Geometry and Analysis on Nonholonomic structures on manifolds
流形上非完整结构的几何与分析
基本信息
- 批准号:1406193
- 负责人:
- 金额:$ 13.02万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2014
- 资助国家:美国
- 起止时间:2014-09-01 至 2017-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
AbstractAward: DMS 1406193, Principal Investigator: Igor ZelenkoStructures of nonholonomic nature such as bracket generating distributions and sub-Riemannian structures appear naturally in Control Theory with applications to Robotics (car-like robots), Mechanics (ball bearing structures) and the models of visual perception. Distributions are also basic objects in the geometric theory of differential equations and in geometry of real submanifolds of complex spaces (Cauchy-Riemann geometry). The knowledge of invariants of a geometric structure up to diffeomorphisms of the ambient manifolds or, informally speaking, of the quantities which are independent of local coordinate representations of such structures often gives the essential information about various qualitative properties of the geometric structures. In the last decade we developed the novel variational approach for the unified construction of the differential invariants for a very wide class of geometric structures. This approach takes its origin in Optimal Control Theory and Symplectic Geometry. It extends significantly the set of the nonholonomic structures for which the canonical frames and differential invariants can be constructed explicitly and uniformly compared with all previously existing approaches (including the classical Cartan method of equivalence and the Tanaka theory from 1970). The aim of the project is to use these invariants in order to solve a number of natural problems that were not considered in such generality before. The results of this project will bring new geometric tools for Control Theory and Mathematical Physics by providing efficient and uniform algorithms for computing state-feedback and gauge invariants and explicit geometric optimality conditions for extremals of the corresponding variational problems. We will develop a special MAPLE based software package for computation of our invariants and apply the theory to the qualitative study of control and mechanical systems of practical interest.The main new point of the aforementioned variational approach is that the study of geometry of nonholonomic structures on manifold can be reduced to a simpler (extrinsic) geometry of curves in flag varieties. In terms of these curves we obtain a new discrete basic invariant of the original structure, called the flag symbol and we have an explicit algorithm for construction of the canonical frame for our original structure that depends only on first fixing this discrete information. However, the main properties of these frames and of the invariants they produce are far of being understood. Among the problems that will be addressed in the project are (1) explicit description of the group of symmetries of the most symmetric models with given flag symbol; (2) exploring when the obtained canonical frames are Cartan connections; (3) distinguishing the fundamental set of invariants among all invariants produced by these frames. To reach these goals we will use various tools and techniques from Representation Theory, Algebraic Geometry, and cohomological theory of Lie algebras. Another main theme of the project is comparison theorems in sub-Riemannian geometry. Several essential differences here compared with the Riemannian case form serious obstacles in obtaining sharp analogs of the classical Rauch and Bonnet-Myers comparison theorems. To overcome these obstacles we propose to work on the infinitesimal version of comparison theorems, i.e. given a sub-Riemannian metric to describe directions in the tangent space at this metric to the space of all sub-Riemannian metrics in which consecutive conjugate points along the corresponding extremals become closer. We also shall study natural flows on the space of sub-Riemannian metrics along which the infinitesimal comparison theorem holds.
