Carleman inequalities for the two-dimensional heat equation in singular domains

Abdelhakim Belghazi; Ferroudja Smadhi; Nawel Zaidi; Ouahiba Zair

doi:10.3934/mcrf.2012.2.331

Article Contents

2012, Volume 2, Issue 4: 331-359. Doi: 10.3934/mcrf.2012.2.331

This issue Previous Article Next Article Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain

Carleman inequalities for the two-dimensional heat equation in singular domains

1.
Faculté de Mathématiques, laboratoire AMNEDP, U.S.T.H.B, B.P. 32, El-Alia, 16111 ALGER

Received: December 31, 2011

Revised: April 30, 2012

Published: November 2012

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Abstract

We consider the Cauchy problem associated to the heat equation firstly in a plane domain with a reentrant corner, then in a cracked domain. By constructing a weight function, we prove a Carleman inequality and we deduce a result of controllability.

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Mathematics Subject Classification: Primary: 95B05, 35J30, 35B65, 35B25, 35J25.

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References

[1]	A-H. Belghazi, Null controllability of three-dimensional heat equation for singular domains and stratified media spectral inequality, in preparation.
[2]	A. H. Belghazi, F. Smadhi, N. Zaidi and O. Zair, Carleman inequalities for the heat equation in singular domains, C. R. Acad. Sci. Paris, 348 (2010), 277-282.doi: 10.1016/j.crma.2010.02.007.
[3]	A. Benabdallah, Y. Dermenjian and J. Le Rousseau, On the controllability of linear parabolic equations with an arbitrary control location for stratified media, C. R. Acad. Sci. Paris, 344 (2007), 357-362.
[4]	R. Bey, J. P Lohéac and M. Moussaoui, Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilization of the wave equation, J. Math. Pures Appl., 78 (1999), 1043-1067.
[5]	L. Bourgeois, A stability estimate for ill-posed elliptic Cauchy problems in a domain with corners, C. R. Acad. Sci. Paris, 345 (2007).
[6]	H. Brézis, "Analyse Fonctionnelle, Théorie et Applications," Masson, 1983.
[7]	N. Burq, Contrôle de l'équation des ondes dans des ouverts contenants des coins, Bull. Soc. math. Franc, 126 (1998), 601-637.
[8]	T. Carleman, Sur un problème d'unicit'e pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys., 26 (1939), 1-9.
[9]	A. Doubova, A. Osses and J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim. Calc. Var., 8 (2002), 621-661.
[10]	H-O. Fattorini and D-L.Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.
[11]	H-O. Fattorini and D-L.Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.
[12]	E. Fernandez-Cara and S. Guerrero, Global Carleman Inequalities For Parabolic Systems And application To Controllability, SIAM J. Control Optim, 45 (2006), 1395-1446.doi: 10.1137/S0363012904439696.
[13]	A. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
[14]	P. Grisvard, Contrôlabilité exacte des solutions de l'équation des ondes en présence de singularités, Journal Maths. Pures et Appl., 68 (1989), 215-259.
[15]	P. Grisvard, "Singularities in Boundary Value Problems," Springer-Verlag, 1992.
[16]	P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Pitman, Boston, London, Melbourne, 1985.
[17]	A. Heibig and M. A. Moussaoui, Exact controllability of the wave equation for domains with slits and for mixed boundary conditions, Discrete and continuous dynamical systems, 2 (1996), 367-386.
[18]	L. Hörmander, "Linear Partial Differential Operators," Springer-verlag, 1963.
[19]	J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, J. Differential Equations, 233 (2007), 417-447.doi: 10.1016/j.jde.2006.10.005.
[20]	G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential. Equations, 20 (1995), 335-356.
[21]	J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, to appear in ESAIM Control Optim. Calc. Var.
[22]	M. A. Moussaoui and B. K. Saadallah, Régularité des coefficients de propagation de singularités de l'équation de la chaleur dans un domaine polygonal plan, C. R. Acad. Sci. Paris, 293 (1981), 297-300.
[23]	S. Nicaise, Exact controllability of a pluridimensional coupled problem, Rev. Math. Univ. Complut. Madrid, 5 (1992), 91-135.
[24]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983.
[25]	J. Zabczyk, "Mathematical Control Theory: An Introduction," Birkhäuser, 1995.