From internal to pointwise control for the 1D heat equation and minimal control time
Résumé
Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $ω_ε = (x_0 − ε, x_0 + ε)$, in the limit $ε → 0$, where $x_0 ∈ (0, 1)$. 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 ε fixed, the heat equation is controllable with control set $ω_ε$ in any time $T > 0$. We relate these two phenomena. We show that the observability constant on $ω_ε$ does not converge to $0$ as $ε → 0$ at the same speed when $T > T_0$ (in which case it is comparable to $ε 1/2$) or $T < T_0$ (in which case it converges faster to $0$). We also describe the behavior of optimal $L^2$ null-controls on $ω_ε$ in the limit $ε → 0$.
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