On the null-controllability of diffusion equations

Gérald Tenenbaum; Marius Tucsnak

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 1088-1100
  • ISSN: 1292-8119

Abstract

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This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check the Lebeau-Robbiano spectral condition. We then show that the sophisticated Carleman and interpolation inequalities used in previous literature may be replaced by a simple result of Turán. In this case, we provide explicit values for the constants involved in the above mentioned spectral condition. As far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates.

How to cite

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Tenenbaum, Gérald, and Tucsnak, Marius. "On the null-controllability of diffusion equations." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 1088-1100. <http://eudml.org/doc/276329>.

@article{Tenenbaum2011,
abstract = { This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check the Lebeau-Robbiano spectral condition. We then show that the sophisticated Carleman and interpolation inequalities used in previous literature may be replaced by a simple result of Turán. In this case, we provide explicit values for the constants involved in the above mentioned spectral condition. As far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates. },
author = {Tenenbaum, Gérald, Tucsnak, Marius},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Heat equation; controllability; spectral condition; Turán's method; heat equation; Turán’s method},
language = {eng},
month = {11},
number = {4},
pages = {1088-1100},
publisher = {EDP Sciences},
title = {On the null-controllability of diffusion equations},
url = {http://eudml.org/doc/276329},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Tenenbaum, Gérald
AU - Tucsnak, Marius
TI - On the null-controllability of diffusion equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/11//
PB - EDP Sciences
VL - 17
IS - 4
SP - 1088
EP - 1100
AB - This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check the Lebeau-Robbiano spectral condition. We then show that the sophisticated Carleman and interpolation inequalities used in previous literature may be replaced by a simple result of Turán. In this case, we provide explicit values for the constants involved in the above mentioned spectral condition. As far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates.
LA - eng
KW - Heat equation; controllability; spectral condition; Turán's method; heat equation; Turán’s method
UR - http://eudml.org/doc/276329
ER -

References

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  2. H.O. Fattorini and D.L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quart. Appl. Math.32 (1974) 45–69.  
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  8. L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Preprint, available at (2009).  URIhttp://hal.archives-ouvertes.fr/hal-00411846/en/
  9. H.L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics84. Published for the Conference Board of the Mathematical Sciences, Washington (1994).  
  10. T.I. Seidman, How violent are fast controls. III. J. Math. Anal. Appl.339 (2008) 461–468.  
  11. M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel (2009).  
  12. P. Turán, On a theorem of Littlewood. J. London Math. Soc.21 (1946) 268–275.  

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