Partial Differential Equations/Optimal Control
A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation
Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 591-595.

In this Note, we explain how results on the cost of the null-controllability of the one-dimensional heat equation in small time can be used to bound from above the cost of the null-controllability of a one-dimensional transport-diffusion equation in the vanishing viscosity limit. We improve previous results about the minimal time needed to obtain the exponential decrease of the cost of the control and explain what would provide the usual conjecture concerning the cost of fast controls for the heat equation.

Dans cette Note, on explique comment des résultats sur le coût de la contrôlabilité à 0 de lʼéquation de la chaleur en temps petit peuvent être utilisés pour majorer le coût de la contrôlabilité à 0 dʼune équation unidimensionelle de transport-diffusion dans la limite de viscosité évanescente. On améliore des résultats précédemment connus concernant le temps minimal nécessaire pour obtenir la décroissance exponentielle du coût du contrôle et on explique ce que donnerait en plus la conjecture habituelle concernant le coût du contrôle en temps petit de lʼéquation de la chaleur.

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Accepted:
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DOI: 10.1016/j.crma.2012.06.004
Lissy, Pierre 1

1 UPMC Univ Paris 06, UMR 7598, laboratoire Jacques-Louis Lions, 75005, Paris, France
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Lissy, Pierre. A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 591-595. doi : 10.1016/j.crma.2012.06.004. http://www.numdam.org/articles/10.1016/j.crma.2012.06.004/

[1] Coron, Jean-Michel; Guerrero, Sergio Singular optimal control: a linear 1-D parabolic–hyperbolic example, Asymptot. Anal., Volume 44 (2005) no. 3–4, pp. 237-257

[2] Ervedoza, Sylvain; Zuazua, Enrique Sharp observability estimates for heat equations, Arch. Ration. Mech. Anal., Volume 202 (2011), pp. 975-1017

[3] Fattorini, Hector O.; Russell, David L. Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 272-292

[4] Fursikov, Andrei V.; Imanuvilov, Oleg Yu. Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996

[5] Glass, Olivier A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit, J. Funct. Anal., Volume 258 (2010) no. 3, pp. 852-868

[6] Guerrero, Sergio; Lebeau, Gilles Singular optimal control for a transport-diffusion equation, Comm. Partial Differential Equations, Volume 32 (2007) no. 10–12, pp. 1813-1836

[7] Lebeau, Gilles; Robbiano, Luc Contrôle exact de lʼéquation de la chaleur, Comm. Partial Differential Equations, Volume 20 (1995) no. 1–2, pp. 335-356

[8] Miller, Luc Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, Volume 204 (2004) no. 1, pp. 202-226

[9] Salem, Ali A numerical study of the null boundary controllability of a convection diffusion equation, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009) no. 15–16, pp. 927-932

[10] Tenenbaum, Gérald; Tucsnak, Marius New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, Volume 243 (2007) no. 1, pp. 70-100

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