C_0 \int_0^{T_0} {\left\Vert{{\mathcal{B}}}y(t)\right\Vert}_{{\mathcal{U}}}^2 \, dt.$$ Estimate is the so-called observability estimate for . As it has been established in the classical works [@DoleckiRussell; @Lions], this property is essentially equivalent to the controllability of the adjoint system: $$\label{ControlledSystem}
    y_c' = {{\mathcal{A}}}y_c + {{\mathcal{B}}}^* v, \quad t \in (0,T), \qquad y_c(0) = y_c^0,$$ where $v$ is a control function in $L^2(0,T; {{\mathcal{U}}})$. Actually the control for can be built by minimizing a suitable quadratic functional over the class of solutions . We do not give details on these links in this article and we refer the interested reader to [@Lions; @TWbook]. Also note that the observability property is also closely linked to some inverse problems, see e.g. the work [@AlvesSilvestreTakahashiTucsnak] for precise statements in a setting similar to ours and the references therein.

We are thus interested in deriving discrete versions of for solutions $y_\tau$ of . Namely, we are asking if, given $\delta \in (0,R)$, there exist a time $T>0$ and constants $C>0$ and $\tau_0>0$ independent of $\tau>0$ such that for all $\tau \in (0,\tau_0)$, solutions $y_\tau$ of with initial data $y^0 \in \mathfrak{C}(\delta/\tau)$ satisfy $$\label{Fully-Obs}
        {\left\Verty^0\right\Vert}_{{\mathcal{X}}}^2 \leq C \tau \sum_{ k \tau \in (0,T)} {\left\Vert{{\mathcal{B}}}y_\tau^k\right\Vert}_{{\mathcal{U}}}^2.$$ This is a discrete observability 