Null controllability of Kolmogorov-type equations

We study the null controllability of Kolmogorov-type equations in a rectangle , under an additive control supported in an open subset of . For , with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support . This improves the previou...

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Bibliographic Details
Published in:Mathematics of control, signals, and systems signals, and systems, 2014-03, Vol.26 (1), p.145-176
Main Author: Beauchard, K.
Format: Article
Language:English
Subjects:
Analysis of PDEs
Boundary conditions
Cauchy problems
Communications Engineering
Control
Control systems
Controllability
Dirichlet problem
Fourier analysis
Horizontal
Mathematical analysis
Mathematics
Mathematics and Statistics
Mechatronics
Networks
Original Article
Robotics
Segments
Strip
Systems Theory
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Summary:We study the null controllability of Kolmogorov-type equations in a rectangle , under an additive control supported in an open subset of . For , with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support . This improves the previous result by Beauchard and Zuazua (Ann Ins H Poincaré Anal Non Linéaire 26:1793–1815, 2009 ), in which the control support was a horizontal strip. With Dirichlet boundary conditions and a horizontal strip as control support, we prove that null controllability holds in any positive time if or if and contains the segment , and only in large time if and does not contain the segment . Our approach, inspired from Benabdallah et al. (C R Math Acad Sci Paris 344(6):357–362, 2007 ), Lebeau and Robbiano (Commun Partial Differ Equ 20:335–356, 1995 ), is based on two key ingredients: the observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components.
ISSN:0932-4194
1435-568X
DOI:10.1007/s00498-013-0110-x