The covering semigroup of invariant control systems on Lie groups

Ayala, Víctor, and Kizil, Eyüp. "The covering semigroup of invariant control systems on Lie groups." Kybernetika 52.6 (2016): 837-847. <http://eudml.org/doc/287874>.

@article{Ayala2016,
abstract = {It is well known that the class of invariant control systems is really relevant both from theoretical and practical point of view. This work was an attempt to connect an invariant systems on a Lie group $G$ with its covering space. Furthermore, to obtain algebraic properties of this set. Let $G$ be a Lie group with identity $e$ and $\Sigma \subset \mathfrak \{g\}$ a cone in the Lie algebra $\mathfrak \{g\}$ of $G$ that satisfies the Lie algebra rank condition. We use a formalism developed by Sussmann, to obtain an algebraic structure on the covering space $\mathbf \{\Gamma \}(\Sigma ,x),x\in G$ introduced by Colonius, Kizil and San Martin. This formalism provides a group $\widehat\{G\}(X)$ of exponential of Lie series and a subsemigroup $ \widehat\{S\}(\{X\})\subset \widehat\{G\}(X)$ that parametrizes the space of controls by means of a map due to Chen, which assigns to each control a noncommutative formal power series. Then we prove that $\Gamma (\Sigma ,e)$ is the intersection of $\widehat\{S\}(X)$ with the congruence classes determined by the kernel of a homomorphism of $\widehat\{S\}(X)$.},
author = {Ayala, Víctor, Kizil, Eyüp},
journal = {Kybernetika},
keywords = {control systems; homotopy of trajectories; covering semigroup},
language = {eng},
number = {6},
pages = {837-847},
publisher = {Institute of Information Theory and Automation AS CR},
title = {The covering semigroup of invariant control systems on Lie groups},
url = {http://eudml.org/doc/287874},
volume = {52},
year = {2016},
}

TY - JOUR
AU - Ayala, Víctor
AU - Kizil, Eyüp
TI - The covering semigroup of invariant control systems on Lie groups
JO - Kybernetika
PY - 2016
PB - Institute of Information Theory and Automation AS CR
VL - 52
IS - 6
SP - 837
EP - 847
AB - It is well known that the class of invariant control systems is really relevant both from theoretical and practical point of view. This work was an attempt to connect an invariant systems on a Lie group $G$ with its covering space. Furthermore, to obtain algebraic properties of this set. Let $G$ be a Lie group with identity $e$ and $\Sigma \subset \mathfrak {g}$ a cone in the Lie algebra $\mathfrak {g}$ of $G$ that satisfies the Lie algebra rank condition. We use a formalism developed by Sussmann, to obtain an algebraic structure on the covering space $\mathbf {\Gamma }(\Sigma ,x),x\in G$ introduced by Colonius, Kizil and San Martin. This formalism provides a group $\widehat{G}(X)$ of exponential of Lie series and a subsemigroup $ \widehat{S}({X})\subset \widehat{G}(X)$ that parametrizes the space of controls by means of a map due to Chen, which assigns to each control a noncommutative formal power series. Then we prove that $\Gamma (\Sigma ,e)$ is the intersection of $\widehat{S}(X)$ with the congruence classes determined by the kernel of a homomorphism of $\widehat{S}(X)$.
LA - eng
KW - control systems; homotopy of trajectories; covering semigroup
UR - http://eudml.org/doc/287874
ER -