Finite-Timeless Local Observability for Linear Control Systems on Lie Groups

In this work, we introduce finite-timeless observability for linear control systems on Lie groups. Observability problem is to distinguish the elements of the state space by looking the images of the solutions passing through those elements in the output space. For this aim, we define almost indistinguishability and show that it is an equivalence relation. We consider linear control systems on Lie groups, where the output function is a Lie group homomorphism and study properties of the image of almost indistinguishable points from the neutral element. Thus, we relate local observability of this kind of systems to its finite-timeless local observability.