On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 712-747.

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.

DOI : https://doi.org/10.1051/cocv/2011168
Classification : 35B60,  35J15,  35K05,  93B05,  93B07
Mots clés : Carleman estimates, semiclassical analysis, elliptic operators, parabolic operators, controllability, observability
@article{COCV_2012__18_3_712_0,
author = {Le Rousseau, J\'er\^ome and Lebeau, Gilles},
title = {On {Carleman} estimates for elliptic and parabolic operators. {Applications} to unique continuation and control of parabolic equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {712--747},
publisher = {EDP-Sciences},
volume = {18},
number = {3},
year = {2012},
doi = {10.1051/cocv/2011168},
zbl = {1262.35206},
mrnumber = {3041662},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2011168/}
}
TY  - JOUR
AU  - Le Rousseau, Jérôme
AU  - Lebeau, Gilles
TI  - On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
DA  - 2012///
SP  - 712
EP  - 747
VL  - 18
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2011168/
UR  - https://zbmath.org/?q=an%3A1262.35206
UR  - https://www.ams.org/mathscinet-getitem?mr=3041662
UR  - https://doi.org/10.1051/cocv/2011168
DO  - 10.1051/cocv/2011168
LA  - en
ID  - COCV_2012__18_3_712_0
ER  - 
Le Rousseau, Jérôme; Lebeau, Gilles. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 712-747. doi : 10.1051/cocv/2011168. http://www.numdam.org/articles/10.1051/cocv/2011168/

[1] S. Agmon, Lectures on Elliptic Boundary Values Problems. Van Nostrand (1965). | MR 178246 | Zbl 0142.37401

[2] S. Alinhac and P. Gérard, Opérateurs Pseudo-Différentiels et Théorème de Nash-Moser. Éditions du CNRS (1991). | Zbl 0791.47044

[3] J.-P. Aubin and I. Ekeland, Applied Non Linear Analysis. John Wiley & Sons, New York (1984). | MR 749753 | Zbl 0641.47066

[4] V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73-89. | MR 1751309 | Zbl 0964.93046

[5] M. Bellassoued, Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization. Asymptotic Anal. 35 (2003) 257-279. | MR 2011790 | Zbl 1137.35388

[6] A. Benabdallah and M.G. Naso, Null controllability of a thermoelastic plate. Abstr. Appl. Anal. 7 (2002) 585-599. | MR 1945447 | Zbl 1013.35008

[7] A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem. J. Math. Anal. Appl. 336 (2007) 865-887. | MR 2352986 | Zbl 1189.35349

[8] 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, Ser. I 344 (2007) 357-362. | MR 2310670 | Zbl 1115.35055

[9] M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM Control Optim. Calc. Var. 14 (2008) 1-42. | Numdam | MR 2375750 | Zbl 1149.35068

[10] H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983). | MR 697382 | Zbl 0511.46001

[11] T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astr. Fys. 26B (1939) 1-9. | Zbl 0022.34201

[12] L. De Teresa, Insensitizing controls for a semilinear heat equation. Comm. Partial Differential Equations 25 (2000) 39-72. | MR 1737542 | Zbl 0942.35028

[13] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series 268. Cambridge University Press, Cambridge (1999). | MR 1735654 | Zbl 0926.35002

[14] A. Doubova, E. Fernandez-Cara, M. Gonzales-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (2002) 798-819. | MR 1939871 | Zbl 1038.93041

[15] A. Doubova, A. Osses and J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM : COCV 8 (2002) 621-661. | Numdam | MR 1932966 | Zbl 1092.93006

[16] E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and application to controllability. SIAM J. Control Optim. 45 (2006) 1395-1446. | MR 2257228 | Zbl 1121.35017

[17] E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations : the linear case. Adv. Differential Equations 5 (2000) 465-514. | MR 1750109 | Zbl 1007.93034

[18] 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. | Numdam | MR 1791879 | Zbl 0970.93023

[19] E. Fernández-Cara, S. Guerrero, O.Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | MR 2103189 | Zbl 1267.93020

[20] E. Fernández-Cara, S. Guerrero, O.Yu. Imanuvilov and J.-P. Puel, Some controllability results for the N-dimensional Navier-Stokes and Boussinesq systems with N − 1 scalar controls. SIAM J. Control Optim. 45 (2006) 146-173. | MR 2225301 | Zbl 1109.93006

[21] C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes. Comm. Partial Differential Equations 21 (1996) 573-596. | MR 1387461 | Zbl 0849.35098

[22] A. Fursikov and O.Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes 34. Seoul National University, Korea (1996). | MR 1406566 | Zbl 0862.49004

[23] M. González-Burgos and R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force. Asymptotic Anal. 46 (2006) 123-162. | MR 2205238 | Zbl 1124.35026

[24] A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators. Cambridge University Press, Cambridge (1994). | MR 1269107 | Zbl 0804.35001

[25] L. Hörmander, Linear Partial Differential Operators. Springer-Verlag, Berlin (1963). | Zbl 0321.35001

[26] L. Hörmander, The Analysis of Linear Partial Differential Operators IV. Springer-Verlag (1985). | Zbl 0612.35001

