The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear term grows slower than at infinity. In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry.
Mots clés : Carleman inequalities, controllability, transmission problems
@article{COCV_2002__8__621_0,
author = {Doubova, Anna and Osses, A. and Puel, J.-P.},
title = {Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {621--661},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
doi = {10.1051/cocv:2002047},
mrnumber = {1932966},
zbl = {1092.93006},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2002047/}
}
TY - JOUR AU - Doubova, Anna AU - Osses, A. AU - Puel, J.-P. TI - Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 SP - 621 EP - 661 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002047/ DO - 10.1051/cocv:2002047 LA - en ID - COCV_2002__8__621_0 ER -
%0 Journal Article %A Doubova, Anna %A Osses, A. %A Puel, J.-P. %T Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2002 %P 621-661 %V 8 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2002047/ %R 10.1051/cocv:2002047 %G en %F COCV_2002__8__621_0
Doubova, Anna; Osses, A.; Puel, J.-P. Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 621-661. doi : 10.1051/cocv:2002047. http://www.numdam.org/articles/10.1051/cocv:2002047/
[1] and, Null controllability of nonlinear convective heat equation. ESAIM: COCV 5 (2000) 157-173. | Numdam | MR | Zbl
[2] and, Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal. 25 (1967) 81-122. | MR | Zbl
[3] and, A maximum principle for nonlinear parabolic equations. Ann. Scuola Norm. Sup. Pisa 3 (1967) 291-305 | Numdam | MR | Zbl
[4] , L'analyse non linéaire et ses motivations économiques. Masson (1984). | Zbl
[5] , Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73-89. | MR | Zbl
[6] and, Introduction aux problèmes d'évolution semi-linéaires. Ellipses, Paris, Mathématiques & Applications (1990). | Zbl
[7] and, The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44 (1995) 545-573. | MR | Zbl
[8] ,, and, On the controllability of parabolic system with a nonlinear term involving the state and the gradient. SIAM: SICON (to appear). | Zbl
[9] , and, (a) Approximate controllability for the semilinear heat equation. C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) 807-812; (b) Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31-61. | MR | Zbl
[10] , and, Approximate controllability for the linear heat equation with controls of minimal norm. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 679-684. | MR | Zbl
[11] , Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-107. | Numdam | Zbl
[12] and, The cost of approximate controllability for heat equations: The linear case. Adv. Differential Equations 5 (2000) 465-514. | Zbl
[13] and, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 583-616. | Numdam | Zbl
[14] and, On the null controllability of the one-dimensional heat equation with BV coefficients (to appear). | Zbl
[15] and, Controllability of Evolution Equations. Seoul National University, Korea, Lecture Notes 34 (1996). | MR | Zbl
[16] , Controllability of parabolic equations. Mat. Sb. 186 (1995) 102-132. | MR | Zbl
[17] and, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications. Lecture Notes in Pure Appl. Math. 218 (2001) 113-137. | MR | Zbl
[18] , and, Linear and Quasilinear Equations of Parabolic Type. Nauka, Moskow (1967). | Zbl
[19] , Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983). | MR | Zbl
[20] , A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52 (1973) 189-211. | MR | Zbl
[21] , Local existence and nonexistence for semilinear parabolic equations in . Indiana Univ. Math. J. 29 (1980) 79-102. | MR | Zbl
[22] , Semilinear evolution equations in Banach spaces. J. Funct. Anal. 32 (1979) 277-296. | MR | Zbl
[23] , Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and their Applications, Vol. X, edited by H. Brezis and J.-L. Lions. Pitman (1991) 357-391. | MR | Zbl
[24] , Finite dimensional controllability for the semilinear heat equations. J. Math. Pures 76 (1997) 570-594. | MR | Zbl
[25] , Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Control and Cybernetics 28 (1999) 665-683. | MR | Zbl
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