#### Vol. 15, No. 4, 2022

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Sharp reachability results for the heat equation in one space dimension

### Karim Kellay, Thomas Normand and Marius Tucsnak

Vol. 15 (2022), No. 4, 891–920
##### Abstract

This paper gives a complete characterization of the reachable space for a system described by the 1-dimensional heat equation with ${L}^{2}$ (with respect to time) Dirichlet boundary controls at both ends. More precisely, we prove that this space coincides with the sum of two spaces of analytic functions (of Bergman type). These results are then applied to give a complete description of the reachable space via inputs which are $n$-times differentiable functions of time. Moreover, we establish a connection between the norm in the obtained sum of Bergman spaces and the cost of null controllability in small time. Finally we show that our methods yield new complex analytic results on the sums of Bergman spaces in infinite sectors.

##### Keywords
reachable space, null controllability, Bergman spaces, smooth inputs, control cost
##### Mathematical Subject Classification 2010
Primary: 93B03, 35K08, 30H20
Secondary: 93B05