Abstract
We prove Carleman estimates for elliptic and parabolic operators, using several methods: a microlocal approach where the main tool is the Gårding inequality and a more computational direct approach. Carleman estimates are proven locally and we describe how they can be patched together to form a global estimate. We expose how they can be used to provide unique continuation properties, as well as approximate and null controllability results.
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Notes
- 1.
In the sense that only constants are affected. In Theorem 4.3.4 below the constants C and τ1 change but not the form of the estimate.
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Acknowledgements
The author was partially supported by l’Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01. The author is grateful to Assia Benabdallah, Gilles Lebeau, Nicolas Lerner and Luc Robbiano for manifold fruitful discussions on Carleman estimates. The authors wishes to deeply thank the organizers of the CIME school in July 2010, Piermaco Cannarsa and Jean-Michel Coron, for giving him the opportunity to present the material of these course notes. The authors also expresses his appreciation to the great work of the CIME staff that made his stay in Cetraro most enjoyable.
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Le Rousseau, J. (2012). Carleman Estimates and Some Applications to Control Theory. In: Control of Partial Differential Equations. Lecture Notes in Mathematics(), vol 2048. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27893-8_4
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DOI: https://doi.org/10.1007/978-3-642-27893-8_4
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