Geometry and topology of nonholonomic structures

标题的非完整结构是在速度约束不是由位置约束引起的物理系统上建模的。这种系统最简单的例子是一个球在坚硬的表面上滚动而不滑动。非完整结构，特别是与速度的线性约束相对应的分布，在机器人学(车式机器人)和神经科学(关节物体的运动模型)以及物理和数学领域都很常见。该项目的一个重点是逆最优控制问题，将提供从给定的最优轨迹集恢复代价函数的显式算法。该项目的另一个重点，致力于分布的Gromovsh-原理，将回答几个具有指定微分性质的新分布类的全局存在性以及通过连续变形避免此类分布奇异性的可能性等重要问题。该项目的其他主旨是致力于新的分布类、CR结构和其他过滤结构的等价性问题。首席研究员将与他的学生合作，继续开发一个基于Maple的特殊软件包，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，以及项目所需的田中延长的新变种，这超出了现有标准软件包的范围。更广泛的影响将包括对几名博士生的培训和建议，以及关于该项目主题的本科生研究项目。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.