Control of hyperbolic and parabolic equations on networks and singular limits

We study the controllability properties of transport equations and of parabolic equations with vanishing diffusivity posed on a tree-shaped network. Using a control localized on the exterior nodes, we obtain a null-controllability result for both systems. The hyperbolic proof relies on the method of...

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Authors: Barcena Petisco, Jon Asier, Cavalcante, Márcio, Coclite, Giuseppe María, De Nitti, Nicola, Zuazua Iriondo, Enrique
Format: article
Publication Date:2025
Country:España
Institution:Universidad del País Vasco
Repository:Addi. Archivo Digital para la Docencia y la Investigación
OAI Identifier:oai:addi.ehu.eus:10810/73759
Online Access:http://hdl.handle.net/10810/73759
Access Level:Open access
Keyword:Controllability
Cost of controllability
Advection-diffusion equations
Vanishing viscosity
Networks
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spelling Control of hyperbolic and parabolic equations on networks and singular limitsBarcena Petisco, Jon AsierCavalcante, MárcioCoclite, Giuseppe MaríaDe Nitti, NicolaZuazua Iriondo, EnriqueControllabilityCost of controllabilityAdvection-diffusion equationsVanishing viscosityNetworksWe study the controllability properties of transport equations and of parabolic equations with vanishing diffusivity posed on a tree-shaped network. Using a control localized on the exterior nodes, we obtain a null-controllability result for both systems. The hyperbolic proof relies on the method of characteristics; while the parabolic one on duality arguments and Carleman inequalities. In particular, we estimate the cost of the null-controllability of advection-diffusion equations with diffusivity ε > 0 and study its asymptotic behavior when ε → 0+. More specifically, we show that the cost of null-controllability decays exponentially for a time sufficiently large and explodes for short times. The core of the proof consists in proving an observability estimate keeping track of the viscosity parameter by relying on a suitable Carleman inequality.This work has received funding from the Alexander von Humboldt-Professorship program, the Transregio 154 Project "Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks" of the DFG, the grant PID2020-112617GB-C22 of MINECO (Spain), and the COST Action grant CA18232, "Mathematical models for interacting dynamics on networks" (MAT-DYN-NET). J. A. Bárcena-Petisco is funded by the Grant PID2021-126813NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe", and by the grant IT1615-22 funded by the Basque Government. M. Cavalcante has been partially funded by CAPES-MATHAMSUD 88887.368708/2019. G. M. Coclite and N. De Nitti are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). G. M. Coclite has been partially supported by the Project funded under the National Recovery and Resilience Plan (NRRP), Mission 4, Component 2, Investment 1.4 (Call for tender No. 3138 of 16/12/2021), of Italian Ministry of University and Research funded by the European Union (NextGenerationEU Award, No. CN000023, Concession Decree No. 1033 of 17/06/2022) adopted by the Italian Ministry of University and Research (CUP D93C22000410001), Centro Nazionale per la Mobilità Sostenibile. He has also been supported by the Italian Ministry of Education, University and Research under the Programme "Department of Excellence" Legge 232/2016 (CUP D93C23000100001), and by the Research Project of National Relevance "Evolution problems involving interacting scales" granted by the Italian Ministry of Education, University and Research (MIUR PRIN 2022, project code 2022M9BKBC, CUP D53D23005880006). N. De Nitti has been funded by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00034 through the project TENSE.American Institute of Mathematical Sciences202520252025info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10810/73759reponame:Addi. Archivo Digital para la Docencia y la Investigacióninstname:Universidad del País VascoInglésinfo:eu-repo/grantAgreement/MINECO/PID2020-112617GB-C22/info:eu-repo/grantAgreement/MCIN/PID2021-126813NB-I00/https://doi.org/10.3934/mcrf.2024015info:eu-repo/semantics/openAccess© 2025 American Institute of Mathematical Sciencesoai:addi.ehu.eus:10810/737592026-06-18T09:23:17Z
dc.title.none.fl_str_mv Control of hyperbolic and parabolic equations on networks and singular limits
title Control of hyperbolic and parabolic equations on networks and singular limits
spellingShingle Control of hyperbolic and parabolic equations on networks and singular limits
Barcena Petisco, Jon Asier
Controllability
Cost of controllability
Advection-diffusion equations
Vanishing viscosity
Networks
title_short Control of hyperbolic and parabolic equations on networks and singular limits
title_full Control of hyperbolic and parabolic equations on networks and singular limits
title_fullStr Control of hyperbolic and parabolic equations on networks and singular limits
title_full_unstemmed Control of hyperbolic and parabolic equations on networks and singular limits
title_sort Control of hyperbolic and parabolic equations on networks and singular limits
dc.creator.none.fl_str_mv Barcena Petisco, Jon Asier
Cavalcante, Márcio
Coclite, Giuseppe María
De Nitti, Nicola
Zuazua Iriondo, Enrique
author Barcena Petisco, Jon Asier
author_facet Barcena Petisco, Jon Asier
Cavalcante, Márcio
Coclite, Giuseppe María
De Nitti, Nicola
Zuazua Iriondo, Enrique
author_role author
author2 Cavalcante, Márcio
Coclite, Giuseppe María
De Nitti, Nicola
Zuazua Iriondo, Enrique
author2_role author
author
author
author
dc.subject.none.fl_str_mv Controllability
Cost of controllability
Advection-diffusion equations
Vanishing viscosity
Networks
topic Controllability
Cost of controllability
Advection-diffusion equations
Vanishing viscosity
Networks
description We study the controllability properties of transport equations and of parabolic equations with vanishing diffusivity posed on a tree-shaped network. Using a control localized on the exterior nodes, we obtain a null-controllability result for both systems. The hyperbolic proof relies on the method of characteristics; while the parabolic one on duality arguments and Carleman inequalities. In particular, we estimate the cost of the null-controllability of advection-diffusion equations with diffusivity ε > 0 and study its asymptotic behavior when ε → 0+. More specifically, we show that the cost of null-controllability decays exponentially for a time sufficiently large and explodes for short times. The core of the proof consists in proving an observability estimate keeping track of the viscosity parameter by relying on a suitable Carleman inequality.
publishDate 2025
dc.date.none.fl_str_mv 2025
2025
2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10810/73759
url http://hdl.handle.net/10810/73759
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/MINECO/PID2020-112617GB-C22/
info:eu-repo/grantAgreement/MCIN/PID2021-126813NB-I00/
https://doi.org/10.3934/mcrf.2024015
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
© 2025 American Institute of Mathematical Sciences
eu_rights_str_mv openAccess
rights_invalid_str_mv © 2025 American Institute of Mathematical Sciences
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv American Institute of Mathematical Sciences
publisher.none.fl_str_mv American Institute of Mathematical Sciences
dc.source.none.fl_str_mv reponame:Addi. Archivo Digital para la Docencia y la Investigación
instname:Universidad del País Vasco
instname_str Universidad del País Vasco
reponame_str Addi. Archivo Digital para la Docencia y la Investigación
collection Addi. Archivo Digital para la Docencia y la Investigación
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.812429

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