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Keywords:
control systems; homotopy of trajectories; covering semigroup
control systems; homotopy of trajectories; covering semigroup
Summary:
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 $\widehat{S}(X)$.
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