equation and its time-discrete counterparts for . This problem, inspired in control theoretical issues, concerns the possibility of recovering the full energy of solutions out of partial measurements. The question is relevant both in the continuous and in the time-discrete frame, and in the later, a natural question arising in numerical analysis is to know whether the observability property is uniform with respect to the time-step. Indeed, this problem is the dual version of the classical controllability one and the uniformity (with respect to time-step) of the observability inequality is equivalent to the convergence of numerical controls towards the continuous ones as the time-step tends to zero (see [@Zua05Survey; @ErvZuaCime]).

We thus consider an observation operator ${{\mathcal{B}}}$ taking value in some Hilbert space ${{\mathcal{U}}}$ and assumed to belong to $ \mathfrak{L}(\mathcal{D}({{\mathcal{A}}}^p), {{\mathcal{U}}})$ for some $p\in {\mathbb{N}}$. To be more precise, we assume that there exists $C_{p}>0$ such that $$\label{C-p-boundedness}
        {\left\Vert{{\mathcal{B}}}y\right\Vert}_{{{\mathcal{U}}}}^2 \leq C_{p}^2 \left({\left\Vert{{\mathcal{A}}}^p y\right\Vert}_{{{\mathcal{X}}}}^2  + {\left\Verty\right\Vert}_{{{\mathcal{X}}}}^2 \right), \quad \forall y \in \mathcal{D}({{\mathcal{A}}}^p).$$ We furthermore assume that equation is observable through ${{\mathcal{B}}}$ at time $T_0>0$, meaning that there exists a constant $C_0$ such that, for all $y^0 \in \mathcal{D}({{\mathcal{A}}}^p)$, the solution $y(t)$ of with initial data $y^0$ satisfies $$\label{FD-Obs}
        {\left\Verty^0\right\Vert}_{{\mathcal{X}}}^2 \leq 