Comptes Rendus
Glowinski and numerical control problems
[Glowinski et le contrôle numérique]
Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 411-430.

Dans cet article, on rappelle quelques contributions sur contrôle numérique des EDPs issues du travail de Roland Glowinski. On considérera des problèmes de contrôlabilité nulle pour des équations de la chaleur linéaires et non linéaires et aussi pour des systèmes à frontière libre. Nous regarderons aussi quelques problèmes de contrôle optimal bi-objectif. En outre, quelques méthodes et résultats nouveaux seront annoncés.

This paper is devoted to recall several contributions to the numerical control of PDE’s that have origin in Glowinski’s work. I will consider null controllability problems for linear and nonlinear heat equations and some free-boundary systems. We will also deal with some bi-objective optimal control problems. Additionally, some new methods and results will be announced.

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Révisé le :
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DOI : 10.5802/crmeca.177
Classification : 35B37, 35K55, 35Q30, 93C20
Keywords: Controllability of linear and nonlinear PDE’s, control of free-boundary problems, bi-objective optimal control problems, Nash equilibria, numerical methods
Mot clés : Contrôlabilité des EDPs linéaires et non linéaires, contrôle de problèmes de frontières libre, Problèmes de contrôle bi-objectif, Equilibria de Nash, Méthodes numériques

Enrique Fernández-Cara 1

1 University of Sevilla, Dep. EDAN and IMUS, Univ. of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Enrique Fernández-Cara. Glowinski and numerical control problems. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 411-430. doi : 10.5802/crmeca.177. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.177/

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[2] Enrique Fernández-Cara Sur l’existence de solutions d’un problème d’évolution apparaissant en physique des semi-conducteurs (1981) no. RR-0079 (https://hal.inria.fr/inria-00076482/file/RR-0079.pdf) (Technical report)

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[5] Enrique Fernández-Cara Glowinski and splitting (to appear)

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[11] Enrique Fernández-Cara; Arnaud Münch Strong convergent approximations of null controls for the 1D heat equation, Se MA J., Volume 61 (2013) no. 1, pp. 49-78 | DOI | MR | Zbl

[12] Andreĭ V. Fursikov; Oleg Yu. Imanuvilov Controllability of evolution equations, Lecture Notes Series, Seoul, 34, Seoul National University, Global Analysis Research Center, 1996 | Zbl

[13] Enrique Fernández-Cara; Arnaud Münch; Diego A. Souza On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations, J. Sci. Comput., Volume 70 (2017) no. 2, pp. 819-858 | DOI | MR | Zbl

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[16] Enrique Fernández-Cara; Arnaud Münch Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods, Math. Control Relat. Fields, Volume 2 (2012) no. 3, pp. 217-246 | DOI | MR | Zbl

[17] Enrique Fernández-Cara; Juan Límaco; Irene Marín-Gayte Theoretical and numerical local null controllability of a quasi-linear parabolic equation in dimensions 2 and 3, J. Franklin Inst., Volume 358 (2021) no. 5, pp. 2846-2871 | DOI | MR | Zbl

[18] Enrique Fernández-Cara; Enrique Zuazua Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 17 (2000) no. 5, pp. 583-616 | DOI | Numdam | MR | Zbl

[19] Sylvain Ervedoza; Jérôme Lemoine; Arnaud Münch Exact controllability of semilinear heat equations through a constructive approach, Evol. Equ. Control Theory, Volume 12 (2023) no. 2, pp. 567-599 | DOI | MR

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[21] Raul K. C. Araújo; Enrique Fernández-Cara; Juan Límaco; Diego A. Souza Remarks on the control of two-phase Stefan free-boundary problems, SIAM J. Control Optim., Volume 60 (2022) no. 5, pp. 3078-3099 | DOI | MR | Zbl

[22] Enrique Fernández-Cara; Enrique Zuazua On the theoretical and numerical control of two-phase Stefan free-boundary problems (to appear)

[23] Angel M. Ramos; Roland Glowinski; Jacques F. Periaux Nash Equilibria for the multiobjective control of linear partial differential equations, J. Optim. Theory Appl., Volume 112 (2002) no. 3, pp. 457-498 | DOI | MR

[24] Angel M. Ramos; Roland Glowinski; Jacques F. Periaux Pointwise control of the burgers equation and related nash equilibrium problems: computational approach, J. Optim. Theory Appl., Volume 112 (2002) no. 3, pp. 499-516 | DOI | MR | Zbl

[25] J. I. Diaz; J.-L. Lions On the approximate controllability of Stackelberg-Nash strategies, Ocean circulation and pollution control - A mathematical and numerical investigation (Madrid, 1997), Springer, 2004, pp. 17-27 | DOI

[26] Enrique Fernández-Cara; Irene Marín-Gayte Bi-objective optimal control of some PDEs: Nash equilibria and quasi-equilibria, ESAIM, Control Optim. Calc. Var., Volume 27 (2021), 50, 30 pages | MR | Zbl

[27] Enrique Fernández-Cara; Irene Marín-Gayte Theoretical and numerical results for some bi-objective optimal control problems, Commun. Pure Appl. Anal., Volume 19 (2020) no. 4, pp. 2101-2126 | DOI | MR | Zbl

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