Observability of Linear Difference Equations in Hilbert Spaces and Applications

Here we present a necessary and sufficient conditions for the exact and approximate observability of the following linear difference equation $$z(n+1) = A(n)z(n), n \in {\mathbb N}^*, z(0) = z_0 \in {\mathbb Z},$$

$$y(n) = Cz(n),$$ where $A \in l^\infty({\mathbb N},L({\mathbb Z}))$, $C \in L({\mathbb Z},U)$, ${\mathbb Z}$, $U$ are Hilbert spaces and ${\mathbb N}^* = {\mathbb N} \cup \{0\}$. We apply these results to a flow-discretization of the wave equation and the heat equation.