Numerical null controllability of semi-linear 1-D heat equations:   Fixed point, least squares and Newton methods

Enrique Fernández-Cara; Arnaud Münch

doi:10.3934/mcrf.2012.2.217

Article Contents

2012, Volume 2, Issue 3: 217-246. Doi: 10.3934/mcrf.2012.2.217

This issue Previous Article Next Article Controllability of the cubic Schroedinger equation via a low-dimensional source term

Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods

Enrique Fernández-Cara^1, and
Arnaud Münch^2,

1.
Dpto. EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla
2.
Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand), UMR CNRS 6620, Campus des Cézeaux, 63177 Aubière

Received: November 2011

Revised: May 2012

Published: September 2012

Abstract Related Papers Cited by

Abstract

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.

Keywords:

One-dimensional semi-linear heat equation,
null controllability,
blow up,
numerical solution,
least squares method.

Mathematics Subject Classification: Primary: 35L05, 49J05; Secondary: 65K10.

Citation:

Related Papers

Cited by

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.