On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

Jérôme Le Rousseau; Gilles Lebeau

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 712-747
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Le Rousseau, Jérôme, and 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 18.3 (2012): 712-747. <http://eudml.org/doc/272793>.

@article{LeRousseau2012,
abstract = {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.},
author = {Le Rousseau, Jérôme, Lebeau, Gilles},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Carleman estimates; semiclassical analysis; elliptic operators; parabolic operators; controllability; observability; observability inequality},
language = {eng},
number = {3},
pages = {712-747},
publisher = {EDP-Sciences},
title = {On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations},
url = {http://eudml.org/doc/272793},
volume = {18},
year = {2012},
}

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
PB - EDP-Sciences
VL - 18
IS - 3
SP - 712
EP - 747
AB - 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.
LA - eng
KW - Carleman estimates; semiclassical analysis; elliptic operators; parabolic operators; controllability; observability; observability inequality
UR - http://eudml.org/doc/272793
ER -

References

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  1. [1] S. Agmon, Lectures on Elliptic Boundary Values Problems. Van Nostrand (1965). Zbl0142.37401MR178246
  2. [2] S. Alinhac and P. Gérard, Opérateurs Pseudo-Différentiels et Théorème de Nash-Moser. Éditions du CNRS (1991). Zbl0791.47044
  3. [3] J.-P. Aubin and I. Ekeland, Applied Non Linear Analysis. John Wiley & Sons, New York (1984). Zbl0641.47066MR749753
  4. [4] V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Optim.42 (2000) 73–89. Zbl0964.93046MR1751309
  5. [5] M. Bellassoued, Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization. Asymptotic Anal.35 (2003) 257–279. Zbl1137.35388MR2011790
  6. [6] A. Benabdallah and M.G. Naso, Null controllability of a thermoelastic plate. Abstr. Appl. Anal.7 (2002) 585–599. Zbl1013.35008MR1945447
  7. [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. Zbl1189.35349MR2352986
  8. [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. Zbl1115.35055MR2310670
  9. [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. Zbl1149.35068MR2375750
  10. [10] H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983). Zbl0511.46001MR697382
  11. [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. Zbl0022.34201
  12. [12] L. de Teresa, Insensitizing controls for a semilinear heat equation. Comm. Partial Differential Equations25 (2000) 39–72. Zbl0942.35028MR1737542
  13. [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). Zbl0926.35002MR1735654
  14. [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. Zbl1038.93041MR1939871
  15. [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. Zbl1092.93006MR1932966
  16. [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. Zbl1121.35017MR2257228
  17. [17] E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations : the linear case. Adv. Differential Equations5 (2000) 465–514. Zbl1007.93034MR1750109
  18. [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. Zbl0970.93023MR1791879
  19. [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. Zbl1267.93020MR2103189
  20. [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. Zbl1109.93006MR2225301
  21. [21] C. Fabre and G. Lebeau, Prolongement unique des solutions de l’equation de Stokes. Comm. Partial Differential Equations21 (1996) 573–596. Zbl0849.35098MR1387461
  22. [22] A. Fursikov and O.Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes 34. Seoul National University, Korea (1996). Zbl0862.49004MR1406566
  23. [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. Zbl1124.35026MR2205238
  24. [24] A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators. Cambridge University Press, Cambridge (1994). Zbl0804.35001MR1269107
  25. [25] L. Hörmander, Linear Partial Differential Operators. Springer-Verlag, Berlin (1963). Zbl0321.35001
  26. [26] L. Hörmander, The Analysis of Linear Partial Differential Operators IV. Springer-Verlag (1985). Zbl0612.35001
