Control of Hyperbolic and Parabolic Equations on Networks and Singular limits

Bárcena-Petisco JA, Cavalcante M, Coclite GM, De Nitti N, Zuazua Iriondo E (2024)

Publication Status: Accepted

Publication Type: Unpublished / Preprint

Future Publication Type: Article in Edited Volumes

Publication year: 2024

Journal

Mathematical Control and Related Fields American Institute of Mathematical Sciences (AIMS)

URI: https://hal.science/hal-03233211

Open Access Link: https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/390

Abstract

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.

Authors with CRIS profile

Nicola De Nitti Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur) Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

Universidad Autónoma de Madrid (UAM) ES Spain (ES) Federal University of Alagoas / Universidade Federal de Alagoas BR Brazil (BR) Politecnico di Bari IT Italy (IT)

How to cite

APA:

Bárcena-Petisco, J.A., Cavalcante, M., Coclite, G.M., De Nitti, N., & Zuazua Iriondo, E. (2024). Control of Hyperbolic and Parabolic Equations on Networks and Singular limits. (Unpublished, Accepted).

MLA:

Bárcena-Petisco, Jon Asier, et al. Control of Hyperbolic and Parabolic Equations on Networks and Singular limits. Unpublished, Accepted. 2024.

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