Dynamics, Integrability and Control of Mechanical and Nonholonomic Systems

该项目是首席研究员对包括哈密顿系统和拉格朗日系统、非完整系统和梯度流在内的机械系统的几何、动力学和控制研究的延续。非完整系统是经典哈密顿系统和拉格朗日系统的推广，在这些系统中，系统受到速度上的不可积约束。研究者拟研究具有此类约束的机械系统的动力学和控制问题，包括具有内自由度的非完整系统、具有控制的非完整系统、具有非线性约束的系统、非完整设置中的Hamilton Jacobi方程、离散系统、无限维系统和某些最优控制问题。他还建议研究可积哈密顿系统和非完整系统及其与梯度流的关系。在这项工作的过程中，申请人将研究对称性在非完整运动方程及其可积性中的作用，当一个非完整系统与流体相互作用时产生的新动力学，以及具有非线性非完整约束的系统如何表现，包括恒温器（与热浴相互作用的系统）。他还将研究柔性非完整系统。以及哈密顿化（转换为哈密顿形式）在理解非完整系统的可积性中的作用。此外，他将分析有限维和无限维的梯度流，包括环上流、群上流和其他无限维空间，以及它们与可积性的关系。机械系统的动力学在技术上是非常重要的。非完整动力学理论是研究受速度约束的机械系统的理论。非完整系统经常出现在许多实际系统中，包括轮式车辆，如汽车（特别是自动转向汽车）和机器人。这类系统的控制是一个重要的技术问题。此外，非完整系统控制背后的数学在一般非线性系统的控制中起着关键作用，例如飞机或水下航行器的控制。同样重要的是耗散（或摩擦）如何影响这些系统的行为和稳定性。我们感兴趣的是描述这些系统的行为，如何控制和稳定它们，以及如何在计算机上模拟动力学和控制。希望这项研究将导致工程上的进步，提案人将与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。这项研究将涉及博士生和本科生的工作。

This project is a continuation of the principal investigator's study of the geometry, dynamics and control of mechanical systems including Hamiltonian and Lagrangian systems, nonholonomic systems, and gradient flows. Nonholonomic systems are a generalization of classical Hamiltonian and Lagrangian systems where the system is subject to nonintegrable constraints on the velocities. The investigator proposes to study the dynamics and control of mechanical systems with such constraints, including nonholonomic systems with internal degrees of freedom, nonholonomic systems with controls, systems with nonlinear constraints, the Hamilton Jacobi equation in the nonholonomic setting, discrete systems, infinite-dimensional systems, and certain optimal control problems. He also proposes to study integrable Hamiltonian and nonholonomic systems and their relation to gradient flows. In the course of this work the proposer will study the role that symmetries play in the formulation of the nonholonomic equations of motion and in their integrability, the new dynamics that arises when one has a nonholonomic system interacting with a fluid, and how systems with nonlinear nonholonomic constraints behave, including thermostats (systems interacting with a heat bath). He will also study flexible nonholonomic systems. and the role of Hamiltonization (transformation to Hamiltonian form) in understanding the integrability of a nonholonomic system. In addition he will analyze gradients flows in finite and infinite dimensions including flows on loops groups and other infinite-dimensional spaces, and their relationship to integrability.The dynamics of mechanical systems is of great importance in technology. The theory of nonholonomic dynamics in particular is the study of mechanical systems subject to constraints imposed on velocities. Such constraints arise for example in systems consisting of rigid bodies rolling on surfaces without slipping Nonholonomic systems occur frequently in many practical systems including wheeled vehicles such as cars (in particular self steering cars) and robots. Control of such systems is an important technological problem. In addition, the mathematics behind the control of nonholonomic systems plays a key role in control of nonlinear systems in general, such as the control of aircraft or underwater vehicles. Also important is how dissipation (or friction) affects the behavior and stability of such systems We are interested in describing the behavior of these systems, how to control and stabilize them and how to simulate the dynamics and control on a computer. It is hoped that this research will lead to advances in engineering and the proposer will collaborate with 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 involve the work of Ph.D students and undergraduates.