Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the $N$-dimensional boundary null-controllability in cylindrical domains

Abstract : In 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 $0^+$.
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Assia Benabdallah, Franck Boyer, Manuel Gonzalez-Burgos, Guillaume Olive. Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the $N$-dimensional boundary null-controllability in cylindrical domains. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (5), pp.2970-3001. ⟨10.1137/130929680⟩. ⟨hal-00845994⟩

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