HARTMANN, Andreas; KELLAY, Karim; TUCSNAK, Marius

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HARTMANN, Andreas
Institut de Mathématiques de Bordeaux [IMB]

KELLAY, Karim
Institut de Mathématiques de Bordeaux [IMB]

TUCSNAK, Marius
Institut de Mathématiques de Bordeaux [IMB]

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Article de revue

Ce document a été publié dans

Journal of the European Mathematical Society. 2020-06-15

European Mathematical Society

Résumé en anglais

This work considers systems described by the heat equation on the interval [0, π] with L^2 boundary controls and it studies the reachable space at some instant τ > 0. The main results assert that this space is generally sandwiched between two Hilbert spaces of holomorphic functions defined on a square in the complex plane and which has [0, π] as one of the diagonals. More precisely, in the case Dirichlet boundary controls acting at both ends we prove that the reachable space contains the Smirnov space and it is contained in the Bergman space associated to the above mentioned square. The methodology, quite different of the one employed in previous literature, is a direct one. We first represent the input-to-state map as an integral operator whose kernel is a sum of Gaussians and then we study the range of this operator by combining the theory of Riesz bases for Smirnov spaces in polygons and the theory developed by Aikawa, Hayashi and Saitoh on the range of integral transforms, in particular those associated with the heat kernel.< Réduire

Mots clés en anglais

reachable space

controllability

heat equation

Bergman spaces

Smirnov spaces

Riesz basis

URI

https://oskar-bordeaux.fr/handle/20.500.12278/193605

Project ANR

Impacts marchands, non marchands et structurels des réformes des politiques agricoles et agri-environnementales - ANR-05-PADD-0015

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Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251