Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 262, Issue 8, Pages 4475-4521Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2016.12.021
Keywords
Degenerate parabolic equations; Carleman estimates; Null controllability; Observability; Lipschitz stability; Heisenberg operator
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Funding
- Agence Nationale de la Recherche Blanc (ANR) [ANR-2011-BS01-017-01]
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We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain Omega=(-1,1) Chi T Chi T taking as observation regions slices of the form w = (a, b) Chi T Chi T or tubes w = (a, b) Chi omega(y) Chi T, with -1 < a < b < 1. 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 and the boundary of Omega: T-min >= 1/8 min{(1 + a)(2), (1 - b)(2)}. 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) Chi T Chi R (C)2017 Published by Elsevier Inc.
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