Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-22T06:52:52.353Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  21 October 2011

Jérôme Le Rousseau  and
Gilles Lebeau
Jérôme Le Rousseau
Affiliation:
Universitéd’Orléans, Laboratoire Mathématiques et Applications, Physique Mathématique d’Orléans, CNRS UMR 6628, Fédération Denis Poisson, FR CNRS 2964, B.P. 6759, 45067 Orléans Cedex 2, France. jlr@univ-orleans.fr
Gilles Lebeau
Affiliation:
Universitéde Nice Sophia-Antipolis, Laboratoire Jean Dieudonné, UMR CNRS 6621, Parc Valrose, 06108 Nice Cedex 02, France; gilles.lebeau@unice.fr
Get access

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.

Keywords

Carleman estimatessemiclassical analysiselliptic operatorsparabolic operatorscontrollabilityobservability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 712 - 747
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

S. Agmon, Lectures on Elliptic Boundary Values Problems. Van Nostrand (1965).
S. Alinhac and P. Gérard, Opérateurs Pseudo-Différentiels et Théorème de Nash-Moser. Éditions du CNRS (1991).
J.-P. Aubin and I. Ekeland, Applied Non Linear Analysis. John Wiley & Sons, New York (1984).
Barbu, V., Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73–89. Google Scholar
Bellassoued, M., Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization. Asymptotic Anal. 35 (2003) 257–279. Google Scholar
Benabdallah, A. and Naso, M.G., Null controllability of a thermoelastic plate. Abstr. Appl. Anal. 7 (2002) 585–599. Google Scholar
Benabdallah, A., Dermenjian, Y. and Le Rousseau, J., 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. Google Scholar
Benabdallah, A., Dermenjian, Y. and Le Rousseau, J., 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. Google Scholar
Boulakia, M. and Osses, A., Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM Control Optim. Calc. Var. 14 (2008) 1–42. Google Scholar
H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983).
Carleman, T., 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. Google Scholar
de Teresa, L., Insensitizing controls for a semilinear heat equation. Comm. Partial Differential Equations 25 (2000) 39–72. Google Scholar
M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series 268. Cambridge University Press, Cambridge (1999).
Doubova, A., Fernandez-Cara, E., Gonzales-Burgos, M. and Zuazua, E., On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (2002) 798–819. Google Scholar
Doubova, A., Osses, A. and Puel, J.-P., Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM : COCV 8 (2002) 621–661. Google Scholar
Fernández-Cara, E. and Guerrero, S., Global Carleman inequalities for parabolic systems and application to controllability. SIAM J. Control Optim. 45 (2006) 1395–1446. Google Scholar
Fernández-Cara, E. and Zuazua, E., The cost of approximate controllability for heat equations : the linear case. Adv. Differential Equations 5 (2000) 465–514. Google Scholar
Fernández-Cara, E. and Zuazua, E., Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. Henri Poincaré, Analyse non linéaire 17 (2000) 583–616. Google Scholar
Fernández-Cara, E., Guerrero, S., Imanuvilov, O.Yu. and Puel, J.-P., Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. Google Scholar
Fernández-Cara, E., Guerrero, S., Imanuvilov, O.Yu. and Puel, J.-P., 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. Google Scholar
Fabre, C. and Lebeau, G., Prolongement unique des solutions de l’equation de Stokes. Comm. Partial Differential Equations 21 (1996) 573–596. Google Scholar
A. Fursikov and O.Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes 34. Seoul National University, Korea (1996).
González-Burgos, M. and Pérez-García, R., Controllability results for some nonlinear coupled parabolic systems by one control force. Asymptotic Anal. 46 (2006) 123–162. Google Scholar
A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators. Cambridge University Press, Cambridge (1994).
L. Hörmander, Linear Partial Differential Operators. Springer-Verlag, Berlin (1963).
