Here, we prove the uniform observability of a two-grid method for the semi-discretization of the 1D-wave equation for a time $T>2\sqrt{2}$ ; this time, if the observation is made in $(-T/2,T/2)$ , is optimal and this result improves an earlier work of Negreanu and Zuazua [C. R. Acad. Sci. Paris Sér. I338 (2004) 413–418].Our proof follows an Ingham type approach.

This paper deals with the observability analysis and the observer synthesis of a class of nonlinear systems. In the single output case, it is known [4-6] that systems which are observable independently of the inputs, admit an observable canonical form. These systems are called uniformly observable systems. Moreover, a high gain observer for these systems can be designed on the basis of this canonical form. In this paper, we extend the above results to multi-output uniformly observable systems. Corresponding canonical forms are presented and sufficient conditions which permit the design of constant and high gain observers for these systems are given.