Partial differential equations/Optimal control
Internal null-controllability of the N-dimensional heat equation in cylindrical domains
Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 925-930.

In this Note, we consider the internal null-controllability of the N-dimensional heat equation on domains of the form Ω=Ω1×Ω2, with Ω1=(0,1) and Ω2 a smooth domain of RN1,N>1. When the control is exerted on γ={x0}×ω2Ω, with x0 an algebraic real number of order d>1 and ω2Ω2 a non-empty open subset, we show the null-controllability, for all time T>0. This result is obtained through the Lebeau–Robbiano strategy and requires an upper bound of the cost of the one-dimensional null-control.

Dans cette note, on considère la contrôlabilité à zéro interne de l'équation de la chaleur N-dimensionnelle, sur des domaines de la forme Ω=Ω1×Ω2, avec Ω1=(0,1) et Ω2 un domaine borné et régulier de RN1,N>1. Lorsque le contrôle est exercé sur γ={x0}×ω2Ω, avec x0 un nombre réel algébrique de degré d>1 et ω2Ω2 un ouvert non vide, on montre la contrôlabilité à zéro, en tout temps T>0. Ce résultat s'appuie sur la stratégie de Lebeau–Robbiano et exige une estimation du coût du contrôle monodimensionnel.

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Published online:
DOI: 10.1016/j.crma.2015.04.021
Samb, El Hadji 1

1 Laboratoire I2M, Institut de mathématiques de Marseille, UMR 7373, 13453 Marseille, France
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Samb, El Hadji. Internal null-controllability of the N-dimensional heat equation in cylindrical domains. Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 925-930. doi : 10.1016/j.crma.2015.04.021. http://www.numdam.org/articles/10.1016/j.crma.2015.04.021/

[1] Benabdallah, A.; Boyer, F.; González-Burgos, M.; Olive, G. Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the N-dimensional boundary null-controllability in cylindrical domains, SIAM J. Control Optim., Volume 52 (2014) no. 5, pp. 2970-3001

[2] Dolecki, S. Observability for the one-dimensional heat equation, Stud. Math., Volume 48 (1973), pp. 291-305

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

[4] Lebeau, G.; Robbiano, L. Contrôle exact de l'équation de la chaleur, Commun. Partial Differ. Equ., Volume 20 (1995) no. 1–2, pp. 335-356

[5] Luxemburg, W.A.J.; Korevaar, J. Entire functions and Müntz–Szasz type approximation, Trans. Amer. Math. Soc., Volume 157 (1971), pp. 23-37

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

[7] Valiron, G. Théorie générale des séries de Dirichlet, Mém. Math., Volume 17 (1926), p. 3

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