Pontryagin maximum principle, (co)adjoint representation, and normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups

具有抖振的 Finsler 和 sub-Finsler 测地线

Finsler and sub-Finsler geodesics with chattering

本文构造了 Finsler 和 sub-Finsler 流形及 Carnot 群上的左不变结构，证明其测地线存在抖振现象，从而否定回答了 Le Donne 问题，并给出了极值发生抖振的充分条件。

Sub-Finsler Lie 群中的正规曲线：强凸范数的分支与多面体范数的面稳定性

Normal Curves in Sub-Finsler Lie Groups: Branching for Strongly Convex Norms and Face Stability for Polyhedral Norms

本文研究了 sub-Finsler Lie 群中的正规曲线，利用凸分析工具重新表述了 Pontryagin Maximum Principle，并证明了强凸范数下的分支现象以及多面体范数下控制的面稳定性。

Co-adjoint orbits and time-optimal problems for two-step free-nilpotent Lie groups

We describe co-adjoint orbits and Casimir functions for two-step free-nilpotent Lie algebras. The symplectic foliation consists of affine subspaces of the Lie coalgebra of different dimensions. Further, we consider left-invariant time-optimal problems on two-step Carnot groups, for which the set of admissible velocities is a strictly convex compactum in the first layer of the Lie algebra containing the origin in its interior. We describe integrals for the vertical subsystem of the Hamiltonian system of Pontryagin maximum principle. Further, we describe constancy and periodicity of solutions to…

On curvatures of homogeneous sub-Riemannian manifolds

The author discusses in some detail the old definitions of the curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov'ev. To calculate the Solov'ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant completely non- holonomic distributions on manifolds. As a justification, we find a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin-Hamilton func- tion. This function was ap…

Finsler geometry and actions of the p-Schatten unitary groups

Let $p$ be an even positive integer and $U_p(H)$ be the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. Let ${\cal O}$ be a smooth manifold on which $U_p(H)$ acts transitively and smoothly. Then one can endow ${\cal O}$ with a natural Finsler metric in terms of the $p$-Schatten norm and the action of $U_p(H)$. Our main result establishes that for any pair of given initial conditions $$ x\in {\cal O}\hbox{and} X\in (T{\cal O})_x $$ there exists a curve $δ(t)=e^{tz}\cdot x$ in ${\cal O}$, with $z$ a skew-hermitian element in the $p$-S…

Extremal curves in nilpotent Lie groups

We classify extremal curves in free nilpotent Lie groups. The classification is obtained via an explicit integration of the adjoint equation in Pontryagin Maximum Principle. It turns out that abnormal extremals are precisely the horizontal curves contained in algebraic varieties of a specific type. We also extend the results to the nonfree case.

Sub-Riemannian geodesics on the free Carnot group with the growth vector (2,3,5,8)

We consider the free nilpotent Lie algebra $L$ with 2 generators, of step 4, and the corresponding connected simply connected Lie group $G$. We study the left-invariant sub-Riemannian structure on $G$ defined by the generators of $L$ as an orthonormal frame. We compute two vector field models of $L$ by polynomial vector fields in $R^8$, and find an infinitesimal symmetry of the sub-Riemannian structure. Further, we compute explicitly the product rule in $G$, the right-invariant frame on $G$, linear on fibers Hamiltonians corresponding to the left-invariant and right-invariant frames on $G$, Ca…

Canonical sphere bundles of the Grassmann manifold

For a given Hilbert space $\mathcal H$, consider the space of self-adjoint projections $\mathcal P(\mathcal H)$. In this paper we study the differentiable structure of a canonical sphere bundle over $\mathcal P(\mathcal H)$ given by $$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \|f\|=1\, \}. $$ We establish the smooth action on $\mathcal R$ of the group of unitary operators of $\mathcal H$, therefore $\mathcal R$ is an homogeneous space. Then we study the metric structure of $\mathcal R$ by endowing it first with the uniform quotient metric, which is a F…

On sub-Riemannian geodesics in $SE(3)$ whose spatial projections do not have cusps

We consider the problem $\mathbf{P_{curve}}$ of minimizing $\int \limits_0^L \sqrt{ξ^2 + κ^2(s)} \, {\rm d}s$ for a curve $\mathbf{x}$ on $\mathbb R$ with fixed boundary points and directions. Here the total length $L\geq 0$ is free, $s$ denotes the arclength parameter, $κ$ denotes the absolute curvature of $\mathbf{x}$, and $ξ>0$ is constant. We lift problem $\mathbf{P_{curve}}$ on $\mathbb R^3$ to a sub-Riemannian problem $\mathbf{P_{mec}}$ on $\operatorname{SE(3)}\nolimits/(\{\mathbf{0}\}\times \operatorname{SO(2)}\nolimits)$. Here, for admissible boundary conditions, the spatial projection…