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Pré-Publication, Document De Travail Année : 2021

Control of hyperbolic and parabolic equations on networks and singular limits

Jon Asier Bárcena-Petisco
  • Fonction : Auteur
Márcio Cavalcante
  • Fonction : Auteur
Giuseppe Maria Coclite
  • Fonction : Auteur
Nicola de Nitti
  • Fonction : Auteur
  • PersonId : 1099507
Enrique Zuazua
  • Fonction : Auteur

Résumé

We study the controllability properties of the transport equation and of parabolic equations posed on a tree. Using a control localized on the exterior nodes, we prove that the hyperbolic and the parabolic systems are null-controllable. The hyperbolic proof relies on the method of characteristics, the parabolic one on duality arguments and Carleman inequalities. We also show that the parabolic system may not be controllable if we do not act on all exterior vertices because of symmetries. Moreover, we estimate the cost of the null-controllability of transport-diffusion equations with diffusivity ε > 0 and study its asymptotic behavior when ε → 0+. We prove that the cost of the controllability decays for a time sufficiently large and explodes for short times. This is done by duality arguments allowing to reduce the problem to obtain observability estimates which depend on the viscosity parameter. These are derived by using Agmon and Carleman inequalities.
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Dates et versions

hal-03233211 , version 1 (24-05-2021)

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  • HAL Id : hal-03233211 , version 1

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Jon Asier Bárcena-Petisco, Márcio Cavalcante, Giuseppe Maria Coclite, Nicola de Nitti, Enrique Zuazua. Control of hyperbolic and parabolic equations on networks and singular limits. 2021. ⟨hal-03233211⟩

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