Boundary null controllability as the limit of internal controllability: The heat case
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 91.

It is well known that for the heat equation with Dirichlet boundary condition both internal and boundary null controllability hold with controls applied to any open subset of the domain and any open subset of the boundary, respectively. The purpose of this paper is to show that for the heat equation the boundary null controllability can be obtained as the limit of distributed null controllability.

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DOI : 10.1051/cocv/2020013
Classification : 35B25, 49E25, 93B05, 93B07
Mots-clés : Internal null controllability, boundary null controllability, Carleman estimates, singular pertubations 
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     author = {Chaves-Silva, F.W. and Puel, J.-P. and Santos, M.C.},
     title = {Boundary null controllability as the limit of internal controllability: {The} heat case},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
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     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2020013/}
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Chaves-Silva, F.W.; Puel, J.-P.; Santos, M.C. Boundary null controllability as the limit of internal controllability: The heat case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 91. doi : 10.1051/cocv/2020013. http://www.numdam.org/articles/10.1051/cocv/2020013/

[1] O.Y. Emanuilov, Boundary controllability of parabolic equations. Russ. Math. Surv. 48 (1993) 192–194. | DOI | MR | Zbl

[2] O.Y. Emanuilov, Controllability of parabolic equations. Sb. Math. 186 (1995) 879–900. | DOI | MR | Zbl

[3] C. Fabre, Exact boundary controllability of the wave equation as the limit of internal controllability. SIAM J. Control Optim. 30 (1992) 1066–1086. | DOI | MR | Zbl

[4] C. Fabre and J.-P. Puel, Behavior near the boundary for solutions of the wave equation. J. Differ. Equ. 106 (1993) 186–213. | DOI | MR | Zbl

[5] H.O. Fattorini and D.L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43 (1971) 272–292. | DOI | MR | Zbl

[6] 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. Q. Appl. Math. 32 (1974/75) 45–69. | DOI | MR | Zbl

[7] E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1399–1446. | DOI | MR | Zbl

[8] E. Fernández-Cara, S. Guerrero, O.Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. | DOI | MR | Zbl

[9] A.V. Fursikov, O.Y. Imanuvilov, Controllability of evolution equations. Lecture Notes Series. Seoul National University (1996). | MR | Zbl

[10] S.W. Hansen and M. Tucsnak, Some new applications of Russell’s principle to infinite dimensional vibrating systems. Ann. Rev. Control 44 (2017) 184–198. | DOI

[11] L. Hörmander, Pseudo-differential operators. The Analysis of Linear Partial Differential Operators III. Springer, Berlin, Heidelberg (2007) 63–179. | DOI | MR | Zbl

[12] R. Joly, Convergence of the wave equation damped on the interior to the one damped on the boundary. J. Differ. Equ. 229 (2006) 588–653. | DOI | MR | Zbl

[13] J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM: COCV 18 (2012) 712–747. | Numdam | MR | Zbl

[14] G. Lebeau and L. Robbiano, Contrõle exact de l’équation de la chaleur. Comm. Partial Differ. Equ. 20 (1995) 335–356. | DOI | MR | Zbl

[15] C. Letrouit, From internal to pointwise control for the 1D heat equation and minimal control time. Syst. Control Lett. 133 (2019) 104549. | DOI | MR

[16] J.-P. Puel, Controllability of Navier-Stokes equations. Optimization with PDE Constraints. Lecture Notes Comput. Sci. Eng. Springer, Cham (2014) 101. | MR

[17] D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52 (1973) 189–211. | DOI | MR | Zbl

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