Geometry and topology of nonholonomic structures

非完整结构的几何和拓扑

• 批准号: 2105528
• 负责人: Igor Zelenko
• 金额: $ 23.53万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-03-01 至 2025-02-28
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105528&HistoricalAwards=false
• 关键词: Geometry; topology; nonholonomic; structures

The nonholonomic structures of the title are modeled on physical systems with velocity constraints not arising from position constraints. The simplest example of such a system is a ball rolling without slipping on a hard surface. Nonholonomic structures and especially distributions that correspond to linear constraints for velocities, are common in robotics (car-like robots) and neuroscience (models of the motion of articulated bodies), as well as in areas of physics and mathematics.One of the thrusts of the project, devoted to inverse optimal control problems, will provide explicit algorithms for recovering a cost functional from the given set of optimal trajectories. Another thrust of the project, devoted to the Gromovs h-principle for distributions, will answer important questions of global existence of several new classes of distributions with prescribed differential properties and the possibility to avoid singularities of such distributions by continuous deformations. Other thrusts of the project are devoted to equivalence problems for new classes of distributions, CR structures, and other filtered structures. The principal investigator, in collaboration with his students, will continue to develop a special Maple based software package for computation of state-feedback and gauge invariants of various control and mechanical systems of practical interest, and for new variants of Tanaka prolongation needed in the project, which are beyond the scope of the existing standard packages. Broader impacts will include training and advising of several PhD students, and undergraduate research projects on topics of this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

标题中的非完整结构是在物理系统上建模的，速度约束不是由位置约束引起的。这种系统最简单的例子是一个在坚硬表面上滚动而不滑动的球。非完整结构，特别是对应于速度线性约束的分布，在机器人（类车机器人）和神经科学（关节体运动模型）以及物理和数学领域中很常见。该项目的重点之一是逆最优控制问题，将提供从给定的最优轨迹集恢复成本泛函的显式算法。 该项目的另一个重点，致力于Gromovs h-原则的分布，将回答全球存在的几个新类的分布与规定的微分性质和可能性，以避免奇异性的连续变形等分布的重要问题。该项目的其他重点致力于新的分布类，CR结构和其他过滤结构的等价问题。首席研究员，与他的学生合作，将继续开发一个特殊的基于Maple的软件包，用于计算各种实际感兴趣的控制和机械系统的状态反馈和规范不变量，以及该项目所需的Tanaka延拓的新变体，这些都超出了现有标准软件包的范围。更广泛的影响将包括培训和建议几个博士生，和本科生的研究项目的主题，这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

项目成果

On geometry of 2-nondegenerate CR structures of hypersurface type and flag structures on leaf spaces of Levi foliations

超曲面型2-非简并CR结构和Levi叶空间上旗形结构的几何

• DOI: 10.1016/j.aim.2022.108850
• 发表时间: 2023
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Sykes, David;Zelenko, Igor
• 通讯作者: Zelenko, Igor

Maximal Dimension of Groups of Symmetries of Homogeneous 2-nondegenerate CR Structures of Hypersurface Type with a 1-dimensional Levi Kernel

具有一维Levi核的超曲面型齐次2-非简并CR结构对称群的最大维数

• DOI: 10.1007/s00031-022-09739-3
• 发表时间: 2022
• 期刊: Transformation Groups
• 影响因子: 0.7
• 作者: Sykes, David;Zelenko, Igor
• 通讯作者: Zelenko, Igor