# American Institute of Mathematical Sciences

September  2012, 2(3): 217-246. doi: 10.3934/mcrf.2012.2.217

## 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

Received  November 2011 Revised  May 2012 Published  August 2012

This paper deals with the numerical computation of distributed null controls for semi-linear 1D heat equations, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [Fernandez-Cara $\&$ Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] via a fixed point reformulation; see also [Barbu, Exact controllability of the superlinear heat equation, 2000]. More precisely, Carleman estimates and Kakutani's Theorem together ensure the existence of solutions to fixed points for an equivalent fixed point reformulated problem. A nontrivial difficulty appears when we want to extract from the associated Picard iterates a convergent (sub)sequence. In this paper, we introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points. We also formulate and apply a Newton-Raphson algorithm in this context. Several numerical experiments that make it possible to test and compare these methods are performed.
Citation: Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control and Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217
##### 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: http://www.freefem.org/ff++. [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: http://www.freefem.org/ff++. [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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