estimate for the time-discrete equations which is uniform with respect to the time-discretization parameter, provided the initial state is suitably filtered. As it is well-known, see e.g. the survey articles [@Zua05Survey; @ErvZuaCime], this is needed to derive algorithms to compute convergent sequences of discrete controls approximating the control of the continuous dynamics.

Our second main result, whose proof is given in Section \[Sec-Rho-tau\], is the following one:

\[Thm-Main\] Assume that ${{\mathcal{B}}}\in \mathfrak{L}(\mathcal{D}({{\mathcal{A}}}^p), {{\mathcal{U}}})$ for some $p\in {\mathbb{N}}$ and ${{\mathcal{U}}}$ an Hilbert space, and that ${{\mathcal{B}}}$ satisfies with constant $C_p$.

Assume that equation is observable through ${{\mathcal{B}}}$ at time $T_0$ with constant $C_0$, i.e. for all $y^0 \in \mathcal{D}({{\mathcal{A}}}^p)$, the solution $y(t)$ of with initial data $y^0$ satisfies .

Let $f$ be a smooth function describing the time discrete operator ${\mathbb{T}}_\tau$ as in , assume ––, and fix $\delta \in (0,R)$. Then, for all $$\label{TimeConditionTau}
        T > \frac{T_0}{\underset{|\alpha| \leq \delta}{\inf} \{f'(\alpha) \}},$$ there exist positive constants $C$ and $\tau_0>0$ depending on $f, T, \delta, p, C_p, C_0,T_0$ such that for any $\tau \in (0,\tau_0)$, solutions $y_\tau$ of lying in $\mathfrak{C}(\delta/\tau)$ satisfy .

Theorem \[Thm-Main\] was already proved in [@je3] under an additional admissibility property for the operator ${{\mathcal{B}}}$ (see [@Lions] for 