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Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport–diffusion equation

Lissy, Pierre (2015), Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport–diffusion equation, Journal of Differential Equations, 259, 10, p. 5331-5352. 10.1016/j.jde.2015.06.031

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-01133526
Date
2015
Journal name
Journal of Differential Equations
Volume
259
Number
10
Publisher
Elsevier
Pages
5331-5352
Publication identifier
10.1016/j.jde.2015.06.031
Metadata
Show full item record
Author(s)
Lissy, Pierre
Abstract (EN)
In this paper, we prove explicit lower bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type involving the spectral fractional Laplace operator. We notably deduce the following striking result: in the case of the heat equation controlled on the boundary, Miller's conjecture formulated in Miller (2004) [16] is not verified. Moreover, we also give a new lower bound for the minimal time needed to ensure the uniform controllability of the one-dimensional convection–diffusion equation with negative speed controlled on the left boundary, proving that the conjecture formulated in Coron and Guerrero (2005) [2] concerning this problem is also not verified at least for negative speeds.The proof is based on complex analysis, and more precisely on a representation formula for entire functions of exponential type, and is quite related to the moment method.
Subjects / Keywords
linear equations of parabolic or dispersive type

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