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Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods
1. | Dpto. EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain |
2. | Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand), UMR CNRS 6620, Campus des Cézeaux, 63177 Aubière, France |
References:
[1] |
V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim., 42 (2000), 73-89. |
[2] |
F. Ben Belgacem and S. M. Kaber, On the Dirichlet boundary controllability of the one-dimensional heat equation: Semi-analytical calculations and ill-posedness degree, Inverse Problems, 27 (2011), 055012, 19 pp.
doi: 10.1088/0266-5611/27/5/055012. |
[3] |
F. Boyer, F. Hubert and J. Le Rousseau, Uniform controllability properties for space/time-discretized parabolic equations, Numerische Mathematik, 118 (2011), 601-661.
doi: 10.1007/s00211-011-0368-1. |
[4] |
C. Carthel, R. Glowinski and J.-L. Lions, On exact and approximate boundary controllabilities for the heat equation: A numerical approach, J. Optimization, Theory and Applications, 82 (1994), 429-484. |
[5] |
T. Cazenave and A. Haraux, "Introduction aux Problèmes d'Évolution Semi-Linéaires," Mathématiques & Applications, 1, Ellipses, Paris, 1990. |
[6] |
I. Charpentier and Y. Maday, Identifications numériques de contrôles distribués pour l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 779-784. |
[7] |
J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, AMS, Providence, RI, 2007. |
[8] |
J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim., 43 (2004), 549-569.
doi: 10.1137/S036301290342471X. |
[9] |
C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 31-61. |
[10] |
E. Fernández-Cara, Null controllability of the semilinear heat equation, ESAIM Control Optim. Calc. Var., 2 (1997), 87-103.
doi: 10.1051/cocv:1997104. |
[11] |
E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446. |
[12] |
E. Fernández-Cara and A. Münch, Numerical null controllability of the 1D heat equation: Primal algorithms, preprint, 2010. Available from: http://hal.archives-ouvertes.fr/hal-00687884. |
[13] |
E. Fernández-Cara and A. Münch, Numerical null controllability of the 1D heat equation: Dual algorithms, preprint, 2010. Avalable from: http://hal.archives-ouvertes.fr/hal-00687887. |
[14] |
E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 17 (2000), 583-616. |
[15] |
X. Fu, J. Yong and X. Zhang, Exact controllability for the multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), 1578-1614.
doi: 10.1137/040610222. |
[16] |
S. Ervedoza and J. Valein, On the observability of abstract time-discrete linear parabolic equations, Rev. Mat. Complut., 23 (2010), 163-190.
doi: 10.1007/s13163-009-0014-y. |
[17] |
A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34, Seoul National University, Research Institute of Mathemtics, Global Analysis Research Center, Seoul, (1996). |
[18] |
R. Glowinski, J. He and J.-L. Lions, "Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach," Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008. |
[19] |
F. Hecht, A. Le Hyaric, J. Morice, K. Ohtsuka and O. Pironneau, FreeFem++, Third edition, Version 3.12., Available from: , ().
|
[20] |
O. Yu. Imanuvilov, Controllability of parabolic equations, (Russian), Mat. Sb., 186 (1995), 109-132; translation in Sb. Math., 186 (1995), 879-900. |
[21] |
S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems, Systems Control Lett., 55 (2006), 597-609. |
[22] |
A. Lopez and E. Zuazua, Some new results to the null controllability of the 1-d heat equation, in "Séminaire sur les Équations aux Dérivées Partielles," 1997-1998, Exp. No. VIII, École Polytech., Palaiseau, (1998), 22 pp. |
[23] |
I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with applications to waves and plates boundary control, Appl. Math. Optim., 23 (1991), 109-154.
doi: 10.1007/BF01442394. |
[24] |
G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.
doi: 10.1080/03605309508821097. |
[25] |
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Contrôlabilité exacte, Recherches en Mathématiques Appliquées, 8, Masson, Paris 1988. |
[26] |
A. Münch and E. Zuazua, Numerical approximation of null controls for the heat equation: Ill-posedness and remedies, Inverse Problems, 26 (2010), 085018, 39 pp. |
[27] |
D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl. Math., 52 (1973), 189-211. |
[28] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[29] |
X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, "Proceedings of the International Congress of Mathematicians," Vol. IV, Hindustan Book Agency, New Delhi, (2010), 3008-3034. |
[30] |
E. Zuazua, Exact boundary controllability for the semilinear wave equation, in "Nonlinear Partial Differential Equations and Their Applications,'' Vol. X (eds. H. Brezis and J.-L. Lions), Pitman, New York, 1991. |
show all references
References:
[1] |
V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim., 42 (2000), 73-89. |
[2] |
F. Ben Belgacem and S. M. Kaber, On the Dirichlet boundary controllability of the one-dimensional heat equation: Semi-analytical calculations and ill-posedness degree, Inverse Problems, 27 (2011), 055012, 19 pp.
