  The input-constrained erasure channel with feedback is considered, where the
binary input sequence contains no consecutive ones, i.e., it satisfies the
$(1,\infty)$-RLL constraint. We derive the capacity for this setting, which can
be expressed as $C_{\epsilon}=\max_{0 \leq p \leq
\frac{1}{2}}\frac{H_{b}(p)}{p+\frac{1}{1-\epsilon}}$, where $\epsilon$ is the
erasure probability and $ H_{b}(\cdot)$ is the binary entropy function.
Moreover, we prove that a-priori knowledge of the erasure at the encoder does
not increase the feedback capacity. The feedback capacity was calculated using
an equivalent dynamic programming (DP) formulation with an optimal
average-reward that is equal to the capacity. Furthermore, we obtained an
optimal encoding procedure from the solution of the DP, leading to a
capacity-achieving, zero-error coding scheme for our setting. DP is thus shown
to be a tool not only for solving optimization problems such as capacity
calculation, but also for constructing optimal coding schemes. The derived
capacity expression also serves as the only non-trivial upper bound known on
the capacity of the input-constrained erasure channel without feedback, a
problem that is still open.
