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Minimal controllability time for the heat equation under unilateral state or control constraints

Authors :
Emmanuel Trélat
Enrique Zuazua
Jérôme Lohéac
Laboratoire des Sciences du Numérique de Nantes (LS2N)
IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique)
Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST)
Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)
Commande (Commande)
Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique)
Département Automatique, Productique et Informatique (IMT Atlantique - DAPI)
Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)
Control And GEometry (CaGE )
Laboratoire Jacques-Louis Lions (LJLL)
Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris
Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
Departamento de Matemáticas [Madrid]
Universidad Autonoma de Madrid (UAM)
Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST)
Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique (IMT Atlantique)
Commande (LS2N - équipe Commande)
IMT Atlantique (IMT Atlantique)
Universidad Autónoma de Madrid (UAM)
Source :
Mathematical Models and Methods in Applied Sciences, Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2017, 27 (9), pp.1587--1644. ⟨10.1142/S0218202517500270⟩, Mathematical Models and Methods in Applied Sciences, 2017, 27 (9), pp.1587--1644. ⟨10.1142/S0218202517500270⟩
Publication Year :
2017
Publisher :
HAL CCSD, 2017.

Abstract

The heat equation with homogeneous Dirichlet boundary conditions is well known to preserve non-negativity. Besides, due to infinite velocity of propagation, the heat equation is null controllable within arbitrary small time, with controls supported in any arbitrarily open subset of the domain (or its boundary) where heat diffuses. The following question then arises naturally: can the heat dynamics be controlled from a positive initial steady state to a positive final one, requiring that the state remains non-negative along the controlled time-dependent trajectory? We show that this state-constrained controllability property can be achieved if the control time is large enough, but that it fails to be true in general if the control time is too short, thus showing the existence of a positive minimal controllability time. In other words, in spite of infinite velocity of propagation, realizing controllability under the unilateral non-negativity state constraint requires a positive minimal time. We establish similar results for unilateral control constraints. We give some explicit bounds on the minimal controllability time, first in 1D by using the sinusoidal spectral expansion of solutions, and then in the multi-dimensional case. We illustrate our results with numerical simulations, and we discuss similar issues for other control problems with various boundary conditions.

Details

Language :
English
ISSN :
02182025 and 17936314
Database :
OpenAIRE
Journal :
Mathematical Models and Methods in Applied Sciences, Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2017, 27 (9), pp.1587--1644. ⟨10.1142/S0218202517500270⟩, Mathematical Models and Methods in Applied Sciences, 2017, 27 (9), pp.1587--1644. ⟨10.1142/S0218202517500270⟩
Accession number :
edsair.doi.dedup.....48f12acc9a558d6a691ba8cd2473dd47
Full Text :
https://doi.org/10.1142/S0218202517500270⟩