  27. [27] L. Hörmander, The Analysis of Linear Partial Differential Operators III. Springer-Verlag (1985). 2nd printing 1994. Zbl0601.35001
  28. [28] L. Hörmander, The Analysis of Linear Partial Differential Operators I. 2nd edition, Springer-Verlag (1990). Zbl1028.35001
  29. [29] O.Yu. Imanuvilov, Remarks on the exact controllability of Navier-Stokes equations. ESAIM : COCV 6 (2001) 39–72. Zbl0961.35104MR1804497
  30. [30] O.Yu. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system. J. Math. Pures Appl.87 (2007) 408–437. Zbl1124.35056MR2317341
  31. [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. Zbl0946.35055MR1743865
  32. [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. Zbl1157.93004MR2226005
  33. [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. Zbl1125.93005MR2148852
  34. [34] J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. J. Differential Equations233 (2007) 417–447. Zbl1128.35020MR2292514
  35. [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. Zbl1202.35336MR2591978
  36. [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. Zbl1218.35054MR2772083
  37. [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. Zbl1185.35153MR2593342
  38. [38] G. Lebeau, Cours sur les inégalités de Carleman, Mastère Equations aux Dérivées Partielles et Applications. Faculté des Sciences de Tunis, Tunisie (2005). 
  39. [39] G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur. Comm. Partial Differential Equations20 (1995) 335–356. Zbl0819.35071MR1312710
  40. [40] G. Lebeau and L. Robbiano, Stabilisation de l’équation des ondes par le bord. Duke Math. J.86 (1997) 465–491. Zbl0884.58093MR1432305
  41. [41] G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal.141 (1998) 297–329. Zbl1064.93501MR1620510
  42. [42] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis. Springer-Verlag (2002). Zbl0994.35003MR1872698
  43. [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. Zbl0991.35010MR1820835
  44. [44] L. Miller, On the null-controllability of the heat equation in unbounded domains. Bull. Sci. Math.129 (2005) 175–185. Zbl1079.35018MR2123266
  45. [45] L. Miller, On the controllability of anomalous diffusions generated by the fractional laplacian. Mathematics of Control, Signals, and Systems 3 (2006) 260–271. Zbl1105.93015MR2272076
  46. [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. [47] L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete Contin. Dyn. Syst. Ser. B14 (2010) 1465–1485. Zbl1219.93017MR2679651
  48. [48] L. Robbiano, Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques. Comm. Partial Differential Equations16 (1991) 789–800. Zbl0735.35086
  49. [49] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptotic Anal.10 (1995) 95–115. Zbl0882.35015MR1324385
  50. [50] D. Robert, Autour de l’Approximation Semi-Classique, Progress in Mathematics 68. Birkhäuser Boston, Boston, MA (1987). Zbl0621.35001MR897108
  51. [51] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differential Equations66 (1987) 118–139. Zbl0631.35044MR871574
  52. [52] M.A. Shubin, Pseudodifferential Operators and Spectral Theory. 2nd edition, Springer-Verlag, Berlin Heidelberg (2001). Zbl0616.47040MR1852334
  53. [53] D. Tataru, Carleman estimates and unique continuation for the Schroedinger equation. Differential Integral Equations8 (1995) 901–905. Zbl0828.35021MR1306599
  54. [54] D. Tataru, Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem. Comm. Partial Differential Equations20 (1995) 855–884. Zbl0846.35021MR1326909
  55. [55] M.E. Taylor, Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey (1981). Zbl0453.47026MR618463
  56. [56] M.E. Taylor, Partial Differential Equations 2 : Qualitative Studies of Linear Equations, Applied Mathematical Sciences 116. Springer-Verlag, New-York (1996). Zbl0869.35003MR1395149
  57. [57] G. Tenenbaum and M. Tucsnak, On the null-controllability of diffusion equations. preprint (2009). Zbl1236.93025MR2859866
  58. [58] F. Trèves, Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967). Zbl0171.10402MR225131
  59. [59] C. Zuily, Uniqueness and Non Uniqueness in the Cauchy Problem. Birkhäuser, Progress in mathematics (1983). Zbl0521.35003MR701544

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