THE COVERING SEMIGROUP OF INVARIANT CONTROL SYSTEMS ON LIE GROUPS

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 of g a cone in the Lie algebra 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 Gamma(Sigma, x), x is an element of G introduced by Colonius, Kizil and San Martin. This formalism provides a group (G) over cap (X) of exponential of Lie series and a subsemigroup (S) over cap (X) subset of (G) over cap (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 (S) over cap (X) with the congruence classes determined by the kernel of a homomorphism of (S) over cap (X).