  Our goal is to study controllability and observability properties of the 1D
heat equation with internal control (or observation) set
$\omega_{\varepsilon}=(x_{0}-\varepsilon, x_{0}+\varepsilon )$, in the limit
$\varepsilon\rightarrow 0$, where $x_{0}\in (0,1)$. It is known that depending
on arithmetic properties of $x_{0}$, there may exist a minimal time $T_{0}$ of
pointwise control at $x_{0}$ of the heat equation. Besides, for any
$\varepsilon$ fixed, the heat equation is controllable with control set
$\omega_{\varepsilon}$ in any time $T>0$. We relate these two phenomena. We
show that the observability constant on $\omega_\varepsilon$ does not converge
to $0$ as $\varepsilon\rightarrow 0$ at the same speed when $T>T_{0}$ (in which
case it is comparable to $\varepsilon^{1/2}$) or $T<T_{0}$ (in which case it
converges faster to $0$). We also describe the behavior of optimal $L^{2}$
null-controls on $\omega_{\varepsilon}$ in the limit $\varepsilon \rightarrow
0$.
