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 $\varepsilon \rightarrow 0$.
Categories: math.AP