L. Hörmander, The Analysis of Linear Partial Differential Operators IV. Springer-Verlag (1985).
L. Hörmander, The Analysis of Linear Partial Differential Operators III. Springer-Verlag (1985). 2nd printing 1994.
L. Hörmander, The Analysis of Linear Partial Differential Operators I. 2nd edition, Springer-Verlag (1990).
Imanuvilov, O.Yu., Remarks on the exact controllability of Navier-Stokes equations. ESAIM : COCV 6 (2001) 39–72. Google Scholar
Imanuvilov, O.Yu. and Takahashi, T., Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. 87 (2007) 408–437. Google Scholar
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.
Ammar Khodja, F., Benabdallah, A. and Dupaix, C., Null controllability of some reaction-diffusion systems with one control force. J. Math. Anal. Appl. 320 (2006) 928–943. Google Scholar
Ammar Khodja, F., Benabdallah, A., Dupaix, C. and Kostin, I., Null-controllability of some systems of parabolic type by one control force. ESAIM : COCV 11 (2005) 426–448. Google Scholar
Le Rousseau, J., Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. J. Differential Equations 233 (2007) 417–447. Google Scholar
Le Rousseau, J., and Robbiano, L., 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. Google Scholar
Le Rousseau, J. and Robbiano, L., Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces. Invent. Math. 183 (2011) 245–336. Google Scholar
Léautaud, M., Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems. J. Funct. Anal. 258 (2010) 2739–2778. Google Scholar
G. Lebeau, Cours sur les inégalités de Carleman, Mastère Equations aux Dérivées Partielles et Applications. Faculté des Sciences de Tunis, Tunisie (2005).
Lebeau, G. and Robbiano, L., Contrôle exact de l’équation de la chaleur. Comm. Partial Differential Equations 20 (1995) 335–356. Google Scholar
Lebeau, G. and Robbiano, L., Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86 (1997) 465–491. Google Scholar
Lebeau, G. and Zuazua, E., Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal. 141 (1998) 297–329. Google Scholar
A. Martinez, An Introduction to Semiclassical and Microlocal Analysis. Springer-Verlag (2002).
Micu, S. and Zuazua, E., On the lack of null-controllability of the heat equation on the half space. Port. Math. (N.S.) 58 (2001) 1–24. Google Scholar
Miller, L., On the null-controllability of the heat equation in unbounded domains. Bull. Sci. Math. 129 (2005) 175–185. Google Scholar
Miller, L., On the controllability of anomalous diffusions generated by the fractional laplacian. Mathematics of Control, Signals, and Systems 3 (2006) 260–271. Google Scholar
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.
Miller, L., A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1465–1485. Google Scholar
Robbiano, L., Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques. Comm. Partial Differential Equations 16 (1991) 789–800. Google Scholar
Robbiano, L., Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptotic Anal. 10 (1995) 95–115. Google Scholar
D. Robert, Autour de l’Approximation Semi-Classique, Progress in Mathematics 68. Birkhäuser Boston, Boston, MA (1987).
Saut, J.-C. and Scheurer, B., Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118–139. Google Scholar
M.A. Shubin, Pseudodifferential Operators and Spectral Theory. 2nd edition, Springer-Verlag, Berlin Heidelberg (2001).
Tataru, D., Carleman estimates and unique continuation for the Schroedinger equation. Differential Integral Equations 8 (1995) 901–905. Google Scholar
Tataru, D., Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem. Comm. Partial Differential Equations 20 (1995) 855–884. Google Scholar
M.E. Taylor, Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey (1981).
M.E. Taylor, Partial Differential Equations 2 : Qualitative Studies of Linear Equations, Applied Mathematical Sciences 116. Springer-Verlag, New-York (1996).
G. Tenenbaum and M. Tucsnak, On the null-controllability of diffusion equations. preprint (2009).
F. Trèves, Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967).
C. Zuily, Uniqueness and Non Uniqueness in the Cauchy Problem. Birkhäuser, Progress in mathematics (1983).