Partial Differential Equations
On the boundary controllability of non-scalar parabolic systems
Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 763-766.

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability.

Cette Note concerne la contrôlabilité frontière des systèmes paraboliques linéaires non scalaires. Plus précisement, on considère un système de deux équations paraboliques linéaires de dimension 1 en espace. Nous montrons qu'il est beaucoup plus compliqué de contrôler sur une partie du bord que de le faire avec des contrôles distribués. Dans notre résultat principal, on donne des conditions nécessaires et suffisantes pour la contrôlabilité exacte à zéro.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.04.020
Fernández-Cara, Enrique 1; González-Burgos, Manuel 1; de Teresa, Luz 2

1 Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
2 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U. 04510 D.F., Mexico
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Fernández-Cara, Enrique; González-Burgos, Manuel; de Teresa, Luz. On the boundary controllability of non-scalar parabolic systems. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 763-766. doi : 10.1016/j.crma.2009.04.020. http://www.numdam.org/articles/10.1016/j.crma.2009.04.020/

[1] Ammar-Khodja, F.; Benabdallah, A.; Dupaix, C. Null controllability of some reaction–diffusion systems with one control force, J. Math. Anal. Appl., Volume 320 (2006) no. 2, pp. 928-943

[2] F. Ammar-Khodja, A. Benabdallah, C. Dupaix, M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, submitted for publication

[3] F. Ammar-Khodja, A. Benabdallah, C. Dupaix, M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, Journal of Evolution Equations, in press, | HAL

[4] Ammar-Khodja, F.; Benabdallah, A.; Dupaix, C.; Kostine, I. Null controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var., Volume 11 (2005) no. 3, pp. 426-448

[5] Bodart, O.; González-Burgos, M.; Pérez-Garcia, R. Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient, Nonlinear Anal., Volume 57 (2004) no. 5–6, pp. 687-711

[6] Fabre, C.; Puel, J.-P.; Zuazua, E. Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, Volume 125 (1995) no. 1, pp. 31-61

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

[8] Fernández-Cara, E.; Zuazua, E. The cost of approximate controllability for heat equations: the linear case, Adv. Differential Equations, Volume 5 (2000) no. 4–6, pp. 465-514

[9] E. Fernández-Cara, M. González-Burgos, L. de Teresa, Boundary controllability of parabolic coupled equations, submitted for publication

[10] Fursikov, A.; Imanuvilov, O.Yu. Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996

[11] González-Burgos, M.; Pérez-García, R. Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal., Volume 46 (2006) no. 2, pp. 123-162

[12] M. González-Burgos, L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force, preprint

[13] Lebeau, G.; Robbiano, L. Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, Volume 20 (1995) no. 1–2, pp. 335-356

[14] Russell, D.L. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., Volume 20 (1978) no. 4, pp. 639-739

[15] de Teresa, L. Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, Volume 25 (2000) no. 1–2, pp. 39-72

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