We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain (−1,1)×T×T taking as observation regions slices of the form ω = (a,b) × T × T, with −1 < a < b < 1, or tubes. We prove that observability fails for an arbitrary time T > 0 but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between ω and the boundary. Our proof follows a mixed strategy which combines the approach by Lebeau and Robbiano, which relies on Fourier decomposition, with Carleman inequalities for the heat equations that are solved by the Fourier modes. We extend the analysis to the unbounded domain (−1, 1) × T × R.

Beauchard, K., Cannarsa, P. (2017). Heat equation on the Heisenberg group: Observability and applications. JOURNAL OF DIFFERENTIAL EQUATIONS, 262(8), 4475-4521 [10.1016/j.jde.2016.12.021].

Heat equation on the Heisenberg group: Observability and applications

CANNARSA, PIERMARCO
2017-01-01

Abstract

We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain (−1,1)×T×T taking as observation regions slices of the form ω = (a,b) × T × T, with −1 < a < b < 1, or tubes. We prove that observability fails for an arbitrary time T > 0 but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between ω and the boundary. Our proof follows a mixed strategy which combines the approach by Lebeau and Robbiano, which relies on Fourier decomposition, with Carleman inequalities for the heat equations that are solved by the Fourier modes. We extend the analysis to the unbounded domain (−1, 1) × T × R.
2017
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Degenerate parabolic equations; Carleman estimates; Null controllability; Observability; Lipschitz stability; Heisenberg operator
This research has been performed in the framework of the GDRE CONEDP. The first author was partially supported by the “Agence Nationale de la Recherche” (ANR).
http://www.sciencedirect.com/science/article/pii/S0022039617300013
Beauchard, K., Cannarsa, P. (2017). Heat equation on the Heisenberg group: Observability and applications. JOURNAL OF DIFFERENTIAL EQUATIONS, 262(8), 4475-4521 [10.1016/j.jde.2016.12.021].
Beauchard, K; Cannarsa, P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/181776
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