# Null controllability of the heat equation with boundary Fourier conditions: the linear case

Enrique Fernández-Cara; Manuel González-Burgos; Sergio Guerrero; Jean-Pierre Puel

ESAIM: Control, Optimisation and Calculus of Variations (2006)

- Volume: 12, Issue: 3, page 442-465
- ISSN: 1292-8119

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topFernández-Cara, Enrique, et al. "Null controllability of the heat equation with boundary Fourier conditions: the linear case." ESAIM: Control, Optimisation and Calculus of Variations 12.3 (2006): 442-465. <http://eudml.org/doc/249619>.

@article{Fernández2006,

abstract = {
In this paper, we prove the global null controllability of
the linear heat equation completed with linear Fourier
boundary conditions of the form
$\{\partial y\over\partial n\} + \beta\,y = 0$.
We consider distributed controls with support in a small set and
nonregular coefficients $\beta=\beta(x,t)$.
For the proof of null controllability, a crucial tool will be a new
Carleman estimate for the weak solutions of the classical heat
equation with
nonhomogeneous Neumann boundary conditions.
},

author = {Fernández-Cara, Enrique, González-Burgos, Manuel, Guerrero, Sergio, Puel, Jean-Pierre},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Controllability; heat equation; Fourier conditions.; controllability; Fourier conditions},

language = {eng},

month = {6},

number = {3},

pages = {442-465},

publisher = {EDP Sciences},

title = {Null controllability of the heat equation with boundary Fourier conditions: the linear case},

url = {http://eudml.org/doc/249619},

volume = {12},

year = {2006},

}

TY - JOUR

AU - Fernández-Cara, Enrique

AU - González-Burgos, Manuel

AU - Guerrero, Sergio

AU - Puel, Jean-Pierre

TI - Null controllability of the heat equation with boundary Fourier conditions: the linear case

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2006/6//

PB - EDP Sciences

VL - 12

IS - 3

SP - 442

EP - 465

AB -
In this paper, we prove the global null controllability of
the linear heat equation completed with linear Fourier
boundary conditions of the form
${\partial y\over\partial n} + \beta\,y = 0$.
We consider distributed controls with support in a small set and
nonregular coefficients $\beta=\beta(x,t)$.
For the proof of null controllability, a crucial tool will be a new
Carleman estimate for the weak solutions of the classical heat
equation with
nonhomogeneous Neumann boundary conditions.

LA - eng

KW - Controllability; heat equation; Fourier conditions.; controllability; Fourier conditions

UR - http://eudml.org/doc/249619

ER -

## References

top- V. Barbu, Controllability of parabolic and Navier-Stokes equations. Sci. Math. Jpn56 (2002) 143–211.
- A. Doubova, E. Fernández-Cara and M. González-Burgos, On the controllability of the heat equation with nonlinear boundary Fourier conditions. J. Diff. Equ.196 (2004) 385–417.
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- E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: the linear case. Adv. Diff. Equ.5 (2000) 465–514.
- A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes no. 34, Seoul National University, Korea, 1996.
- O.Yu. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Dekker, New York. Lect. Notes Pure Appl. Math.218 (2001).
- G. Lebeau and L. Robbiano, Contrôle exacte de l'equation de la chaleur (French). Comm. Partial Differ. Equat.20 (1995) 335–356.
- D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Studies Appl. Math.52 (1973) 189–211.

## Citations in EuDML Documents

top- Sergio Guerrero, Controllability of systems of Stokes equations with one control force : existence of insensitizing controls
- Enrique Fernández-Cara, Manuel González-Burgos, Sergio Guerrero, Jean-Pierre Puel, Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case

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