摘要奖:DMS 1406193,首席研究员:伊戈尔·泽伦科在控制理论中自然地出现了非完整性质的结构,如支架生成分布和次黎曼结构,并应用于机器人学(类似汽车的机器人)、力学(滚珠结构)和视觉感知模型。分布也是微分方程组几何理论和复空间实子流形几何(柯西-黎曼几何)中的基本对象。几何结构的不变量的知识,直到环境流形的微分同胚,或者,非正式地说,与这种结构的局部坐标表示无关的量的知识,往往给出关于几何结构的各种定性性质的基本信息。在过去的十年里,我们发展了一种新的变分方法来统一构造一类非常广泛的几何结构的微分不变量。这种方法起源于最优控制理论和辛几何。与已有的方法(包括经典的Cartan等价方法和1970年以来的Tanaka理论)相比,它极大地扩展了非完整结构的集合,其中的正则框架和微分不变量可以显式且一致地构造出来。该项目的目的是使用这些不变量来解决一些以前没有在这种一般性方面考虑的自然问题。该项目的结果将为控制论和数学物理带来新的几何工具,为计算状态反馈和规范不变量提供有效和统一的算法,并为相应变分问题的极值提供显式的几何最优性条件。我们将开发一个基于Maple的计算不变量的专用软件包,并将该理论应用于具有实际意义的控制和机械系统的定性研究。上述变分方法的主要新点是,流形上非完整结构的几何研究可以归结为旗型簇中更简单的(外部)曲线几何。对于这些曲线,我们得到了原始结构的一个新的离散基本不变量,称为标志符号,并且我们有一个显式算法来构造原始结构的规范框架,它只需要首先确定这个离散信息。然而,这些框架及其产生的不变量的主要性质还远未被理解。该项目将解决的问题包括:(1)明确描述具有给定标志符号的最对称模型的对称性群;(2)探索所获得的正则框架何时是Cartan连接;(3)区分由这些框架产生的所有不变量中的基本不变量集。为了达到这些目标,我们将使用表示论、代数几何和李代数的上同调理论中的各种工具和技巧。该项目的另一个主要主题是次黎曼几何中的比较定理。与Riemannian情形相比,这里的几个本质差异构成了获得经典Rauch和Bonnet-Myers比较定理的尖锐类比的严重障碍。为了克服这些障碍,我们建议研究无穷小版本的比较定理,即给出一个次黎曼度量来描述在该度量处的切线空间中的方向与所有次黎曼度量空间的方向,在该空间中,沿相应极值的连续共轭点变得更近。我们还将研究次黎曼度量空间上的自然流,沿着该度量空间,无穷小比较定理成立。
项目成果
期刊论文数量(0)
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Igor Zelenko其他文献
Bipedal Robotic Walking on Flat-Ground, Up-Slope and Rough Terrain with Human-Inspired Hybrid Zero Dynamics
采用受人类启发的混合零动力学在平地、上坡和崎岖地形上行走的双足机器人
- DOI:
- 发表时间:
2012 - 期刊:
- 影响因子:0
- 作者:
N. Yadukumar;Aaron D. Ames;Shankar P. Bhattacharyya;Mehrdad Ehsani;Igor Zelenko;Ohannes Eknoyan;Shishir Nadubettu Yadukumar;Aaron D. Ames - 通讯作者:
Aaron D. Ames
Morse inequalities for ordered eigenvalues of generic self-adjoint families
- DOI:
10.1007/s00222-024-01284-y - 发表时间:
2024-08-12 - 期刊:
- 影响因子:3.600
- 作者:
Gregory Berkolaiko;Igor Zelenko - 通讯作者:
Igor Zelenko
On projective and affine equivalence of sub-Riemannian metrics
- DOI:
10.1007/s10711-019-00437-1 - 发表时间:
2019-03-16 - 期刊:
- 影响因子:0.500
- 作者:
Frédéric Jean;Sofya Maslovskaya;Igor Zelenko - 通讯作者:
Igor Zelenko
On Eisenhart’s Type Theorem for Sub-Riemannian Metrics on Step $$2$$ Distributions with $$\mathrm{ad}$$-Surjective Tanaka Symbols
关于具有 $$mathrm{ad}$$-满射田中符号的步 $$2$$ 分布的亚黎曼度量的艾森哈特类型定理
- DOI:
10.1134/s1560354724020023 - 发表时间:
2024 - 期刊:
- 影响因子:1.4
- 作者:
Zaifeng Lin;Igor Zelenko - 通讯作者:
Igor Zelenko
On Weyl’s type theorems and genericity of projective rigidity in sub-Riemannian geometry
- DOI:
10.1007/s10711-020-00581-z - 发表时间:
2020-11-01 - 期刊:
- 影响因子:0.500
- 作者:
Frédéric Jean;Sofya Maslovskaya;Igor Zelenko - 通讯作者:
Igor Zelenko
Igor Zelenko的其他文献
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{{ truncateString('Igor Zelenko', 18)}}的其他基金
Geometry and topology of nonholonomic structures
非完整结构的几何和拓扑
- 批准号:
2105528 - 财政年份:2022
- 资助金额:
$ 13.02万 - 项目类别:
Standard Grant
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