Genericity of Observability and the Existence of Asymptotic Observers

In this paper, we deal with the genericity of the observability property and the existence of asymptotic observers for nonlinear systems. In the case where the number of outputs is larger than the number of inputs and the state space is compact, we prove that observability in a very strong sense (more or less, observability for each sufficiently differentiable input) is generic. This is obtained by using standard (but not easy) transversality arguments. For the inputs that are bounded with their derivatives up to some order, we prove the generic existence of an asymptotic observer with arbitrary exponential decay of the error.