control of quantum mechanical systems. A general area of interest that encompasses several parts of this proposal is the relationship between energy preservation and asymptotic behavior in dynamical systems and between reversible and irreversible behavior. In the integrable systems area the proposer is interested in the nonabelian Toda lattice, infinite-dimensional generalizations of Toda, and generalized smooth and discrete rigid body systems. Finally he intends to analyze the dynamics and control of various coupled mechanical systems in both the classical and quantum regimes.This proposal is aimed at studying the behavior of various mechanical systems that are important for applications in science and engineering. These include systems such as wheeled or articulated robots, aerospace systems, submarine vehicles, and quantum (microscopic) devices. Quantum devices, where the laws of motion of atoms and molecules play a key role, have become increasingly important in such areas as communication and coding theory. In addition to studying the behavior of these systems the proposer intends to analyze their control and optimal control -- that is, to provide methods for using such systems in engineering applications in a practical, stable, and efficient manner. The proposerintends to analyze various key examples of such mechanical systems with mathematical structure which is particularly amenable to detailedqualitative and quantitative analysis. Further, the proposer intends toanalyze the transition between the behavior of systems at the quantum(microscopic) level and the large scale (macroscopic) level. This transition is important for the application of physical devices in the real world as well as interesting from the scientific point of view.