Observability of real analytic vector fields on compact manifolds

A smooth vector field on a smooth compact manifold is said to be observable if there exists a smooth output function that distinguishes initial states on any time interval. In this paper we prove that a real analytic vector field on a compact manifold is observable if and only if its singularities are isolated. The result remains true for functions. In this case one has to assume that the set of singular points is (p − 1)-dimensional.