# 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

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topTenenbaum, 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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- L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian. Math. Control Signals Systems18 (2006) 260–271. Zbl1105.93015
- L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Preprint, available at (2009). Zbl1219.93017URIhttp://hal.archives-ouvertes.fr/hal-00411846/en/
- 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). Zbl0814.11001
- T.I. Seidman, How violent are fast controls. III. J. Math. Anal. Appl.339 (2008) 461–468. Zbl1129.93025
- M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel (2009).
- P. Turán, On a theorem of Littlewood. J. London Math. Soc.21 (1946) 268–275. Zbl0061.06504

## Citations in EuDML Documents

top- Yuning Liu, Takéo Takahashi, Marius Tucsnak, Single input controllability of a simplified fluid-structure interaction model
- Yang Liu, Jianquan Lu, Bo Wu, Some necessary and sufficient conditions for the output controllability of temporal Boolean control networks
- Jérôme Le Rousseau, Matthieu Léautaud, Luc Robbiano, Controllability of a parabolic system with a diffusive interface

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