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On the lack of null-controllability of the heat equation on the half-line


Authors: Sorin Micu and Enrique Zuazua
Journal: Trans. Amer. Math. Soc. 353 (2001), 1635-1659
MSC (2000): Primary 35B37, 35K05
Published electronically: November 21, 2000
MathSciNet review: 1806726
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Abstract:

We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the $L^2$ boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials $\{e^{jt}\}_{j\geq1}$ in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.


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Additional Information

Sorin Micu
Affiliation: Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain
Email: sorin@sunma4.mat.ucm.es

Enrique Zuazua
Affiliation: Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain
Email: zuazua@eucmax.sim.ucm.es

DOI: https://doi.org/10.1090/S0002-9947-00-02665-9
Keywords: Heat equation, similarity variables, control, moments
Received by editor(s): July 21, 1999
Received by editor(s) in revised form: October 22, 1999
Published electronically: November 21, 2000
Additional Notes: The first author was supported by grants PB96-0663 and 303/1999 of CNCSU (Romania)
The second author was supported by grant PB96-0663 of the DGES (Spain)
Article copyright: © Copyright 2000 American Mathematical Society