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

Null controllability of the heat equation with boundary Fourier conditions: the linear case. (English) Zbl 1106.93009

ESAIM, Control Optim. Calc. Var. 12, 442-465 (2006).
Summary: 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.

Cited in 68 Documents

MSC:

93B05 Controllability
35K20 Initial-boundary value problems for second-order parabolic equations
49J20 Existence theories for optimal control problems involving partial differential equations

Keywords:

controllability; heat equation; Fourier conditions
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References:

[1] V. Barbu , Controllability of parabolic and Navier-Stokes equations . Sci. Math. Jpn 56 ( 2002 ) 143 - 211 . Zbl 1010.93054 · Zbl 1010.93054
[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 . Zbl 1049.35042 · Zbl 1049.35042 · doi:10.1016/j.jde.2003.09.002
[3] C. Fabre , J.P. Puel and E. Zuazua , Approximate controllability of the semilinear heat equation . Proc. Roy. Soc. Edinburgh 125A ( 1995 ) 31 - 61 . Zbl 0818.93032 · Zbl 0818.93032 · doi:10.1017/S0308210500030742
[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 . Zbl 1007.93034 · Zbl 1007.93034
[5] A. Fursikov and O.Yu. Imanuvilov , Controllability of Evolution Equations . Lecture Notes no. 34, Seoul National University, Korea, 1996. MR 1406566 | Zbl 0862.49004 · Zbl 0862.49004
[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 ). MR 1817179 | Zbl 0977.93041 · Zbl 0977.93041
[7] G. Lebeau and L. Robbiano , Contrôle exacte de l’equation de la chaleur (French). Comm. Partial Differ. Equat. 20 ( 1995 ) 335 - 356 . Zbl 0819.35071 · Zbl 0819.35071 · doi:10.1080/03605309508821097
[8] D.L. Russell , A unified boundary controllability theory for hyperbolic and parabolic partial differential equations . Studies Appl. Math. 52 ( 1973 ) 189 - 211 . Zbl 0274.35041 · Zbl 0274.35041
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