doi: 10.1088/0266-5611/27/5/055012. |
[3] |
F. Boyer, F. Hubert and J. Le Rousseau, Uniform controllability properties for space/time-discretized parabolic equations, Numerische Mathematik, 118 (2011), 601-661.
doi: 10.1007/s00211-011-0368-1. |
[4] |
C. Carthel, R. Glowinski and J.-L. Lions, On exact and approximate boundary controllabilities for the heat equation: A numerical approach, J. Optimization, Theory and Applications, 82 (1994), 429-484. |
[5] |
T. Cazenave and A. Haraux, "Introduction aux Problèmes d'Évolution Semi-Linéaires," Mathématiques & Applications, 1, Ellipses, Paris, 1990. |
[6] |
I. Charpentier and Y. Maday, Identifications numériques de contrôles distribués pour l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 779-784. |
[7] |
J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, AMS, Providence, RI, 2007. |
[8] |
J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim., 43 (2004), 549-569.
doi: 10.1137/S036301290342471X. |
[9] |
C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 31-61. |
[10] |
E. Fernández-Cara, Null controllability of the semilinear heat equation, ESAIM Control Optim. Calc. Var., 2 (1997), 87-103.
doi: 10.1051/cocv:1997104. |
[11] |
E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446. |
[12] |
E. Fernández-Cara and A. Münch, Numerical null controllability of the 1D heat equation: Primal algorithms, preprint, 2010. Available from: http://hal.archives-ouvertes.fr/hal-00687884. |
[13] |
E. Fernández-Cara and A. Münch, Numerical null controllability of the 1D heat equation: Dual algorithms, preprint, 2010. Avalable from: http://hal.archives-ouvertes.fr/hal-00687887. |
[14] |
E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 17 (2000), 583-616. |
[15] |
X. Fu, J. Yong and X. Zhang, Exact controllability for the multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), 1578-1614.
doi: 10.1137/040610222. |
[16] |
S. Ervedoza and J. Valein, On the observability of abstract time-discrete linear parabolic equations, Rev. Mat. Complut., 23 (2010), 163-190.
doi: 10.1007/s13163-009-0014-y. |
[17] |
A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34, Seoul National University, Research Institute of Mathemtics, Global Analysis Research Center, Seoul, (1996). |
[18] |
R. Glowinski, J. He and J.-L. Lions, "Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach," Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008. |
[19] |
F. Hecht, A. Le Hyaric, J. Morice, K. Ohtsuka and O. Pironneau, FreeFem++, Third edition, Version 3.12., Available from: , ().
|
[20] |
O. Yu. Imanuvilov, Controllability of parabolic equations, (Russian), Mat. Sb., 186 (1995), 109-132; translation in Sb. Math., 186 (1995), 879-900. |
[21] |
S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems, Systems Control Lett., 55 (2006), 597-609. |
[22] |
A. Lopez and E. Zuazua, Some new results to the null controllability of the 1-d heat equation, in "Séminaire sur les Équations aux Dérivées Partielles," 1997-1998, Exp. No. VIII, École Polytech., Palaiseau, (1998), 22 pp. |
[23] |
I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with applications to waves and plates boundary control, Appl. Math. Optim., 23 (1991), 109-154.
doi: 10.1007/BF01442394. |
[24] |
G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.
doi: 10.1080/03605309508821097. |
[25] |
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Contrôlabilité exacte, Recherches en Mathématiques Appliquées, 8, Masson, Paris 1988. |
[26] |
A. Münch and E. Zuazua, Numerical approximation of null controls for the heat equation: Ill-posedness and remedies, Inverse Problems, 26 (2010), 085018, 39 pp. |
[27] |
D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl. Math., 52 (1973), 189-211. |
[28] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[29] |
X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, "Proceedings of the International Congress of Mathematicians," Vol. IV, Hindustan Book Agency, New Delhi, (2010), 3008-3034. |
[30] |
E. Zuazua, Exact boundary controllability for the semilinear wave equation, in "Nonlinear Partial Differential Equations and Their Applications,'' Vol. X (eds. H. Brezis and J.-L. Lions), Pitman, New York, 1991. |
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