On Optimal Observability of Lipschitz Systems

Abstract

Let X,Y be two metric spaces. Let d_X,d_Y denote the metrics in X and Y. Let C be a mapping from X into Y. Suppose that f is a real valued Lipschitz function defined on X. The problem of Lipschitz-observability is to find a Lipschitz function φ defined on Y such that

$$ {\rm{\varphi }}\left( {{\rm{Cx}}} \right){\rm{ = f}}\left( {\rm{x}} \right){\rm{ for x}} \in {\rm{x}} $$

((1))

If such a φ exists we say that f is Lipschitz-observable and is called a method of observation.