From the reachable space of the heat equation to Hilbert spaces of holomorphic functions

Andreas Hartmann (1) , Karim Kellay (1) , Marius Tucsnak (1)

1 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: reachable space; controllability; heat equation; Bergman spaces; Smirnov spaces; Riesz basis

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]; Variables complexes [math.CV]; Analyse fonctionnelle [math.FA]

Andreas Hartmann, Karim Kellay, Marius Tucsnak. From the reachable space of the heat equation to Hilbert spaces of holomorphic functions. Journal of the European Mathematical Society, 2020. ⟨hal-01569695⟩