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A non-controllability result for the half-heat equation on the whole line based on the prolate spheroidal wave functions and its application to the Grushin equation

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PSWF1D.pdf (362.6Kb)
Date
2020
Publisher city
Paris
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Publishing date
02-2020
Link to item file
https://hal.archives-ouvertes.fr/hal-02420212
Dewey
Analyse
Sujet
Controllability; observability; fractional parabolic equations,; prolate spheroidalwave functions
URI
https://basepub.dauphine.fr/handle/123456789/20675
Collections
  • CEREMADE : Publications
Metadata
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Author
Lissy, Pierre
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Document de travail / Working paper
Item number of pages
20
Abstract (EN)
In this article, we revisit a result by A. Koenig concerning the non-controllability of the half-heat equation posed on R, with a control domain that is an open set whose exterior contains an interval. The main novelty of the present article is to disprove the corresponding observability inequality by using as an initial condition a family of prolate spheroidal wave function (PSWF) translated in the Fourier space, associated to a parameter c that goes to ∞. The proof is essentially based on the dual nature of the PSWF together with direct computations, showing that the solution "does not spread out" too much during time. As a consequence, we obtain a new non-controllability result on the Grushin equation posed on R × R.

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