On the regularity of null-controls of the linear 1-d heat equation

Micu S, Zuazua E (2011)

Publication Type: Journal article

Publication year: 2011

Journal

Comptes Rendus Mathematique Elsevier Masson

Book Volume: 349

Pages Range: 673-677

Journal Issue: 11-12

DOI: 10.1016/j.crma.2011.05.005

Abstract

The 1-d heat equation in a bounded interval is null-controllable from the boundary. More precisely, for each initial data y(0) is an element of L-2(0.1) there corresponds a unique boundary control of minimal L-2(0. T)-norm which drives the state of the 1-d linear heat equation to zero in time T > 0. This control is given as the normal derivative of a solution of the homogeneous adjoint equation whose initial data (phi) over cap (0) minimizes a suitable quadratic cost functional. In this Note we analyze the relation between the regularity of the initial datum to be controlled y(0) and that of (phi) over cap (0). We show that, if y(0) has only one Fourier mode, the corresponding (phi) over cap (0) does not belong to any Sobolev space of negative exponent. This explains the severe ill-posedness of the problem and the lack of efficiency of most of the existing numerical algorithms for the numerical approximation of the controls, and in particular the slow convergence rate of Tychonoff regularization procedures. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

University of Craiova RO Romania (RO) Ikerbasque (Basque Foundation for Science) ES Spain (ES)

How to cite

APA:

Micu, S., & Zuazua, E. (2011). On the regularity of null-controls of the linear 1-d heat equation. Comptes Rendus Mathematique, 349(11-12), 673-677. https://dx.doi.org/10.1016/j.crma.2011.05.005

MLA:

Micu, Sorin, and Enrique Zuazua. "On the regularity of null-controls of the linear 1-d heat equation." Comptes Rendus Mathematique 349.11-12 (2011): 673-677.

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