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Averaged controllability of parameter dependent conservative semigroups

Authors :
Enrique Zuazua
Jérôme Lohéac
Laboratoire des Sciences du Numérique de Nantes (LS2N)
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)
Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)
Commande (LS2N - équipe Commande)
Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST)
Robotique Et Vivant (LS2N - équipe ReV)
Département Automatique, Productique et Informatique (IMT Atlantique - DAPI)
IMT Atlantique (IMT Atlantique)
Departamento de Matemáticas [Madrid]
Universidad Autónoma de Madrid (UAM)
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)
IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique)
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)
Robotique Et Vivant (ReV)
Universidad Autonoma de Madrid (UAM)
Source :
Journal of Differential Equations, Journal of Differential Equations, 2017, 262 (3), pp.1540-1574. ⟨10.1016/j.jde.2016.10.017⟩, Journal of Differential Equations, Elsevier, 2017, 262 (3), pp.1540-1574. ⟨10.1016/j.jde.2016.10.017⟩
Publication Year :
2017
Publisher :
HAL CCSD, 2017.

Abstract

International audience; We consider the problem of averaged controllability for parameter depending (either in a discrete or continuous fashion) control systems, the aim being to find a control, independent of the unknown parameters, so that the average of the states is controlled. We do it in the context of conservative models, both in an abstract setting and also analysing the specific example of the wave equation. Our first result is of perturbative nature. Assuming the averaging probability measure to be a small parameter-dependent perturbation (in a sense that we make precise) of an atomic measure given by a Dirac mass corresponding to a specific realisation of the system, we show that the averaged controllability property is achieved whenever the system corresponding to the support of the Dirac is controllable. Similar tools can be employed to obtain averaged versions of the so-called Ingham inequalities. Particular attention is devoted to 1d wave and Schrödinger equations in which the time-periodicity of solutions can be exploited to obtain more precise results, provided the parameters involved satisfy Diophantine conditions ensuring the lack of resonances.

Details

Language :
English
ISSN :
00220396 and 10902732
Database :
OpenAIRE
Journal :
Journal of Differential Equations, Journal of Differential Equations, 2017, 262 (3), pp.1540-1574. ⟨10.1016/j.jde.2016.10.017⟩, Journal of Differential Equations, Elsevier, 2017, 262 (3), pp.1540-1574. ⟨10.1016/j.jde.2016.10.017⟩
Accession number :
edsair.doi.dedup.....ef05d7e6dbdd6f17b54ffe6755e36a15
Full Text :
https://doi.org/10.1016/j.jde.2016.10.017⟩