Observability and symmetries of linear control systems

The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space Rⁿ and, in a more general setup, on a connected Lie group G. For that, we establish well-known results. The symmetries involved in this theory allow characterizing the observability property on Euclidean spaces and the local observability property on Lie groups.