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Preprints, Working Papers, ... (Preprint) Year : 2023

Approximate control of parabolic equations with on-off shape controls by Fenchel duality

Abstract

We consider the internal control of linear parabolic equations through \textit{on-off shape} controls, \ie controls of the form $M(t) \chi_{\omega(t)}$ with $M(t) \geq 0$ and $\omega(t)$ with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative controls. We manage to build controls with constant amplitude~$M(t) \equiv \overline{M}$. In contrast, if the moving control set $\omega(t)$ is confined to evolve in some region of the whole domain, we prove that approximate controllability fails to hold for small times. The method of proof is constructive. Using Fenchel-Rockafellar duality and the bathtub principle, the on-off shape control is obtained as the bang-bang solution of an optimal control problem, which we design by relaxing the constraints. Our optimal control approach is outlined in a rather general form for linear constrained control problems, paving the way for generalisations and applications to other PDEs and constraints.
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Dates and versions

hal-03889865 , version 1 (08-12-2022)
hal-03889865 , version 2 (11-12-2023)

Identifiers

  • HAL Id : hal-03889865 , version 2

Cite

Camille Pouchol, Emmanuel Trélat, Christophe Zhang. Approximate control of parabolic equations with on-off shape controls by Fenchel duality. 2023. ⟨hal-03889865v2⟩
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