[27] L. Hörmander, The Analysis of Linear Partial Differential Operators III. Springer-Verlag (1985). 2nd printing 1994. | Zbl 0601.35001

[28] L. Hörmander, The Analysis of Linear Partial Differential Operators I. 2nd edition, Springer-Verlag (1990). | Zbl 1028.35001

[29] O.Yu. Imanuvilov, Remarks on the exact controllability of Navier-Stokes equations. ESAIM : COCV 6 (2001) 39-72. | Numdam | MR 1804497 | Zbl 0961.35104

[30] O.Yu. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. 87 (2007) 408-437. | MR 2317341 | Zbl 1124.35056

[31] D. Jerison and G. Lebeau, Harmonic analysis and partial differential equations (Chicago, IL, 1996). chapter Nodal sets of sums of eigenfunctions, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago (1999) 223-239. | MR 1743865 | Zbl 0946.35055

[32] F. Ammar Khodja, A. Benabdallah and C. Dupaix, Null controllability of some reaction-diffusion systems with one control force. J. Math. Anal. Appl. 320 (2006) 928-943. | MR 2226005 | Zbl 1157.93004

[33] F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Null-controllability of some systems of parabolic type by one control force. ESAIM : COCV 11 (2005) 426-448. | Numdam | MR 2148852 | Zbl 1125.93005

[34] J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. J. Differential Equations 233 (2007) 417-447. | MR 2292514 | Zbl 1128.35020

[35] J. Le Rousseau, and L. Robbiano, Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations. Arch. Rational Mech. Anal. 105 (2010) 953-990. | MR 2591978 | Zbl 1202.35336

[36] J. Le Rousseau and L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces. Invent. Math. 183 (2011) 245-336. | MR 2772083 | Zbl 1218.35054

[37] M. Léautaud, Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems. J. Funct. Anal. 258 (2010) 2739-2778. | MR 2593342 | Zbl 1185.35153

[38] G. Lebeau, Cours sur les inégalités de Carleman, Mastère Equations aux Dérivées Partielles et Applications. Faculté des Sciences de Tunis, Tunisie (2005).

[39] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur. Comm. Partial Differential Equations 20 (1995) 335-356. | MR 1312710 | Zbl 0819.35071

[40] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997) 465-491. | MR 1432305 | Zbl 0884.58093

[41] G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal. 141 (1998) 297-329. | MR 1620510 | Zbl 1064.93501

[42] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis. Springer-Verlag (2002). | MR 1872698 | Zbl 0994.35003

[43] S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half space. Port. Math. (N.S.) 58 (2001) 1-24. | MR 1820835 | Zbl 0991.35010

[44] L. Miller, On the null-controllability of the heat equation in unbounded domains. Bull. Sci. Math. 129 (2005) 175-185. | MR 2123266 | Zbl 1079.35018

[45] L. Miller, On the controllability of anomalous diffusions generated by the fractional laplacian. Mathematics of Control, Signals, and Systems 3 (2006) 260-271. | MR 2272076 | Zbl 1105.93015

[46] L. Miller, Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones. Preprint (2008). http://hal.archives-ouvertes.fr/hal-00411840/fr.

[47] L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1465-1485. | MR 2679651 | Zbl 1219.93017

[48] L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques. Comm. Partial Differential Equations 16 (1991) 789-800. | Zbl 0735.35086

[49] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptotic Anal. 10 (1995) 95-115. | MR 1324385 | Zbl 0882.35015

[50] D. Robert, Autour de l'Approximation Semi-Classique, Progress in Mathematics 68. Birkhäuser Boston, Boston, MA (1987). | MR 897108 | Zbl 0621.35001

[51] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118-139. | MR 871574 | Zbl 0631.35044

[52] M.A. Shubin, Pseudodifferential Operators and Spectral Theory. 2nd edition, Springer-Verlag, Berlin Heidelberg (2001). | MR 1852334 | Zbl 0616.47040

[53] D. Tataru, Carleman estimates and unique continuation for the Schroedinger equation. Differential Integral Equations 8 (1995) 901-905. | MR 1306599 | Zbl 0828.35021

[54] D. Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem. Comm. Partial Differential Equations 20 (1995) 855-884. | MR 1326909 | Zbl 0846.35021

[55] M.E. Taylor, Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey (1981). | MR 618463 | Zbl 0453.47026

[56] M.E. Taylor, Partial Differential Equations 2 : Qualitative Studies of Linear Equations, Applied Mathematical Sciences 116. Springer-Verlag, New-York (1996). | MR 1395149 | Zbl 0869.35003

[57] G. Tenenbaum and M. Tucsnak, On the null-controllability of diffusion equations. preprint (2009). | MR 2859866 | Zbl 1236.93025

[58] F. Trèves, Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967). | MR 225131 | Zbl 0171.10402

[59] C. Zuily, Uniqueness and Non Uniqueness in the Cauchy Problem. Birkhäuser, Progress in mathematics (1983). | MR 701544 | Zbl 0521.35003

Cité par Sources :