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

Abstract

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In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form ∂ y ∂ n + β y = 0 . We consider distributed controls with support in a small set and nonregular coefficients β = β ( 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.

How to cite

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Fernández-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

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  1. V. Barbu, Controllability of parabolic and Navier-Stokes equations. Sci. Math. Jpn56 (2002) 143–211.  
  2. 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.  
  3. C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh125A (1995) 31–61.  
  4. E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: the linear case. Adv. Diff. Equ.5 (2000) 465–514.  
  5. A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes no. 34, Seoul National University, Korea, 1996.  
  6. 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).  
  7. G. Lebeau and L. Robbiano, Contrôle exacte de l'equation de la chaleur (French). Comm. Partial Differ. Equat.20 (1995) 335–356.  
  8. D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Studies Appl. Math.52 (1973) 189–211.  

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