左不变控制系统在SO上的时间最优综合（3）

Symmetries in left-invariant optimal control problems

We consider left-invariant optimal control problems on connected Lie groups such that generic stabilizer of the coadjoint action is connected and has dimension not more than 1. We introduce a construction for symmetries of the exponential map. These symmetries play a key role in investigation of optimality of extremal trajectories.

Semigroups and Controllability of Invariant Control Systems on $\mathrm{Sl}\left(n,\mathbb{H}\right)$

Let $\mathrm{Sl}\left( n,\mathbb{H}\right)$ be the Lie group of $n\times n$ quaternionic matrices $g$ with $\left\vert \det g\right\vert =1$. We prove that a subsemigroup $S \subset \mathrm{Sl}\left( n,\mathbb{H}\right)$ with nonempty interior is equal to $\mathrm{Sl}\left( n,\mathbb{H}\right)$ if $S$ contains a subgroup isomorphic to $\mathrm{Sl}\left( 2,\mathbb{H}\right)$. As application we give sufficient conditions on $A,B\in \mathfrak{sl}\left( n,\mathbb{H}\right)$ to ensuring that the invariant control system $\dot{g}=Ag+uBg$ is controllable on $\mathrm{Sl}\left( n,\mathbb{H}\right)$. We…

On an isoperimetric problem on the Lobachevsky hyperbolic plane with left-invariant Finsler structure

We deal with an isoperimetric problem on the Finsler hyperbolic plane. The space is defined as the Lie group of proper affine transformations of the line with a left-invariant Finsler structure. To state the problem, we use the left-invariant volume form. The optimal isoperimetric loops are found in terms of convex trigonometry functions. We also propose a generalization of the isoperimetric inequality in parametric form.

Left-invariant optimal control problems on Lie groups

Left-invariant optimal control problems on Lie groups form an important class of problems with big symmetry group. They are interesting from the theoretical point of view since they often can be completely studied, and general features can be investigated on these model problems. In particular, problems on nilpotent Lie groups give a fundamental nilpotent approximation of general problems. Left-invariant problems also often arise in applications: in classical and quantum mechanics, geometry, robotics, models of vision and image processing. This work aims to review the main notions, methods and…

Explicit formulae for geodesics in left invariant sub-Finsler problems on Heisenberg groups via convex trigonometry

In the present paper, we obtain explicit formulae for geodesics in some left-invariant sub-Finsler problems on Heisenberg groups $\mathbb{H}_{2n+1}$. Our main assumption is the following: the compact convex set of unit velocities at identity admits a generalization of spherical coordinates. This includes convex hulls and sums of coordinate 2-dimensional sets, all left-invariant sub-Riemannian structures on $\mathbb{H}_{2n+1}$, and unit balls in $L_p$-metric for $1\le p\le\infty$. In the last case, extremals are obtained in terms of incomplete Euler integral of the first kind.

幂零李群上左不变控制系统的属性集和Carnot-Caratheodory度量

Attainable sets for left invariant control systems and Carnot-Caratheodory metrics on nilpotent Lie groups

设$G$是单连通幂零李群与$\R$的半直积.对于以凸锥为控制区域的$G$上的左不变控制系统，证明了当系统与某些特殊子空间的接触度足够高时，系统的可达集与“半空间”重合.