Remarks on the existence of a Kazantzis-Kravaris/Luenberger observer

We state sufficient conditions for the existence, on a given open set, of the extension, to nonlinear systems, of the Luenberger observer as it has been proposed by Kazantzis and Kravaris. To weaken these conditions, the observer is modified in a way which induces a time rescaling and which follows from a forward unboundedness observability property. Also, we state it is sufficient to choose the dimension of the dynamic system, giving the observer, less than or equal to 2 + twice the dimension of the state to be observed. Finally we show how approximation is allowed and we establish a link with high gain observers.