Finite singularities of nonlinear systems. Output stabilization, observability and observers

We consider observed, either controlled or uncontrolled analytic nonlinear systems having the property that the kth derivatives of the outputs can be smoothly expressed in terms of the outputs, their k-1 first derivatives, the inputs and their k first derivatives. We show that, in the uncontrolled case, the finiteness of the map which to the state associates the output with its first derivatives, with additional standard observability assumptions, implies this property. More generally, in both the controlled and uncontrolled cases, this property together with some observability assumptions allows the construction of asymptotic observers, even if the state is a continuous function only of the inputs, the outputs and their derivatives up to a certain order (at least in restriction to compacta). We show that our systems verifying the stated property, if state feedback stabilizable, are also dynamic output feedback stabilizable (with an additional assumption). Therefore, our result contains a number of previous ones. Proofs are not given.