Sharp Estimates of the One-Dimensional Boundary Control Cost for Parabolic Systems and Application to the $N$-Dimensional Boundary Null Controllability in Cylindrical Domains

International audienceIn this paper we consider the boundary null-controllability of a system of $n$ parabolic equations on domains of the form $\Omega =(0,\pi)\times \Omega_2$ with $\Omega_2$ a smooth domain of $\R^{N-1}$, $N>1$. When the control is exerted on $\{0\}\times \omega_2$ with $\omega_2\subset \Omega_2$, we obtain a necessary and sufficient condition that completely characterizes the null-controllability. This result is obtained through the Lebeau-Robbiano strategy and require an upper bound of the cost of the one-dimensional boundary null-control on $(0,\pi)$. This latter is obtained using the moment method and it is shown to be bounded by $Ce^{C/T}$ when $T$ goes to $0^+$