On the reachable set of the 1-d heat equation

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Résumé : The goal of this talk is to present a new result concerning the reachable set of the heat equation in space dimension 1. Our approach characterizes the target states in terms of the set on which an analytic expansion exists. To be more precise, when considering the heat equation on a space interval (-L, L) and controlled from both boundaries x = -L and x = L, we show that a function y= y(x) belongs to the reachable set when it admits an analytic extension in a square of the form {z = x_1 + ix_2, with |x_1| + |x_2| < L'} for some L' >L. This result is mainly sharp as one can show that a reachable state necessarily admits an analytic extension in {z = x_1 + i x_2, with |x_1| + |x_2| < L}. Our approach is based on a suitable "limiting" Carleman estimate for the heat equation, a duality result and Cauchy's formula. Let us finally note that this result improves previous ones, namely the pioneering work [Fattorini-Russell 1971], or more recently the work [Martin-Rosier-Rouchon, 2016] which was, to our knowledge, the first work considering the characterization of the reachable states through the set where they admit an holomorphic extension.
This is a joint work with Jérémi Dardé.